BusyBeaverWiki - User contributions [en]

Instruction-Limited Busy Beaver

2025-10-02T20:28:25Z

MrBrain: Spacing

An '''n-instruction Turing machine''' is a [[Turing machine]] having an arbitrary number of states and symbols, but having only ''n'' defined transitions/instructions in its transition table (all others are undefined).

The '''Instruction-Limited Busy Beaver function (BBi(n))''' is the maximum steps taken by any n-instruction Turing machines before eventually halting (when started on an initially blank tape). Similarly, the '''Instruction-Limited Symbols Busy Beaver function (Σi(n))''' is the maximum [[sigma score]] (number of non-blank symbols left on the tape at halting) for all halting n-instruction Turing machines (when started on an initially blank tape).

* The currently known values of BBi(n) for n>=0 are: 0, 1, 3, 5, 16, 37, 123, 3932963, .... BBi(8) is at least 6.889 x 101565, which is the number of steps taken by the current 8-instruction champion machine, discovered by Nick Drozd<ref>Drozd, N. (Sept. 2025). "[https://nickdrozd.github.io/2025/09/30/shape-of-a-turing-machine.html The Shape of a Turing Machine | Something Something Programming]"</ref>.
* The currently known values of Σi(n) for n>=0 are: 0, 1, 2, 4, 5, 9, 14, 2050, .... Σi(8) is at least 1.355 x 10783, which is the number of non-blank symbols written to tape by the same 8-instruction champion machine.

As with the traditional Busy Beaver sequences, BBi(n) and Σi(n) are both uncomputable, with each growing faster than any computable function.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''')<ref>Tibor Radó (May 1962). "[https://computation4cognitivescientists.weebly.com/uploads/6/2/8/3/6283774/rado-on_non-computable_functions.pdf On non-computable functions]" (PDF). ''Bell System Technical Journal''. '''41''' (3): 877–884. https://doi.org/10.1002%2Fj.1538-7305.1962.tb00480.x</ref>. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n = 1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m = 1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? The current BBi(8) champion is a 3-state, 4-symbol machine, but this may still be surpassed by a 3-state, 3-symbol machine yet to be found.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!Step Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|0RH|halt}} {{TM|1RH---|halt}}
|BB(1,2)
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''3'''
|{{TM|0RB---_1LA---|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''5'''
|{{TM|1RB1LB_1LA---|halt}} (and 13 others)
|[[BB(2)|BB(2,2)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''16'''
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''37'''
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|[[BB(2,3)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''123'''
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''3,932,963'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|[[BB(2,4)]] - 1, and
A [[BB(3,3)]] high-ranking machine
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 6.889 \times 10^{1565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://nickdrozd.github.io/2025/09/30/shape-of-a-turing-machine.html Blog] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

{| class="wikitable"
!n
!'''Σ'''i(n)
!Sigma Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|1RH---|halt}}
|Σ(1,2)
|[[oeis:A384766|A384766]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''2'''
|{{TM|1RB---_1LA---|halt}} (and 7 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''4'''
|{{TM|1RB1LB_1LA---|halt}} (and 4 others)
|[[BB(2)|Σ(2,2)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''5'''
|{{TM|1RB0LB---_1LA2RA---|halt}} (and 40 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''9'''
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|Σ[[BB(2,3)|(2,3)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''14'''
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''2,050'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|Σ[[BB(2,4)|(2,4)]], and
A Σ[[BB(3,3)|(3,3)]] high-ranking machine
|[[oeis:A384766|A384766]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 1.355 \times 10^{783}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://nickdrozd.github.io/2025/09/30/shape-of-a-turing-machine.html Blog] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

Notes:
* Proven values for BBi(n) and Σi(n) are shown as bold in the above two tables.
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue. With the BBi(8) champion it appears that this trend has been broken (it is believed that <math>BB(3,3) < 10^{1565}</math>)
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5) -1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* The convention for writing TMs in standard text format here is to include as many states and symbols as are referenced. So for example, we write {{TM|1RH---|halt}} instead of {{TM|1RH|halt}} since this TM references 2 symbols (including the implicit blank symbol 0). If one state has no instructions leaving it, the convention is to treat it as the halt state <code>H</code>.

== Tree Normal Form (TNF) for BBi ==
Shown below are the total number of n-instruction Turing machines for each number of instructions n>=0. The numbers of halting and non-halting machines for n = 8 are not yet known. The function Tree Normal Form for BBi is denoted BBiTNF(n)
{| class="wikitable"
!n
!All Machines
!All Cumulative
!Halting Machines
!Halting Cumulative
!Non-Halting Machines
!Non-Halting Cumulative
|-
| style="text-align: right;" |0
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |0
| style="text-align: right;" |0
|-
| style="text-align: right;" |1
| style="text-align: right;" |4
| style="text-align: right;" |5
| style="text-align: right;" |2
| style="text-align: right;" |3
| style="text-align: right;" |2
| style="text-align: right;" |2
|-
| style="text-align: right;" |2
| style="text-align: right;" |30
| style="text-align: right;" |35
| style="text-align: right;" |17
| style="text-align: right;" |20
| style="text-align: right;" |13
| style="text-align: right;" |15
|-
| style="text-align: right;" |3
| style="text-align: right;" |378
| style="text-align: right;" |413
| style="text-align: right;" |260
| style="text-align: right;" |280
| style="text-align: right;" |118
| style="text-align: right;" |133
|-
| style="text-align: right;" |4
| style="text-align: right;" |7,640
| style="text-align: right;" |8,053
| style="text-align: right;" |5,581
| style="text-align: right;" |5,861
| style="text-align: right;" |2,059
| style="text-align: right;" |2,192
|-
| style="text-align: right;" |5
| style="text-align: right;" |205,580
| style="text-align: right;" |213,633
| style="text-align: right;" |160,952
| style="text-align: right;" |166,813
| style="text-align: right;" |44,628
| style="text-align: right;" |46.820
|-
| style="text-align: right;" |6
| style="text-align: right;" |7,149,820
| style="text-align: right;" |7,363,453
| style="text-align: right;" |5,843,696
| style="text-align: right;" |6,010,509
| style="text-align: right;" |1,306,124
| style="text-align: right;" |1,352,944
|-
| style="text-align: right;" |7
| style="text-align: right;" |305,333,128
| style="text-align: right;" |312,696,581
| style="text-align: right;" |258,044,501
| style="text-align: right;" |264,055,010
| style="text-align: right;" |47,288,627
| style="text-align: right;" |48,641,571
|-
| style="text-align: right;" |8
| style="text-align: right;" |15,561,793,526
| style="text-align: right;" |15,874,490,107
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
|}
Enumeration of Turing machines in Tree Normal Form ([[TNF]]) for the Limited-Instruction Busy Beaver problem uses the following procedure:

* The "null Turing machine" with 0 instructions is the root of the tree. This "machine" halts after 0 steps, writing 0 non-blank symbols to the tape.
* The first instruction starts from A, the implied starting state, reads a 0 on the tape (which is blank at machine start) and must shift right (to avoid consideration of trivial mirror-image machines).
** This allows the following four possibilities: A0→0RA, A0→0RB, A0→1RA, and A0→1RB. The two machines transitioning to state A never halt, while the two machines transitioning to state B halt. The two halting machines are next expecting an instruction for B0, but since no such instruction exists, the machines halt after a single step.
* Following the enumeration of machines for each previous number of instructions, each of the halting machines is extended from the instruction at which it halted. For n = 2, this instruction must be B0. (Non-halting machines cannot be extended since they can never reach a new instruction to be added.)
* As per normal TNF procedure, the new instruction can write any symbol to tape, shift left or right, and transition to any state as long as:
** The symbol written can be at most one greater than the highest symbol previously read or written, and
** The state to which the instruction transitions can be at most one state greater than the current maximum state.
* For the BBi problem, any instruction that transitions to a "new" state (one not yet containing any instructions) can be interpreted as a "halting instruction". (e.g. The one-instruction machine A0→1RB is exactly the same as A0→1RH.) Similarly, this new state can be considered a "halt state", as there is functionally no difference between a halt state and an operational state having no instructions. (B = H.)
** Unlike the traditional BB problem, where a halting instruction is normally used in the last remaining cell in an n x m transition table, it is necessary to consider all possible halting transitions for BBi, (not just those of the form →1RH), as this allows the most possible extensions of n-instruction machines to (n+1)-instruction machines.

== Domain-Restricted BBi ==
In addition to the standard Instruction-Limited Busy Beaver function BBi(n), where the numbers of states and symbols a Turing machine may have is unlimited, it may also be useful to consider the “Domain-Restricted BBi”, where the numbers of instructions, states, and/or symbols are specified values or ranges.

Following are some example three-argument functions within the Domain-Restricted BBi. For each function, the value is always the greatest number of steps taken by a Turing machine satisfying the corresponding restrictions:

* BBi(i, n, m): An n-state, m-symbol Turing machine having i instructions
* BBi(i, <=n, m): An i-instruction, m-symbol Turing machine having no greater than n states
* BBi(i, -, m): An i-instruction, m-symbol Turing machine having any number of states
* BBi(i, n, <=m): An i-instruction, n-state Turing machine having no greater than m instructions
* Etc.

Note that we have replaced the argument n from the standard BBi function with i in this three-argument version, primarily to avoid confusion with the n normally used to denote the number of states in a standard BB Turing machine.

For each of the three arguments, we may simply list a value, in which case the Turing machine must have exactly that number of instructions, states, or symbols. Another option is to specify a range, such as “<=n”, or “<=m”, in which case the Turing machine must have n or fewer states, or m or fewer symbols, respectively. And the final option is to give an argument as “-“, in which case, there is no restriction to the Turing machine’s number of instructions, states, or symbols.

'''Example Questions:'''

The main reason for having the flexibility in the way that the arguments are presented within the Domain-Restricted BBi function is that some problems relating to BB and BBi are best suited to examining Turing machines having specific numbers of states and symbols, while others are more practically examined within a range of state and symbol values instead. For example, consider the following two questions, along with two possible ways to interpret and answer each question:

'''Question 1:''' What are the maximum numbers of steps an i-instruction, 5-state, 2-symbol Turing machine can take before halting?

* Interpretation #1: For i = 1 through 10, BBi(i, 5, 2) = -, -, -, -, 9, 31, 65, 793, 47176869, 47176870. In this interpretation, there are no values for 1<=i<=4, because there is no way to construct a 5-state Turing machine using fewer than 5 instructions. Also, the values for i = 5, 6, and 7 are lower than those in the following second interpretation.
* Interpretation #2: For i = 1 through 10, BBi(i, <=5, 2) = 1, 3, 5, 16, 20, 37, 106, 793, 47176869, 47176870. For this interpretation, we do not take the questions wording of “5-state” so literally, but rather we allow the consideration of any Turing machine having no greater than 5 states. For example, BBi(7, <=5, 2) = 106 = BB(4,2) – 1.

The second interpretation, despite being less literally adherent to the question wording than the first, seems more useful for this particular question, specifically because we are probably interested in seeing how the number of steps grows for the best possible arrangement of i instructions within the 5 x 2 state table, and not just among the arrangements that specifically include defined transitions for all five states.

'''Question 2:''' For all n>1, will the 2n-1 instruction, n-state, 2-symbol BBi champion (one undefined transition) always exceed the 2n-1 instruction, n+1 state, 2-symbol BBi champion (3 undefined transitions)? (This is currently conjectured to be false.)

* Interpretation #1: Compare values of BBi(2n-1, n, 2) with those of BBi(2n-1, n+1, 2) for n = 2 through 5:
** For n=2 through 5, BBi(2n-1, n, 2) = 5, 20, 106, 47176869.
** For n=2 through 5, BBi(2n-1, n+1, 2) = 4, 19, 65, ???.
* Interpretation #2: Compare values of BBi(2n-1, <=n, 2) with those of BBi(2n-1, <=n+1, 2) for n=2 through 5:
** For n=2 through 5, BBi(2n-1, <=n, 2) = 5, 20, 106, 47176869.
** For n=2 through 5, BBi(2n-1, <=n+1, 2) = 5, 20, 106, ???.

For this second question, the first interpretation clearly seems to be the more useful one, as we are probably interested in comparing the steps differences between machines having strictly n states (with one undefined transition) with machines having strictly n+1 states (with three undefined transitions). Using the second interpretation, information about these steps differences is lost since we have replaced the steps values of the machines with n+1 states with the values from the stronger n-state machines.

'''Selected Results:'''

Following are some results for Domain-Restricted BBi, some of which follow logically from their definitions, and others which have been computed so far:

* BBi(i, -, -) = BBi(i). If we allow any number of states and symbols, then this is the standard BBi function.
* BBi(-, n, m) = BBi(nm, n, m) = BB(n, m) for n>=2, m>=2. If we allow any number of instructions, or if we specify nm instructions, then this is equivalent to the standard BB(n, m) function.
* BBi(nm-1, n, m) = BBi(nm, n, m) - 1 = BB(n, m) - 1 for n>=2, m>=2. With one undefined transition, the number of steps is one less than having a completely filled n x m transition table including a halting instruction, which adds one final step.
* For i>=1, BBi(i, -, 2) = 1, 3, 5, 16, 20, 37, 106, 793, …. BBi(9, -, 2) is at least 47176869, but it may be even larger if a more powerful 6-state, 2-symbol machine having three undefined transitions can be found. BBi(10, -, 2) will almost certainly be greater than the current lower bound of 47176870 = BB(5,2).
* For i>=1, BBi(i, <=2, -) = 1, 3, 5, 13, 37, 123, 3232963, …. The number of states is specified here as “<=2” rather than “2” so that we can have a valid value for one instruction. (A 1-instruction Turing machine has only one operating state.) BBi(8, <=2, -) will almost certainly be greater than the current lower bound of 3232964 = BB(2,4).

== Instruction-Limited Busy Beaver Variants ==
The concept of classifying a Turing machine's size by number of instructions, rather than by the number of states and symbols in its transition table, can also be utilized for Busy Beaver variants. The notation for the instruction-limited version of a variant can conveniently be denoted by appending an "i" to the already existing notation for the variant. Furthermore, the name of the Instruction-Limited version of the variant may be more conveniently started with "I-L" for brevity. Some known results for Instruction-Limited BB variants are as follows:

* I-L Greedy Busy Beaver. gBBi(n) is the largest number of steps taken by an n-instruction Turing machine, but where its first n-1 instructions (in the order encountered during running the machine) must also be a machine taking gBBi(n-1) steps. In other words, once the greedy n-instruction champion machines are identified, all machines having n+1 instructions must be extended from one of those n-instruction champions.
** gBBi(n) for n = 1 through n = 13 are: 1, 3, 5, 13, 19, 25, 41, 55, 238, 941, 1341, 10465, 10675. gBBi(14) must be at least 9,874,580 steps for current champion machine {{TM|0RB6RB1LB---3LA1RB7RB2LB_1LA2RB3LA4RB5RB1LB5LA---|halt}}.
* I-L Blanking Busy Beaver. BLBi(n) is the largest amount of steps taken by an n-instruction Turing machine when blanking the tape for the first time after having written a non-blank symbol to tape.
** BLBi(1) and BLBi(2) have no values because it is impossible to blank the tape with one or two instructions. BLBi(n) for n = 3 through n = 7 are: 4, 12, 30, 77, 808. BLBi(8) is at least 1,367,361,263,049 by the current champion {{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}.

== References ==
<references />
[[category:Functions]]

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-10-02T20:27:45Z

MrBrain: Adding reference to blog post about current BBi(8) champion.

An '''n-instruction Turing machine''' is a [[Turing machine]] having an arbitrary number of states and symbols, but having only ''n'' defined transitions/instructions in its transition table (all others are undefined).

The '''Instruction-Limited Busy Beaver function (BBi(n))''' is the maximum steps taken by any n-instruction Turing machines before eventually halting (when started on an initially blank tape). Similarly, the '''Instruction-Limited Symbols Busy Beaver function (Σi(n))''' is the maximum [[sigma score]] (number of non-blank symbols left on the tape at halting) for all halting n-instruction Turing machines (when started on an initially blank tape).

* The currently known values of BBi(n) for n>=0 are: 0, 1, 3, 5, 16, 37, 123, 3932963, .... BBi(8) is at least 6.889 x 101565, which is the number of steps taken by the current 8-instruction champion machine, discovered by Nick Drozd <ref>Drozd, N. (Sept. 2025). "[https://nickdrozd.github.io/2025/09/30/shape-of-a-turing-machine.html The Shape of a Turing Machine | Something Something Programming]"</ref>.
* The currently known values of Σi(n) for n>=0 are: 0, 1, 2, 4, 5, 9, 14, 2050, .... Σi(8) is at least 1.355 x 10783, which is the number of non-blank symbols written to tape by the same 8-instruction champion machine.

As with the traditional Busy Beaver sequences, BBi(n) and Σi(n) are both uncomputable, with each growing faster than any computable function.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') <ref>Tibor Radó (May 1962). "[https://computation4cognitivescientists.weebly.com/uploads/6/2/8/3/6283774/rado-on_non-computable_functions.pdf On non-computable functions]" (PDF). ''Bell System Technical Journal''. '''41''' (3): 877–884. https://doi.org/10.1002%2Fj.1538-7305.1962.tb00480.x</ref>. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n = 1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m = 1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? The current BBi(8) champion is a 3-state, 4-symbol machine, but this may still be surpassed by a 3-state, 3-symbol machine yet to be found.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!Step Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|0RH|halt}} {{TM|1RH---|halt}}
|BB(1,2)
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''3'''
|{{TM|0RB---_1LA---|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''5'''
|{{TM|1RB1LB_1LA---|halt}} (and 13 others)
|[[BB(2)|BB(2,2)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''16'''
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''37'''
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|[[BB(2,3)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''123'''
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''3,932,963'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|[[BB(2,4)]] - 1, and
A [[BB(3,3)]] high-ranking machine
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 6.889 \times 10^{1565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://nickdrozd.github.io/2025/09/30/shape-of-a-turing-machine.html Blog] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

{| class="wikitable"
!n
!'''Σ'''i(n)
!Sigma Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|1RH---|halt}}
|Σ(1,2)
|[[oeis:A384766|A384766]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''2'''
|{{TM|1RB---_1LA---|halt}} (and 7 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''4'''
|{{TM|1RB1LB_1LA---|halt}} (and 4 others)
|[[BB(2)|Σ(2,2)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''5'''
|{{TM|1RB0LB---_1LA2RA---|halt}} (and 40 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''9'''
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|Σ[[BB(2,3)|(2,3)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''14'''
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''2,050'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|Σ[[BB(2,4)|(2,4)]], and
A Σ[[BB(3,3)|(3,3)]] high-ranking machine
|[[oeis:A384766|A384766]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 1.355 \times 10^{783}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://nickdrozd.github.io/2025/09/30/shape-of-a-turing-machine.html Blog] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

Notes:
* Proven values for BBi(n) and Σi(n) are shown as bold in the above two tables.
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue. With the BBi(8) champion it appears that this trend has been broken (it is believed that <math>BB(3,3) < 10^{1565}</math>)
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5) -1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* The convention for writing TMs in standard text format here is to include as many states and symbols as are referenced. So for example, we write {{TM|1RH---|halt}} instead of {{TM|1RH|halt}} since this TM references 2 symbols (including the implicit blank symbol 0). If one state has no instructions leaving it, the convention is to treat it as the halt state <code>H</code>.

== Tree Normal Form (TNF) for BBi ==
Shown below are the total number of n-instruction Turing machines for each number of instructions n>=0. The numbers of halting and non-halting machines for n = 8 are not yet known. The function Tree Normal Form for BBi is denoted BBiTNF(n)
{| class="wikitable"
!n
!All Machines
!All Cumulative
!Halting Machines
!Halting Cumulative
!Non-Halting Machines
!Non-Halting Cumulative
|-
| style="text-align: right;" |0
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |0
| style="text-align: right;" |0
|-
| style="text-align: right;" |1
| style="text-align: right;" |4
| style="text-align: right;" |5
| style="text-align: right;" |2
| style="text-align: right;" |3
| style="text-align: right;" |2
| style="text-align: right;" |2
|-
| style="text-align: right;" |2
| style="text-align: right;" |30
| style="text-align: right;" |35
| style="text-align: right;" |17
| style="text-align: right;" |20
| style="text-align: right;" |13
| style="text-align: right;" |15
|-
| style="text-align: right;" |3
| style="text-align: right;" |378
| style="text-align: right;" |413
| style="text-align: right;" |260
| style="text-align: right;" |280
| style="text-align: right;" |118
| style="text-align: right;" |133
|-
| style="text-align: right;" |4
| style="text-align: right;" |7,640
| style="text-align: right;" |8,053
| style="text-align: right;" |5,581
| style="text-align: right;" |5,861
| style="text-align: right;" |2,059
| style="text-align: right;" |2,192
|-
| style="text-align: right;" |5
| style="text-align: right;" |205,580
| style="text-align: right;" |213,633
| style="text-align: right;" |160,952
| style="text-align: right;" |166,813
| style="text-align: right;" |44,628
| style="text-align: right;" |46.820
|-
| style="text-align: right;" |6
| style="text-align: right;" |7,149,820
| style="text-align: right;" |7,363,453
| style="text-align: right;" |5,843,696
| style="text-align: right;" |6,010,509
| style="text-align: right;" |1,306,124
| style="text-align: right;" |1,352,944
|-
| style="text-align: right;" |7
| style="text-align: right;" |305,333,128
| style="text-align: right;" |312,696,581
| style="text-align: right;" |258,044,501
| style="text-align: right;" |264,055,010
| style="text-align: right;" |47,288,627
| style="text-align: right;" |48,641,571
|-
| style="text-align: right;" |8
| style="text-align: right;" |15,561,793,526
| style="text-align: right;" |15,874,490,107
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
|}
Enumeration of Turing machines in Tree Normal Form ([[TNF]]) for the Limited-Instruction Busy Beaver problem uses the following procedure:

* The "null Turing machine" with 0 instructions is the root of the tree. This "machine" halts after 0 steps, writing 0 non-blank symbols to the tape.
* The first instruction starts from A, the implied starting state, reads a 0 on the tape (which is blank at machine start) and must shift right (to avoid consideration of trivial mirror-image machines).
** This allows the following four possibilities: A0→0RA, A0→0RB, A0→1RA, and A0→1RB. The two machines transitioning to state A never halt, while the two machines transitioning to state B halt. The two halting machines are next expecting an instruction for B0, but since no such instruction exists, the machines halt after a single step.
* Following the enumeration of machines for each previous number of instructions, each of the halting machines is extended from the instruction at which it halted. For n = 2, this instruction must be B0. (Non-halting machines cannot be extended since they can never reach a new instruction to be added.)
* As per normal TNF procedure, the new instruction can write any symbol to tape, shift left or right, and transition to any state as long as:
** The symbol written can be at most one greater than the highest symbol previously read or written, and
** The state to which the instruction transitions can be at most one state greater than the current maximum state.
* For the BBi problem, any instruction that transitions to a "new" state (one not yet containing any instructions) can be interpreted as a "halting instruction". (e.g. The one-instruction machine A0→1RB is exactly the same as A0→1RH.) Similarly, this new state can be considered a "halt state", as there is functionally no difference between a halt state and an operational state having no instructions. (B = H.)
** Unlike the traditional BB problem, where a halting instruction is normally used in the last remaining cell in an n x m transition table, it is necessary to consider all possible halting transitions for BBi, (not just those of the form →1RH), as this allows the most possible extensions of n-instruction machines to (n+1)-instruction machines.

== Domain-Restricted BBi ==
In addition to the standard Instruction-Limited Busy Beaver function BBi(n), where the numbers of states and symbols a Turing machine may have is unlimited, it may also be useful to consider the “Domain-Restricted BBi”, where the numbers of instructions, states, and/or symbols are specified values or ranges.

Following are some example three-argument functions within the Domain-Restricted BBi. For each function, the value is always the greatest number of steps taken by a Turing machine satisfying the corresponding restrictions:

* BBi(i, n, m): An n-state, m-symbol Turing machine having i instructions
* BBi(i, <=n, m): An i-instruction, m-symbol Turing machine having no greater than n states
* BBi(i, -, m): An i-instruction, m-symbol Turing machine having any number of states
* BBi(i, n, <=m): An i-instruction, n-state Turing machine having no greater than m instructions
* Etc.

Note that we have replaced the argument n from the standard BBi function with i in this three-argument version, primarily to avoid confusion with the n normally used to denote the number of states in a standard BB Turing machine.

For each of the three arguments, we may simply list a value, in which case the Turing machine must have exactly that number of instructions, states, or symbols. Another option is to specify a range, such as “<=n”, or “<=m”, in which case the Turing machine must have n or fewer states, or m or fewer symbols, respectively. And the final option is to give an argument as “-“, in which case, there is no restriction to the Turing machine’s number of instructions, states, or symbols.

'''Example Questions:'''

The main reason for having the flexibility in the way that the arguments are presented within the Domain-Restricted BBi function is that some problems relating to BB and BBi are best suited to examining Turing machines having specific numbers of states and symbols, while others are more practically examined within a range of state and symbol values instead. For example, consider the following two questions, along with two possible ways to interpret and answer each question:

'''Question 1:''' What are the maximum numbers of steps an i-instruction, 5-state, 2-symbol Turing machine can take before halting?

* Interpretation #1: For i = 1 through 10, BBi(i, 5, 2) = -, -, -, -, 9, 31, 65, 793, 47176869, 47176870. In this interpretation, there are no values for 1<=i<=4, because there is no way to construct a 5-state Turing machine using fewer than 5 instructions. Also, the values for i = 5, 6, and 7 are lower than those in the following second interpretation.
* Interpretation #2: For i = 1 through 10, BBi(i, <=5, 2) = 1, 3, 5, 16, 20, 37, 106, 793, 47176869, 47176870. For this interpretation, we do not take the questions wording of “5-state” so literally, but rather we allow the consideration of any Turing machine having no greater than 5 states. For example, BBi(7, <=5, 2) = 106 = BB(4,2) – 1.

The second interpretation, despite being less literally adherent to the question wording than the first, seems more useful for this particular question, specifically because we are probably interested in seeing how the number of steps grows for the best possible arrangement of i instructions within the 5 x 2 state table, and not just among the arrangements that specifically include defined transitions for all five states.

'''Question 2:''' For all n>1, will the 2n-1 instruction, n-state, 2-symbol BBi champion (one undefined transition) always exceed the 2n-1 instruction, n+1 state, 2-symbol BBi champion (3 undefined transitions)? (This is currently conjectured to be false.)

* Interpretation #1: Compare values of BBi(2n-1, n, 2) with those of BBi(2n-1, n+1, 2) for n = 2 through 5:
** For n=2 through 5, BBi(2n-1, n, 2) = 5, 20, 106, 47176869.
** For n=2 through 5, BBi(2n-1, n+1, 2) = 4, 19, 65, ???.
* Interpretation #2: Compare values of BBi(2n-1, <=n, 2) with those of BBi(2n-1, <=n+1, 2) for n=2 through 5:
** For n=2 through 5, BBi(2n-1, <=n, 2) = 5, 20, 106, 47176869.
** For n=2 through 5, BBi(2n-1, <=n+1, 2) = 5, 20, 106, ???.

For this second question, the first interpretation clearly seems to be the more useful one, as we are probably interested in comparing the steps differences between machines having strictly n states (with one undefined transition) with machines having strictly n+1 states (with three undefined transitions). Using the second interpretation, information about these steps differences is lost since we have replaced the steps values of the machines with n+1 states with the values from the stronger n-state machines.

'''Selected Results:'''

Following are some results for Domain-Restricted BBi, some of which follow logically from their definitions, and others which have been computed so far:

* BBi(i, -, -) = BBi(i). If we allow any number of states and symbols, then this is the standard BBi function.
* BBi(-, n, m) = BBi(nm, n, m) = BB(n, m) for n>=2, m>=2. If we allow any number of instructions, or if we specify nm instructions, then this is equivalent to the standard BB(n, m) function.
* BBi(nm-1, n, m) = BBi(nm, n, m) - 1 = BB(n, m) - 1 for n>=2, m>=2. With one undefined transition, the number of steps is one less than having a completely filled n x m transition table including a halting instruction, which adds one final step.
* For i>=1, BBi(i, -, 2) = 1, 3, 5, 16, 20, 37, 106, 793, …. BBi(9, -, 2) is at least 47176869, but it may be even larger if a more powerful 6-state, 2-symbol machine having three undefined transitions can be found. BBi(10, -, 2) will almost certainly be greater than the current lower bound of 47176870 = BB(5,2).
* For i>=1, BBi(i, <=2, -) = 1, 3, 5, 13, 37, 123, 3232963, …. The number of states is specified here as “<=2” rather than “2” so that we can have a valid value for one instruction. (A 1-instruction Turing machine has only one operating state.) BBi(8, <=2, -) will almost certainly be greater than the current lower bound of 3232964 = BB(2,4).

== Instruction-Limited Busy Beaver Variants ==
The concept of classifying a Turing machine's size by number of instructions, rather than by the number of states and symbols in its transition table, can also be utilized for Busy Beaver variants. The notation for the instruction-limited version of a variant can conveniently be denoted by appending an "i" to the already existing notation for the variant. Furthermore, the name of the Instruction-Limited version of the variant may be more conveniently started with "I-L" for brevity. Some known results for Instruction-Limited BB variants are as follows:

* I-L Greedy Busy Beaver. gBBi(n) is the largest number of steps taken by an n-instruction Turing machine, but where its first n-1 instructions (in the order encountered during running the machine) must also be a machine taking gBBi(n-1) steps. In other words, once the greedy n-instruction champion machines are identified, all machines having n+1 instructions must be extended from one of those n-instruction champions.
** gBBi(n) for n = 1 through n = 13 are: 1, 3, 5, 13, 19, 25, 41, 55, 238, 941, 1341, 10465, 10675. gBBi(14) must be at least 9,874,580 steps for current champion machine {{TM|0RB6RB1LB---3LA1RB7RB2LB_1LA2RB3LA4RB5RB1LB5LA---|halt}}.
* I-L Blanking Busy Beaver. BLBi(n) is the largest amount of steps taken by an n-instruction Turing machine when blanking the tape for the first time after having written a non-blank symbol to tape.
** BLBi(1) and BLBi(2) have no values because it is impossible to blank the tape with one or two instructions. BLBi(n) for n = 3 through n = 7 are: 4, 12, 30, 77, 808. BLBi(8) is at least 1,367,361,263,049 by the current champion {{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}.

== References ==
<references />
[[category:Functions]]

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-10-02T20:23:09Z

MrBrain: Added blog link to Symbols table as well

An '''n-instruction Turing machine''' is a [[Turing machine]] having an arbitrary number of states and symbols, but having only ''n'' defined transitions/instructions in its transition table (all others are undefined).

The '''Instruction-Limited Busy Beaver function (BBi(n))''' is the maximum steps taken by any n-instruction Turing machines before eventually halting (when started on an initially blank tape). Similarly, the '''Instruction-Limited Symbols Busy Beaver function (Σi(n))''' is the maximum [[sigma score]] (number of non-blank symbols left on the tape at halting) for all halting n-instruction Turing machines (when started on an initially blank tape).

* The currently known values of BBi(n) for n>=0 are: 0, 1, 3, 5, 16, 37, 123, 3932963, .... BBi(8) is at least 6.889 x 101565, which is the number of steps taken by the current 8-instruction champion machine, discovered by Nick Drozd.
* The currently known values of Σi(n) for n>=0 are: 0, 1, 2, 4, 5, 9, 14, 2050, .... Σi(8) is at least 1.355 x 10783, which is the number of non-blank symbols written to tape by the same 8-instruction champion machine.

As with the traditional Busy Beaver sequences, BBi(n) and Σi(n) are both uncomputable, with each growing faster than any computable function.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') <ref>Tibor Radó (May 1962). "[https://computation4cognitivescientists.weebly.com/uploads/6/2/8/3/6283774/rado-on_non-computable_functions.pdf On non-computable functions]" (PDF). ''Bell System Technical Journal''. '''41''' (3): 877–884. https://doi.org/10.1002%2Fj.1538-7305.1962.tb00480.x</ref>. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n = 1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m = 1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? The current BBi(8) champion is a 3-state, 4-symbol machine, but this may still be surpassed by a 3-state, 3-symbol machine yet to be found.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!Step Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|0RH|halt}} {{TM|1RH---|halt}}
|BB(1,2)
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''3'''
|{{TM|0RB---_1LA---|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''5'''
|{{TM|1RB1LB_1LA---|halt}} (and 13 others)
|[[BB(2)|BB(2,2)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''16'''
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''37'''
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|[[BB(2,3)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''123'''
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''3,932,963'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|[[BB(2,4)]] - 1, and
A [[BB(3,3)]] high-ranking machine
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 6.889 \times 10^{1565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://nickdrozd.github.io/2025/09/30/shape-of-a-turing-machine.html Blog] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

{| class="wikitable"
!n
!'''Σ'''i(n)
!Sigma Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|1RH---|halt}}
|Σ(1,2)
|[[oeis:A384766|A384766]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''2'''
|{{TM|1RB---_1LA---|halt}} (and 7 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''4'''
|{{TM|1RB1LB_1LA---|halt}} (and 4 others)
|[[BB(2)|Σ(2,2)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''5'''
|{{TM|1RB0LB---_1LA2RA---|halt}} (and 40 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''9'''
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|Σ[[BB(2,3)|(2,3)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''14'''
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''2,050'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|Σ[[BB(2,4)|(2,4)]], and
A Σ[[BB(3,3)|(3,3)]] high-ranking machine
|[[oeis:A384766|A384766]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 1.355 \times 10^{783}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://nickdrozd.github.io/2025/09/30/shape-of-a-turing-machine.html Blog] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

Notes:
* Proven values for BBi(n) and Σi(n) are shown as bold in the above two tables.
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue. With the BBi(8) champion it appears that this trend has been broken (it is believed that <math>BB(3,3) < 10^{1565}</math>)
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5) -1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* The convention for writing TMs in standard text format here is to include as many states and symbols as are referenced. So for example, we write {{TM|1RH---|halt}} instead of {{TM|1RH|halt}} since this TM references 2 symbols (including the implicit blank symbol 0). If one state has no instructions leaving it, the convention is to treat it as the halt state <code>H</code>.

== Tree Normal Form (TNF) for BBi ==
Shown below are the total number of n-instruction Turing machines for each number of instructions n>=0. The numbers of halting and non-halting machines for n = 8 are not yet known. The function Tree Normal Form for BBi is denoted BBiTNF(n)
{| class="wikitable"
!n
!All Machines
!All Cumulative
!Halting Machines
!Halting Cumulative
!Non-Halting Machines
!Non-Halting Cumulative
|-
| style="text-align: right;" |0
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |0
| style="text-align: right;" |0
|-
| style="text-align: right;" |1
| style="text-align: right;" |4
| style="text-align: right;" |5
| style="text-align: right;" |2
| style="text-align: right;" |3
| style="text-align: right;" |2
| style="text-align: right;" |2
|-
| style="text-align: right;" |2
| style="text-align: right;" |30
| style="text-align: right;" |35
| style="text-align: right;" |17
| style="text-align: right;" |20
| style="text-align: right;" |13
| style="text-align: right;" |15
|-
| style="text-align: right;" |3
| style="text-align: right;" |378
| style="text-align: right;" |413
| style="text-align: right;" |260
| style="text-align: right;" |280
| style="text-align: right;" |118
| style="text-align: right;" |133
|-
| style="text-align: right;" |4
| style="text-align: right;" |7,640
| style="text-align: right;" |8,053
| style="text-align: right;" |5,581
| style="text-align: right;" |5,861
| style="text-align: right;" |2,059
| style="text-align: right;" |2,192
|-
| style="text-align: right;" |5
| style="text-align: right;" |205,580
| style="text-align: right;" |213,633
| style="text-align: right;" |160,952
| style="text-align: right;" |166,813
| style="text-align: right;" |44,628
| style="text-align: right;" |46.820
|-
| style="text-align: right;" |6
| style="text-align: right;" |7,149,820
| style="text-align: right;" |7,363,453
| style="text-align: right;" |5,843,696
| style="text-align: right;" |6,010,509
| style="text-align: right;" |1,306,124
| style="text-align: right;" |1,352,944
|-
| style="text-align: right;" |7
| style="text-align: right;" |305,333,128
| style="text-align: right;" |312,696,581
| style="text-align: right;" |258,044,501
| style="text-align: right;" |264,055,010
| style="text-align: right;" |47,288,627
| style="text-align: right;" |48,641,571
|-
| style="text-align: right;" |8
| style="text-align: right;" |15,561,793,526
| style="text-align: right;" |15,874,490,107
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
|}
Enumeration of Turing machines in Tree Normal Form ([[TNF]]) for the Limited-Instruction Busy Beaver problem uses the following procedure:

* The "null Turing machine" with 0 instructions is the root of the tree. This "machine" halts after 0 steps, writing 0 non-blank symbols to the tape.
* The first instruction starts from A, the implied starting state, reads a 0 on the tape (which is blank at machine start) and must shift right (to avoid consideration of trivial mirror-image machines).
** This allows the following four possibilities: A0→0RA, A0→0RB, A0→1RA, and A0→1RB. The two machines transitioning to state A never halt, while the two machines transitioning to state B halt. The two halting machines are next expecting an instruction for B0, but since no such instruction exists, the machines halt after a single step.
* Following the enumeration of machines for each previous number of instructions, each of the halting machines is extended from the instruction at which it halted. For n = 2, this instruction must be B0. (Non-halting machines cannot be extended since they can never reach a new instruction to be added.)
* As per normal TNF procedure, the new instruction can write any symbol to tape, shift left or right, and transition to any state as long as:
** The symbol written can be at most one greater than the highest symbol previously read or written, and
** The state to which the instruction transitions can be at most one state greater than the current maximum state.
* For the BBi problem, any instruction that transitions to a "new" state (one not yet containing any instructions) can be interpreted as a "halting instruction". (e.g. The one-instruction machine A0→1RB is exactly the same as A0→1RH.) Similarly, this new state can be considered a "halt state", as there is functionally no difference between a halt state and an operational state having no instructions. (B = H.)
** Unlike the traditional BB problem, where a halting instruction is normally used in the last remaining cell in an n x m transition table, it is necessary to consider all possible halting transitions for BBi, (not just those of the form →1RH), as this allows the most possible extensions of n-instruction machines to (n+1)-instruction machines.

== Domain-Restricted BBi ==
In addition to the standard Instruction-Limited Busy Beaver function BBi(n), where the numbers of states and symbols a Turing machine may have is unlimited, it may also be useful to consider the “Domain-Restricted BBi”, where the numbers of instructions, states, and/or symbols are specified values or ranges.

Following are some example three-argument functions within the Domain-Restricted BBi. For each function, the value is always the greatest number of steps taken by a Turing machine satisfying the corresponding restrictions:

* BBi(i, n, m): An n-state, m-symbol Turing machine having i instructions
* BBi(i, <=n, m): An i-instruction, m-symbol Turing machine having no greater than n states
* BBi(i, -, m): An i-instruction, m-symbol Turing machine having any number of states
* BBi(i, n, <=m): An i-instruction, n-state Turing machine having no greater than m instructions
* Etc.

Note that we have replaced the argument n from the standard BBi function with i in this three-argument version, primarily to avoid confusion with the n normally used to denote the number of states in a standard BB Turing machine.

For each of the three arguments, we may simply list a value, in which case the Turing machine must have exactly that number of instructions, states, or symbols. Another option is to specify a range, such as “<=n”, or “<=m”, in which case the Turing machine must have n or fewer states, or m or fewer symbols, respectively. And the final option is to give an argument as “-“, in which case, there is no restriction to the Turing machine’s number of instructions, states, or symbols.

'''Example Questions:'''

The main reason for having the flexibility in the way that the arguments are presented within the Domain-Restricted BBi function is that some problems relating to BB and BBi are best suited to examining Turing machines having specific numbers of states and symbols, while others are more practically examined within a range of state and symbol values instead. For example, consider the following two questions, along with two possible ways to interpret and answer each question:

'''Question 1:''' What are the maximum numbers of steps an i-instruction, 5-state, 2-symbol Turing machine can take before halting?

* Interpretation #1: For i = 1 through 10, BBi(i, 5, 2) = -, -, -, -, 9, 31, 65, 793, 47176869, 47176870. In this interpretation, there are no values for 1<=i<=4, because there is no way to construct a 5-state Turing machine using fewer than 5 instructions. Also, the values for i = 5, 6, and 7 are lower than those in the following second interpretation.
* Interpretation #2: For i = 1 through 10, BBi(i, <=5, 2) = 1, 3, 5, 16, 20, 37, 106, 793, 47176869, 47176870. For this interpretation, we do not take the questions wording of “5-state” so literally, but rather we allow the consideration of any Turing machine having no greater than 5 states. For example, BBi(7, <=5, 2) = 106 = BB(4,2) – 1.

The second interpretation, despite being less literally adherent to the question wording than the first, seems more useful for this particular question, specifically because we are probably interested in seeing how the number of steps grows for the best possible arrangement of i instructions within the 5 x 2 state table, and not just among the arrangements that specifically include defined transitions for all five states.

'''Question 2:''' For all n>1, will the 2n-1 instruction, n-state, 2-symbol BBi champion (one undefined transition) always exceed the 2n-1 instruction, n+1 state, 2-symbol BBi champion (3 undefined transitions)? (This is currently conjectured to be false.)

* Interpretation #1: Compare values of BBi(2n-1, n, 2) with those of BBi(2n-1, n+1, 2) for n = 2 through 5:
** For n=2 through 5, BBi(2n-1, n, 2) = 5, 20, 106, 47176869.
** For n=2 through 5, BBi(2n-1, n+1, 2) = 4, 19, 65, ???.
* Interpretation #2: Compare values of BBi(2n-1, <=n, 2) with those of BBi(2n-1, <=n+1, 2) for n=2 through 5:
** For n=2 through 5, BBi(2n-1, <=n, 2) = 5, 20, 106, 47176869.
** For n=2 through 5, BBi(2n-1, <=n+1, 2) = 5, 20, 106, ???.

For this second question, the first interpretation clearly seems to be the more useful one, as we are probably interested in comparing the steps differences between machines having strictly n states (with one undefined transition) with machines having strictly n+1 states (with three undefined transitions). Using the second interpretation, information about these steps differences is lost since we have replaced the steps values of the machines with n+1 states with the values from the stronger n-state machines.

'''Selected Results:'''

Following are some results for Domain-Restricted BBi, some of which follow logically from their definitions, and others which have been computed so far:

* BBi(i, -, -) = BBi(i). If we allow any number of states and symbols, then this is the standard BBi function.
* BBi(-, n, m) = BBi(nm, n, m) = BB(n, m) for n>=2, m>=2. If we allow any number of instructions, or if we specify nm instructions, then this is equivalent to the standard BB(n, m) function.
* BBi(nm-1, n, m) = BBi(nm, n, m) - 1 = BB(n, m) - 1 for n>=2, m>=2. With one undefined transition, the number of steps is one less than having a completely filled n x m transition table including a halting instruction, which adds one final step.
* For i>=1, BBi(i, -, 2) = 1, 3, 5, 16, 20, 37, 106, 793, …. BBi(9, -, 2) is at least 47176869, but it may be even larger if a more powerful 6-state, 2-symbol machine having three undefined transitions can be found. BBi(10, -, 2) will almost certainly be greater than the current lower bound of 47176870 = BB(5,2).
* For i>=1, BBi(i, <=2, -) = 1, 3, 5, 13, 37, 123, 3232963, …. The number of states is specified here as “<=2” rather than “2” so that we can have a valid value for one instruction. (A 1-instruction Turing machine has only one operating state.) BBi(8, <=2, -) will almost certainly be greater than the current lower bound of 3232964 = BB(2,4).

== Instruction-Limited Busy Beaver Variants ==
The concept of classifying a Turing machine's size by number of instructions, rather than by the number of states and symbols in its transition table, can also be utilized for Busy Beaver variants. The notation for the instruction-limited version of a variant can conveniently be denoted by appending an "i" to the already existing notation for the variant. Furthermore, the name of the Instruction-Limited version of the variant may be more conveniently started with "I-L" for brevity. Some known results for Instruction-Limited BB variants are as follows:

* I-L Greedy Busy Beaver. gBBi(n) is the largest number of steps taken by an n-instruction Turing machine, but where its first n-1 instructions (in the order encountered during running the machine) must also be a machine taking gBBi(n-1) steps. In other words, once the greedy n-instruction champion machines are identified, all machines having n+1 instructions must be extended from one of those n-instruction champions.
** gBBi(n) for n = 1 through n = 13 are: 1, 3, 5, 13, 19, 25, 41, 55, 238, 941, 1341, 10465, 10675. gBBi(14) must be at least 9,874,580 steps for current champion machine {{TM|0RB6RB1LB---3LA1RB7RB2LB_1LA2RB3LA4RB5RB1LB5LA---|halt}}.
* I-L Blanking Busy Beaver. BLBi(n) is the largest amount of steps taken by an n-instruction Turing machine when blanking the tape for the first time after having written a non-blank symbol to tape.
** BLBi(1) and BLBi(2) have no values because it is impossible to blank the tape with one or two instructions. BLBi(n) for n = 3 through n = 7 are: 4, 12, 30, 77, 808. BLBi(8) is at least 1,367,361,263,049 by the current champion {{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}.

== References ==
<references />
[[category:Functions]]

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-10-02T20:21:57Z

MrBrain: Removed carriage return

An '''n-instruction Turing machine''' is a [[Turing machine]] having an arbitrary number of states and symbols, but having only ''n'' defined transitions/instructions in its transition table (all others are undefined).

The '''Instruction-Limited Busy Beaver function (BBi(n))''' is the maximum steps taken by any n-instruction Turing machines before eventually halting (when started on an initially blank tape). Similarly, the '''Instruction-Limited Symbols Busy Beaver function (Σi(n))''' is the maximum [[sigma score]] (number of non-blank symbols left on the tape at halting) for all halting n-instruction Turing machines (when started on an initially blank tape).

* The currently known values of BBi(n) for n>=0 are: 0, 1, 3, 5, 16, 37, 123, 3932963, .... BBi(8) is at least 6.889 x 101565, which is the number of steps taken by the current 8-instruction champion machine, discovered by Nick Drozd.
* The currently known values of Σi(n) for n>=0 are: 0, 1, 2, 4, 5, 9, 14, 2050, .... Σi(8) is at least 1.355 x 10783, which is the number of non-blank symbols written to tape by the same 8-instruction champion machine.

As with the traditional Busy Beaver sequences, BBi(n) and Σi(n) are both uncomputable, with each growing faster than any computable function.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') <ref>Tibor Radó (May 1962). "[https://computation4cognitivescientists.weebly.com/uploads/6/2/8/3/6283774/rado-on_non-computable_functions.pdf On non-computable functions]" (PDF). ''Bell System Technical Journal''. '''41''' (3): 877–884. https://doi.org/10.1002%2Fj.1538-7305.1962.tb00480.x</ref>. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n = 1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m = 1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? The current BBi(8) champion is a 3-state, 4-symbol machine, but this may still be surpassed by a 3-state, 3-symbol machine yet to be found.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!Step Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|0RH|halt}} {{TM|1RH---|halt}}
|BB(1,2)
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''3'''
|{{TM|0RB---_1LA---|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''5'''
|{{TM|1RB1LB_1LA---|halt}} (and 13 others)
|[[BB(2)|BB(2,2)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''16'''
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''37'''
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|[[BB(2,3)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''123'''
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''3,932,963'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|[[BB(2,4)]] - 1, and
A [[BB(3,3)]] high-ranking machine
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 6.889 \times 10^{1565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://nickdrozd.github.io/2025/09/30/shape-of-a-turing-machine.html Blog] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

{| class="wikitable"
!n
!'''Σ'''i(n)
!Sigma Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|1RH---|halt}}
|Σ(1,2)
|[[oeis:A384766|A384766]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''2'''
|{{TM|1RB---_1LA---|halt}} (and 7 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''4'''
|{{TM|1RB1LB_1LA---|halt}} (and 4 others)
|[[BB(2)|Σ(2,2)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''5'''
|{{TM|1RB0LB---_1LA2RA---|halt}} (and 40 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''9'''
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|Σ[[BB(2,3)|(2,3)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''14'''
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''2,050'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|Σ[[BB(2,4)|(2,4)]], and
A Σ[[BB(3,3)|(3,3)]] high-ranking machine
|[[oeis:A384766|A384766]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 1.355 \times 10^{783}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

Notes:
* Proven values for BBi(n) and Σi(n) are shown as bold in the above two tables.
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue. With the BBi(8) champion it appears that this trend has been broken (it is believed that <math>BB(3,3) < 10^{1565}</math>)
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5) -1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* The convention for writing TMs in standard text format here is to include as many states and symbols as are referenced. So for example, we write {{TM|1RH---|halt}} instead of {{TM|1RH|halt}} since this TM references 2 symbols (including the implicit blank symbol 0). If one state has no instructions leaving it, the convention is to treat it as the halt state <code>H</code>.

== Tree Normal Form (TNF) for BBi ==
Shown below are the total number of n-instruction Turing machines for each number of instructions n>=0. The numbers of halting and non-halting machines for n = 8 are not yet known. The function Tree Normal Form for BBi is denoted BBiTNF(n)
{| class="wikitable"
!n
!All Machines
!All Cumulative
!Halting Machines
!Halting Cumulative
!Non-Halting Machines
!Non-Halting Cumulative
|-
| style="text-align: right;" |0
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |0
| style="text-align: right;" |0
|-
| style="text-align: right;" |1
| style="text-align: right;" |4
| style="text-align: right;" |5
| style="text-align: right;" |2
| style="text-align: right;" |3
| style="text-align: right;" |2
| style="text-align: right;" |2
|-
| style="text-align: right;" |2
| style="text-align: right;" |30
| style="text-align: right;" |35
| style="text-align: right;" |17
| style="text-align: right;" |20
| style="text-align: right;" |13
| style="text-align: right;" |15
|-
| style="text-align: right;" |3
| style="text-align: right;" |378
| style="text-align: right;" |413
| style="text-align: right;" |260
| style="text-align: right;" |280
| style="text-align: right;" |118
| style="text-align: right;" |133
|-
| style="text-align: right;" |4
| style="text-align: right;" |7,640
| style="text-align: right;" |8,053
| style="text-align: right;" |5,581
| style="text-align: right;" |5,861
| style="text-align: right;" |2,059
| style="text-align: right;" |2,192
|-
| style="text-align: right;" |5
| style="text-align: right;" |205,580
| style="text-align: right;" |213,633
| style="text-align: right;" |160,952
| style="text-align: right;" |166,813
| style="text-align: right;" |44,628
| style="text-align: right;" |46.820
|-
| style="text-align: right;" |6
| style="text-align: right;" |7,149,820
| style="text-align: right;" |7,363,453
| style="text-align: right;" |5,843,696
| style="text-align: right;" |6,010,509
| style="text-align: right;" |1,306,124
| style="text-align: right;" |1,352,944
|-
| style="text-align: right;" |7
| style="text-align: right;" |305,333,128
| style="text-align: right;" |312,696,581
| style="text-align: right;" |258,044,501
| style="text-align: right;" |264,055,010
| style="text-align: right;" |47,288,627
| style="text-align: right;" |48,641,571
|-
| style="text-align: right;" |8
| style="text-align: right;" |15,561,793,526
| style="text-align: right;" |15,874,490,107
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
|}
Enumeration of Turing machines in Tree Normal Form ([[TNF]]) for the Limited-Instruction Busy Beaver problem uses the following procedure:

* The "null Turing machine" with 0 instructions is the root of the tree. This "machine" halts after 0 steps, writing 0 non-blank symbols to the tape.
* The first instruction starts from A, the implied starting state, reads a 0 on the tape (which is blank at machine start) and must shift right (to avoid consideration of trivial mirror-image machines).
** This allows the following four possibilities: A0→0RA, A0→0RB, A0→1RA, and A0→1RB. The two machines transitioning to state A never halt, while the two machines transitioning to state B halt. The two halting machines are next expecting an instruction for B0, but since no such instruction exists, the machines halt after a single step.
* Following the enumeration of machines for each previous number of instructions, each of the halting machines is extended from the instruction at which it halted. For n = 2, this instruction must be B0. (Non-halting machines cannot be extended since they can never reach a new instruction to be added.)
* As per normal TNF procedure, the new instruction can write any symbol to tape, shift left or right, and transition to any state as long as:
** The symbol written can be at most one greater than the highest symbol previously read or written, and
** The state to which the instruction transitions can be at most one state greater than the current maximum state.
* For the BBi problem, any instruction that transitions to a "new" state (one not yet containing any instructions) can be interpreted as a "halting instruction". (e.g. The one-instruction machine A0→1RB is exactly the same as A0→1RH.) Similarly, this new state can be considered a "halt state", as there is functionally no difference between a halt state and an operational state having no instructions. (B = H.)
** Unlike the traditional BB problem, where a halting instruction is normally used in the last remaining cell in an n x m transition table, it is necessary to consider all possible halting transitions for BBi, (not just those of the form →1RH), as this allows the most possible extensions of n-instruction machines to (n+1)-instruction machines.

== Domain-Restricted BBi ==
In addition to the standard Instruction-Limited Busy Beaver function BBi(n), where the numbers of states and symbols a Turing machine may have is unlimited, it may also be useful to consider the “Domain-Restricted BBi”, where the numbers of instructions, states, and/or symbols are specified values or ranges.

Following are some example three-argument functions within the Domain-Restricted BBi. For each function, the value is always the greatest number of steps taken by a Turing machine satisfying the corresponding restrictions:

* BBi(i, n, m): An n-state, m-symbol Turing machine having i instructions
* BBi(i, <=n, m): An i-instruction, m-symbol Turing machine having no greater than n states
* BBi(i, -, m): An i-instruction, m-symbol Turing machine having any number of states
* BBi(i, n, <=m): An i-instruction, n-state Turing machine having no greater than m instructions
* Etc.

Note that we have replaced the argument n from the standard BBi function with i in this three-argument version, primarily to avoid confusion with the n normally used to denote the number of states in a standard BB Turing machine.

For each of the three arguments, we may simply list a value, in which case the Turing machine must have exactly that number of instructions, states, or symbols. Another option is to specify a range, such as “<=n”, or “<=m”, in which case the Turing machine must have n or fewer states, or m or fewer symbols, respectively. And the final option is to give an argument as “-“, in which case, there is no restriction to the Turing machine’s number of instructions, states, or symbols.

'''Example Questions:'''

The main reason for having the flexibility in the way that the arguments are presented within the Domain-Restricted BBi function is that some problems relating to BB and BBi are best suited to examining Turing machines having specific numbers of states and symbols, while others are more practically examined within a range of state and symbol values instead. For example, consider the following two questions, along with two possible ways to interpret and answer each question:

'''Question 1:''' What are the maximum numbers of steps an i-instruction, 5-state, 2-symbol Turing machine can take before halting?

* Interpretation #1: For i = 1 through 10, BBi(i, 5, 2) = -, -, -, -, 9, 31, 65, 793, 47176869, 47176870. In this interpretation, there are no values for 1<=i<=4, because there is no way to construct a 5-state Turing machine using fewer than 5 instructions. Also, the values for i = 5, 6, and 7 are lower than those in the following second interpretation.
* Interpretation #2: For i = 1 through 10, BBi(i, <=5, 2) = 1, 3, 5, 16, 20, 37, 106, 793, 47176869, 47176870. For this interpretation, we do not take the questions wording of “5-state” so literally, but rather we allow the consideration of any Turing machine having no greater than 5 states. For example, BBi(7, <=5, 2) = 106 = BB(4,2) – 1.

The second interpretation, despite being less literally adherent to the question wording than the first, seems more useful for this particular question, specifically because we are probably interested in seeing how the number of steps grows for the best possible arrangement of i instructions within the 5 x 2 state table, and not just among the arrangements that specifically include defined transitions for all five states.

'''Question 2:''' For all n>1, will the 2n-1 instruction, n-state, 2-symbol BBi champion (one undefined transition) always exceed the 2n-1 instruction, n+1 state, 2-symbol BBi champion (3 undefined transitions)? (This is currently conjectured to be false.)

* Interpretation #1: Compare values of BBi(2n-1, n, 2) with those of BBi(2n-1, n+1, 2) for n = 2 through 5:
** For n=2 through 5, BBi(2n-1, n, 2) = 5, 20, 106, 47176869.
** For n=2 through 5, BBi(2n-1, n+1, 2) = 4, 19, 65, ???.
* Interpretation #2: Compare values of BBi(2n-1, <=n, 2) with those of BBi(2n-1, <=n+1, 2) for n=2 through 5:
** For n=2 through 5, BBi(2n-1, <=n, 2) = 5, 20, 106, 47176869.
** For n=2 through 5, BBi(2n-1, <=n+1, 2) = 5, 20, 106, ???.

For this second question, the first interpretation clearly seems to be the more useful one, as we are probably interested in comparing the steps differences between machines having strictly n states (with one undefined transition) with machines having strictly n+1 states (with three undefined transitions). Using the second interpretation, information about these steps differences is lost since we have replaced the steps values of the machines with n+1 states with the values from the stronger n-state machines.

'''Selected Results:'''

Following are some results for Domain-Restricted BBi, some of which follow logically from their definitions, and others which have been computed so far:

* BBi(i, -, -) = BBi(i). If we allow any number of states and symbols, then this is the standard BBi function.
* BBi(-, n, m) = BBi(nm, n, m) = BB(n, m) for n>=2, m>=2. If we allow any number of instructions, or if we specify nm instructions, then this is equivalent to the standard BB(n, m) function.
* BBi(nm-1, n, m) = BBi(nm, n, m) - 1 = BB(n, m) - 1 for n>=2, m>=2. With one undefined transition, the number of steps is one less than having a completely filled n x m transition table including a halting instruction, which adds one final step.
* For i>=1, BBi(i, -, 2) = 1, 3, 5, 16, 20, 37, 106, 793, …. BBi(9, -, 2) is at least 47176869, but it may be even larger if a more powerful 6-state, 2-symbol machine having three undefined transitions can be found. BBi(10, -, 2) will almost certainly be greater than the current lower bound of 47176870 = BB(5,2).
* For i>=1, BBi(i, <=2, -) = 1, 3, 5, 13, 37, 123, 3232963, …. The number of states is specified here as “<=2” rather than “2” so that we can have a valid value for one instruction. (A 1-instruction Turing machine has only one operating state.) BBi(8, <=2, -) will almost certainly be greater than the current lower bound of 3232964 = BB(2,4).

== Instruction-Limited Busy Beaver Variants ==
The concept of classifying a Turing machine's size by number of instructions, rather than by the number of states and symbols in its transition table, can also be utilized for Busy Beaver variants. The notation for the instruction-limited version of a variant can conveniently be denoted by appending an "i" to the already existing notation for the variant. Furthermore, the name of the Instruction-Limited version of the variant may be more conveniently started with "I-L" for brevity. Some known results for Instruction-Limited BB variants are as follows:

* I-L Greedy Busy Beaver. gBBi(n) is the largest number of steps taken by an n-instruction Turing machine, but where its first n-1 instructions (in the order encountered during running the machine) must also be a machine taking gBBi(n-1) steps. In other words, once the greedy n-instruction champion machines are identified, all machines having n+1 instructions must be extended from one of those n-instruction champions.
** gBBi(n) for n = 1 through n = 13 are: 1, 3, 5, 13, 19, 25, 41, 55, 238, 941, 1341, 10465, 10675. gBBi(14) must be at least 9,874,580 steps for current champion machine {{TM|0RB6RB1LB---3LA1RB7RB2LB_1LA2RB3LA4RB5RB1LB5LA---|halt}}.
* I-L Blanking Busy Beaver. BLBi(n) is the largest amount of steps taken by an n-instruction Turing machine when blanking the tape for the first time after having written a non-blank symbol to tape.
** BLBi(1) and BLBi(2) have no values because it is impossible to blank the tape with one or two instructions. BLBi(n) for n = 3 through n = 7 are: 4, 12, 30, 77, 808. BLBi(8) is at least 1,367,361,263,049 by the current champion {{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}.

== References ==
<references />
[[category:Functions]]

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-10-02T20:21:20Z

MrBrain: Added link to blog post about the current BBi(8) champion machine.

An '''n-instruction Turing machine''' is a [[Turing machine]] having an arbitrary number of states and symbols, but having only ''n'' defined transitions/instructions in its transition table (all others are undefined).

The '''Instruction-Limited Busy Beaver function (BBi(n))''' is the maximum steps taken by any n-instruction Turing machines before eventually halting (when started on an initially blank tape). Similarly, the '''Instruction-Limited Symbols Busy Beaver function (Σi(n))''' is the maximum [[sigma score]] (number of non-blank symbols left on the tape at halting) for all halting n-instruction Turing machines (when started on an initially blank tape).

* The currently known values of BBi(n) for n>=0 are: 0, 1, 3, 5, 16, 37, 123, 3932963, .... BBi(8) is at least 6.889 x 101565, which is the number of steps taken by the current 8-instruction champion machine, discovered by Nick Drozd.
* The currently known values of Σi(n) for n>=0 are: 0, 1, 2, 4, 5, 9, 14, 2050, .... Σi(8) is at least 1.355 x 10783, which is the number of non-blank symbols written to tape by the same 8-instruction champion machine.

As with the traditional Busy Beaver sequences, BBi(n) and Σi(n) are both uncomputable, with each growing faster than any computable function.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') <ref>Tibor Radó (May 1962). "[https://computation4cognitivescientists.weebly.com/uploads/6/2/8/3/6283774/rado-on_non-computable_functions.pdf On non-computable functions]" (PDF). ''Bell System Technical Journal''. '''41''' (3): 877–884. https://doi.org/10.1002%2Fj.1538-7305.1962.tb00480.x</ref>. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n = 1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m = 1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? The current BBi(8) champion is a 3-state, 4-symbol machine, but this may still be surpassed by a 3-state, 3-symbol machine yet to be found.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!Step Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|0RH|halt}} {{TM|1RH---|halt}}
|BB(1,2)
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''3'''
|{{TM|0RB---_1LA---|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''5'''
|{{TM|1RB1LB_1LA---|halt}} (and 13 others)
|[[BB(2)|BB(2,2)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''16'''
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''37'''
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|[[BB(2,3)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''123'''
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''3,932,963'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|[[BB(2,4)]] - 1, and
A [[BB(3,3)]] high-ranking machine
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 6.889 \times 10^{1565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://nickdrozd.github.io/2025/09/30/shape-of-a-turing-machine.html Blog]
[https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

{| class="wikitable"
!n
!'''Σ'''i(n)
!Sigma Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|1RH---|halt}}
|Σ(1,2)
|[[oeis:A384766|A384766]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''2'''
|{{TM|1RB---_1LA---|halt}} (and 7 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''4'''
|{{TM|1RB1LB_1LA---|halt}} (and 4 others)
|[[BB(2)|Σ(2,2)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''5'''
|{{TM|1RB0LB---_1LA2RA---|halt}} (and 40 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''9'''
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|Σ[[BB(2,3)|(2,3)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''14'''
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''2,050'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|Σ[[BB(2,4)|(2,4)]], and
A Σ[[BB(3,3)|(3,3)]] high-ranking machine
|[[oeis:A384766|A384766]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 1.355 \times 10^{783}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

Notes:
* Proven values for BBi(n) and Σi(n) are shown as bold in the above two tables.
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue. With the BBi(8) champion it appears that this trend has been broken (it is believed that <math>BB(3,3) < 10^{1565}</math>)
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5) -1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* The convention for writing TMs in standard text format here is to include as many states and symbols as are referenced. So for example, we write {{TM|1RH---|halt}} instead of {{TM|1RH|halt}} since this TM references 2 symbols (including the implicit blank symbol 0). If one state has no instructions leaving it, the convention is to treat it as the halt state <code>H</code>.

== Tree Normal Form (TNF) for BBi ==
Shown below are the total number of n-instruction Turing machines for each number of instructions n>=0. The numbers of halting and non-halting machines for n = 8 are not yet known. The function Tree Normal Form for BBi is denoted BBiTNF(n)
{| class="wikitable"
!n
!All Machines
!All Cumulative
!Halting Machines
!Halting Cumulative
!Non-Halting Machines
!Non-Halting Cumulative
|-
| style="text-align: right;" |0
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |0
| style="text-align: right;" |0
|-
| style="text-align: right;" |1
| style="text-align: right;" |4
| style="text-align: right;" |5
| style="text-align: right;" |2
| style="text-align: right;" |3
| style="text-align: right;" |2
| style="text-align: right;" |2
|-
| style="text-align: right;" |2
| style="text-align: right;" |30
| style="text-align: right;" |35
| style="text-align: right;" |17
| style="text-align: right;" |20
| style="text-align: right;" |13
| style="text-align: right;" |15
|-
| style="text-align: right;" |3
| style="text-align: right;" |378
| style="text-align: right;" |413
| style="text-align: right;" |260
| style="text-align: right;" |280
| style="text-align: right;" |118
| style="text-align: right;" |133
|-
| style="text-align: right;" |4
| style="text-align: right;" |7,640
| style="text-align: right;" |8,053
| style="text-align: right;" |5,581
| style="text-align: right;" |5,861
| style="text-align: right;" |2,059
| style="text-align: right;" |2,192
|-
| style="text-align: right;" |5
| style="text-align: right;" |205,580
| style="text-align: right;" |213,633
| style="text-align: right;" |160,952
| style="text-align: right;" |166,813
| style="text-align: right;" |44,628
| style="text-align: right;" |46.820
|-
| style="text-align: right;" |6
| style="text-align: right;" |7,149,820
| style="text-align: right;" |7,363,453
| style="text-align: right;" |5,843,696
| style="text-align: right;" |6,010,509
| style="text-align: right;" |1,306,124
| style="text-align: right;" |1,352,944
|-
| style="text-align: right;" |7
| style="text-align: right;" |305,333,128
| style="text-align: right;" |312,696,581
| style="text-align: right;" |258,044,501
| style="text-align: right;" |264,055,010
| style="text-align: right;" |47,288,627
| style="text-align: right;" |48,641,571
|-
| style="text-align: right;" |8
| style="text-align: right;" |15,561,793,526
| style="text-align: right;" |15,874,490,107
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
|}
Enumeration of Turing machines in Tree Normal Form ([[TNF]]) for the Limited-Instruction Busy Beaver problem uses the following procedure:

* The "null Turing machine" with 0 instructions is the root of the tree. This "machine" halts after 0 steps, writing 0 non-blank symbols to the tape.
* The first instruction starts from A, the implied starting state, reads a 0 on the tape (which is blank at machine start) and must shift right (to avoid consideration of trivial mirror-image machines).
** This allows the following four possibilities: A0→0RA, A0→0RB, A0→1RA, and A0→1RB. The two machines transitioning to state A never halt, while the two machines transitioning to state B halt. The two halting machines are next expecting an instruction for B0, but since no such instruction exists, the machines halt after a single step.
* Following the enumeration of machines for each previous number of instructions, each of the halting machines is extended from the instruction at which it halted. For n = 2, this instruction must be B0. (Non-halting machines cannot be extended since they can never reach a new instruction to be added.)
* As per normal TNF procedure, the new instruction can write any symbol to tape, shift left or right, and transition to any state as long as:
** The symbol written can be at most one greater than the highest symbol previously read or written, and
** The state to which the instruction transitions can be at most one state greater than the current maximum state.
* For the BBi problem, any instruction that transitions to a "new" state (one not yet containing any instructions) can be interpreted as a "halting instruction". (e.g. The one-instruction machine A0→1RB is exactly the same as A0→1RH.) Similarly, this new state can be considered a "halt state", as there is functionally no difference between a halt state and an operational state having no instructions. (B = H.)
** Unlike the traditional BB problem, where a halting instruction is normally used in the last remaining cell in an n x m transition table, it is necessary to consider all possible halting transitions for BBi, (not just those of the form →1RH), as this allows the most possible extensions of n-instruction machines to (n+1)-instruction machines.

== Domain-Restricted BBi ==
In addition to the standard Instruction-Limited Busy Beaver function BBi(n), where the numbers of states and symbols a Turing machine may have is unlimited, it may also be useful to consider the “Domain-Restricted BBi”, where the numbers of instructions, states, and/or symbols are specified values or ranges.

Following are some example three-argument functions within the Domain-Restricted BBi. For each function, the value is always the greatest number of steps taken by a Turing machine satisfying the corresponding restrictions:

* BBi(i, n, m): An n-state, m-symbol Turing machine having i instructions
* BBi(i, <=n, m): An i-instruction, m-symbol Turing machine having no greater than n states
* BBi(i, -, m): An i-instruction, m-symbol Turing machine having any number of states
* BBi(i, n, <=m): An i-instruction, n-state Turing machine having no greater than m instructions
* Etc.

Note that we have replaced the argument n from the standard BBi function with i in this three-argument version, primarily to avoid confusion with the n normally used to denote the number of states in a standard BB Turing machine.

For each of the three arguments, we may simply list a value, in which case the Turing machine must have exactly that number of instructions, states, or symbols. Another option is to specify a range, such as “<=n”, or “<=m”, in which case the Turing machine must have n or fewer states, or m or fewer symbols, respectively. And the final option is to give an argument as “-“, in which case, there is no restriction to the Turing machine’s number of instructions, states, or symbols.

'''Example Questions:'''

The main reason for having the flexibility in the way that the arguments are presented within the Domain-Restricted BBi function is that some problems relating to BB and BBi are best suited to examining Turing machines having specific numbers of states and symbols, while others are more practically examined within a range of state and symbol values instead. For example, consider the following two questions, along with two possible ways to interpret and answer each question:

'''Question 1:''' What are the maximum numbers of steps an i-instruction, 5-state, 2-symbol Turing machine can take before halting?

* Interpretation #1: For i = 1 through 10, BBi(i, 5, 2) = -, -, -, -, 9, 31, 65, 793, 47176869, 47176870. In this interpretation, there are no values for 1<=i<=4, because there is no way to construct a 5-state Turing machine using fewer than 5 instructions. Also, the values for i = 5, 6, and 7 are lower than those in the following second interpretation.
* Interpretation #2: For i = 1 through 10, BBi(i, <=5, 2) = 1, 3, 5, 16, 20, 37, 106, 793, 47176869, 47176870. For this interpretation, we do not take the questions wording of “5-state” so literally, but rather we allow the consideration of any Turing machine having no greater than 5 states. For example, BBi(7, <=5, 2) = 106 = BB(4,2) – 1.

The second interpretation, despite being less literally adherent to the question wording than the first, seems more useful for this particular question, specifically because we are probably interested in seeing how the number of steps grows for the best possible arrangement of i instructions within the 5 x 2 state table, and not just among the arrangements that specifically include defined transitions for all five states.

'''Question 2:''' For all n>1, will the 2n-1 instruction, n-state, 2-symbol BBi champion (one undefined transition) always exceed the 2n-1 instruction, n+1 state, 2-symbol BBi champion (3 undefined transitions)? (This is currently conjectured to be false.)

* Interpretation #1: Compare values of BBi(2n-1, n, 2) with those of BBi(2n-1, n+1, 2) for n = 2 through 5:
** For n=2 through 5, BBi(2n-1, n, 2) = 5, 20, 106, 47176869.
** For n=2 through 5, BBi(2n-1, n+1, 2) = 4, 19, 65, ???.
* Interpretation #2: Compare values of BBi(2n-1, <=n, 2) with those of BBi(2n-1, <=n+1, 2) for n=2 through 5:
** For n=2 through 5, BBi(2n-1, <=n, 2) = 5, 20, 106, 47176869.
** For n=2 through 5, BBi(2n-1, <=n+1, 2) = 5, 20, 106, ???.

For this second question, the first interpretation clearly seems to be the more useful one, as we are probably interested in comparing the steps differences between machines having strictly n states (with one undefined transition) with machines having strictly n+1 states (with three undefined transitions). Using the second interpretation, information about these steps differences is lost since we have replaced the steps values of the machines with n+1 states with the values from the stronger n-state machines.

'''Selected Results:'''

Following are some results for Domain-Restricted BBi, some of which follow logically from their definitions, and others which have been computed so far:

* BBi(i, -, -) = BBi(i). If we allow any number of states and symbols, then this is the standard BBi function.
* BBi(-, n, m) = BBi(nm, n, m) = BB(n, m) for n>=2, m>=2. If we allow any number of instructions, or if we specify nm instructions, then this is equivalent to the standard BB(n, m) function.
* BBi(nm-1, n, m) = BBi(nm, n, m) - 1 = BB(n, m) - 1 for n>=2, m>=2. With one undefined transition, the number of steps is one less than having a completely filled n x m transition table including a halting instruction, which adds one final step.
* For i>=1, BBi(i, -, 2) = 1, 3, 5, 16, 20, 37, 106, 793, …. BBi(9, -, 2) is at least 47176869, but it may be even larger if a more powerful 6-state, 2-symbol machine having three undefined transitions can be found. BBi(10, -, 2) will almost certainly be greater than the current lower bound of 47176870 = BB(5,2).
* For i>=1, BBi(i, <=2, -) = 1, 3, 5, 13, 37, 123, 3232963, …. The number of states is specified here as “<=2” rather than “2” so that we can have a valid value for one instruction. (A 1-instruction Turing machine has only one operating state.) BBi(8, <=2, -) will almost certainly be greater than the current lower bound of 3232964 = BB(2,4).

== Instruction-Limited Busy Beaver Variants ==
The concept of classifying a Turing machine's size by number of instructions, rather than by the number of states and symbols in its transition table, can also be utilized for Busy Beaver variants. The notation for the instruction-limited version of a variant can conveniently be denoted by appending an "i" to the already existing notation for the variant. Furthermore, the name of the Instruction-Limited version of the variant may be more conveniently started with "I-L" for brevity. Some known results for Instruction-Limited BB variants are as follows:

* I-L Greedy Busy Beaver. gBBi(n) is the largest number of steps taken by an n-instruction Turing machine, but where its first n-1 instructions (in the order encountered during running the machine) must also be a machine taking gBBi(n-1) steps. In other words, once the greedy n-instruction champion machines are identified, all machines having n+1 instructions must be extended from one of those n-instruction champions.
** gBBi(n) for n = 1 through n = 13 are: 1, 3, 5, 13, 19, 25, 41, 55, 238, 941, 1341, 10465, 10675. gBBi(14) must be at least 9,874,580 steps for current champion machine {{TM|0RB6RB1LB---3LA1RB7RB2LB_1LA2RB3LA4RB5RB1LB5LA---|halt}}.
* I-L Blanking Busy Beaver. BLBi(n) is the largest amount of steps taken by an n-instruction Turing machine when blanking the tape for the first time after having written a non-blank symbol to tape.
** BLBi(1) and BLBi(2) have no values because it is impossible to blank the tape with one or two instructions. BLBi(n) for n = 3 through n = 7 are: 4, 12, 30, 77, 808. BLBi(8) is at least 1,367,361,263,049 by the current champion {{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}.

== References ==
<references />
[[category:Functions]]

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-10-02T20:20:16Z

MrBrain:

An '''n-instruction Turing machine''' is a [[Turing machine]] having an arbitrary number of states and symbols, but having only ''n'' defined transitions/instructions in its transition table (all others are undefined).

The '''Instruction-Limited Busy Beaver function (BBi(n))''' is the maximum steps taken by any n-instruction Turing machines before eventually halting (when started on an initially blank tape). Similarly, the '''Instruction-Limited Symbols Busy Beaver function (Σi(n))''' is the maximum [[sigma score]] (number of non-blank symbols left on the tape at halting) for all halting n-instruction Turing machines (when started on an initially blank tape).

* The currently known values of BBi(n) for n>=0 are: 0, 1, 3, 5, 16, 37, 123, 3932963, .... BBi(8) is at least 6.889 x 101565, which is the number of steps taken by the current 8-instruction champion machine, discovered by Nick Drozd.
* The currently known values of Σi(n) for n>=0 are: 0, 1, 2, 4, 5, 9, 14, 2050, .... Σi(8) is at least 1.355 x 10783, which is the number of non-blank symbols written to tape by the same 8-instruction champion machine.

As with the traditional Busy Beaver sequences, BBi(n) and Σi(n) are both uncomputable, with each growing faster than any computable function.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') <ref>Tibor Radó (May 1962). "[https://computation4cognitivescientists.weebly.com/uploads/6/2/8/3/6283774/rado-on_non-computable_functions.pdf On non-computable functions]" (PDF). ''Bell System Technical Journal''. '''41''' (3): 877–884. https://doi.org/10.1002%2Fj.1538-7305.1962.tb00480.x</ref>. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n = 1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m = 1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? The current BBi(8) champion is a 3-state, 4-symbol machine, but this may still be surpassed by a 3-state, 3-symbol machine yet to be found.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!Step Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|0RH|halt}} {{TM|1RH---|halt}}
|BB(1,2)
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''3'''
|{{TM|0RB---_1LA---|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''5'''
|{{TM|1RB1LB_1LA---|halt}} (and 13 others)
|[[BB(2)|BB(2,2)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''16'''
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''37'''
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|[[BB(2,3)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''123'''
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''3,932,963'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|[[BB(2,4)]] - 1, and
A [[BB(3,3)]] high-ranking machine
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 6.889 \times 10^{1565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd]
[https://nickdrozd.github.io/2025/09/30/shape-of-a-turing-machine.html Blog]
[https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

{| class="wikitable"
!n
!'''Σ'''i(n)
!Sigma Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|1RH---|halt}}
|Σ(1,2)
|[[oeis:A384766|A384766]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''2'''
|{{TM|1RB---_1LA---|halt}} (and 7 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''4'''
|{{TM|1RB1LB_1LA---|halt}} (and 4 others)
|[[BB(2)|Σ(2,2)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''5'''
|{{TM|1RB0LB---_1LA2RA---|halt}} (and 40 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''9'''
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|Σ[[BB(2,3)|(2,3)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''14'''
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''2,050'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|Σ[[BB(2,4)|(2,4)]], and
A Σ[[BB(3,3)|(3,3)]] high-ranking machine
|[[oeis:A384766|A384766]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 1.355 \times 10^{783}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

Notes:
* Proven values for BBi(n) and Σi(n) are shown as bold in the above two tables.
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue. With the BBi(8) champion it appears that this trend has been broken (it is believed that <math>BB(3,3) < 10^{1565}</math>)
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5) -1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* The convention for writing TMs in standard text format here is to include as many states and symbols as are referenced. So for example, we write {{TM|1RH---|halt}} instead of {{TM|1RH|halt}} since this TM references 2 symbols (including the implicit blank symbol 0). If one state has no instructions leaving it, the convention is to treat it as the halt state <code>H</code>.

== Tree Normal Form (TNF) for BBi ==
Shown below are the total number of n-instruction Turing machines for each number of instructions n>=0. The numbers of halting and non-halting machines for n = 8 are not yet known. The function Tree Normal Form for BBi is denoted BBiTNF(n)
{| class="wikitable"
!n
!All Machines
!All Cumulative
!Halting Machines
!Halting Cumulative
!Non-Halting Machines
!Non-Halting Cumulative
|-
| style="text-align: right;" |0
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |0
| style="text-align: right;" |0
|-
| style="text-align: right;" |1
| style="text-align: right;" |4
| style="text-align: right;" |5
| style="text-align: right;" |2
| style="text-align: right;" |3
| style="text-align: right;" |2
| style="text-align: right;" |2
|-
| style="text-align: right;" |2
| style="text-align: right;" |30
| style="text-align: right;" |35
| style="text-align: right;" |17
| style="text-align: right;" |20
| style="text-align: right;" |13
| style="text-align: right;" |15
|-
| style="text-align: right;" |3
| style="text-align: right;" |378
| style="text-align: right;" |413
| style="text-align: right;" |260
| style="text-align: right;" |280
| style="text-align: right;" |118
| style="text-align: right;" |133
|-
| style="text-align: right;" |4
| style="text-align: right;" |7,640
| style="text-align: right;" |8,053
| style="text-align: right;" |5,581
| style="text-align: right;" |5,861
| style="text-align: right;" |2,059
| style="text-align: right;" |2,192
|-
| style="text-align: right;" |5
| style="text-align: right;" |205,580
| style="text-align: right;" |213,633
| style="text-align: right;" |160,952
| style="text-align: right;" |166,813
| style="text-align: right;" |44,628
| style="text-align: right;" |46.820
|-
| style="text-align: right;" |6
| style="text-align: right;" |7,149,820
| style="text-align: right;" |7,363,453
| style="text-align: right;" |5,843,696
| style="text-align: right;" |6,010,509
| style="text-align: right;" |1,306,124
| style="text-align: right;" |1,352,944
|-
| style="text-align: right;" |7
| style="text-align: right;" |305,333,128
| style="text-align: right;" |312,696,581
| style="text-align: right;" |258,044,501
| style="text-align: right;" |264,055,010
| style="text-align: right;" |47,288,627
| style="text-align: right;" |48,641,571
|-
| style="text-align: right;" |8
| style="text-align: right;" |15,561,793,526
| style="text-align: right;" |15,874,490,107
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
|}
Enumeration of Turing machines in Tree Normal Form ([[TNF]]) for the Limited-Instruction Busy Beaver problem uses the following procedure:

* The "null Turing machine" with 0 instructions is the root of the tree. This "machine" halts after 0 steps, writing 0 non-blank symbols to the tape.
* The first instruction starts from A, the implied starting state, reads a 0 on the tape (which is blank at machine start) and must shift right (to avoid consideration of trivial mirror-image machines).
** This allows the following four possibilities: A0→0RA, A0→0RB, A0→1RA, and A0→1RB. The two machines transitioning to state A never halt, while the two machines transitioning to state B halt. The two halting machines are next expecting an instruction for B0, but since no such instruction exists, the machines halt after a single step.
* Following the enumeration of machines for each previous number of instructions, each of the halting machines is extended from the instruction at which it halted. For n = 2, this instruction must be B0. (Non-halting machines cannot be extended since they can never reach a new instruction to be added.)
* As per normal TNF procedure, the new instruction can write any symbol to tape, shift left or right, and transition to any state as long as:
** The symbol written can be at most one greater than the highest symbol previously read or written, and
** The state to which the instruction transitions can be at most one state greater than the current maximum state.
* For the BBi problem, any instruction that transitions to a "new" state (one not yet containing any instructions) can be interpreted as a "halting instruction". (e.g. The one-instruction machine A0→1RB is exactly the same as A0→1RH.) Similarly, this new state can be considered a "halt state", as there is functionally no difference between a halt state and an operational state having no instructions. (B = H.)
** Unlike the traditional BB problem, where a halting instruction is normally used in the last remaining cell in an n x m transition table, it is necessary to consider all possible halting transitions for BBi, (not just those of the form →1RH), as this allows the most possible extensions of n-instruction machines to (n+1)-instruction machines.

== Domain-Restricted BBi ==
In addition to the standard Instruction-Limited Busy Beaver function BBi(n), where the numbers of states and symbols a Turing machine may have is unlimited, it may also be useful to consider the “Domain-Restricted BBi”, where the numbers of instructions, states, and/or symbols are specified values or ranges.

Following are some example three-argument functions within the Domain-Restricted BBi. For each function, the value is always the greatest number of steps taken by a Turing machine satisfying the corresponding restrictions:

* BBi(i, n, m): An n-state, m-symbol Turing machine having i instructions
* BBi(i, <=n, m): An i-instruction, m-symbol Turing machine having no greater than n states
* BBi(i, -, m): An i-instruction, m-symbol Turing machine having any number of states
* BBi(i, n, <=m): An i-instruction, n-state Turing machine having no greater than m instructions
* Etc.

Note that we have replaced the argument n from the standard BBi function with i in this three-argument version, primarily to avoid confusion with the n normally used to denote the number of states in a standard BB Turing machine.

For each of the three arguments, we may simply list a value, in which case the Turing machine must have exactly that number of instructions, states, or symbols. Another option is to specify a range, such as “<=n”, or “<=m”, in which case the Turing machine must have n or fewer states, or m or fewer symbols, respectively. And the final option is to give an argument as “-“, in which case, there is no restriction to the Turing machine’s number of instructions, states, or symbols.

'''Example Questions:'''

The main reason for having the flexibility in the way that the arguments are presented within the Domain-Restricted BBi function is that some problems relating to BB and BBi are best suited to examining Turing machines having specific numbers of states and symbols, while others are more practically examined within a range of state and symbol values instead. For example, consider the following two questions, along with two possible ways to interpret and answer each question:

'''Question 1:''' What are the maximum numbers of steps an i-instruction, 5-state, 2-symbol Turing machine can take before halting?

* Interpretation #1: For i = 1 through 10, BBi(i, 5, 2) = -, -, -, -, 9, 31, 65, 793, 47176869, 47176870. In this interpretation, there are no values for 1<=i<=4, because there is no way to construct a 5-state Turing machine using fewer than 5 instructions. Also, the values for i = 5, 6, and 7 are lower than those in the following second interpretation.
* Interpretation #2: For i = 1 through 10, BBi(i, <=5, 2) = 1, 3, 5, 16, 20, 37, 106, 793, 47176869, 47176870. For this interpretation, we do not take the questions wording of “5-state” so literally, but rather we allow the consideration of any Turing machine having no greater than 5 states. For example, BBi(7, <=5, 2) = 106 = BB(4,2) – 1.

The second interpretation, despite being less literally adherent to the question wording than the first, seems more useful for this particular question, specifically because we are probably interested in seeing how the number of steps grows for the best possible arrangement of i instructions within the 5 x 2 state table, and not just among the arrangements that specifically include defined transitions for all five states.

'''Question 2:''' For all n>1, will the 2n-1 instruction, n-state, 2-symbol BBi champion (one undefined transition) always exceed the 2n-1 instruction, n+1 state, 2-symbol BBi champion (3 undefined transitions)? (This is currently conjectured to be false.)

* Interpretation #1: Compare values of BBi(2n-1, n, 2) with those of BBi(2n-1, n+1, 2) for n = 2 through 5:
** For n=2 through 5, BBi(2n-1, n, 2) = 5, 20, 106, 47176869.
** For n=2 through 5, BBi(2n-1, n+1, 2) = 4, 19, 65, ???.
* Interpretation #2: Compare values of BBi(2n-1, <=n, 2) with those of BBi(2n-1, <=n+1, 2) for n=2 through 5:
** For n=2 through 5, BBi(2n-1, <=n, 2) = 5, 20, 106, 47176869.
** For n=2 through 5, BBi(2n-1, <=n+1, 2) = 5, 20, 106, ???.

For this second question, the first interpretation clearly seems to be the more useful one, as we are probably interested in comparing the steps differences between machines having strictly n states (with one undefined transition) with machines having strictly n+1 states (with three undefined transitions). Using the second interpretation, information about these steps differences is lost since we have replaced the steps values of the machines with n+1 states with the values from the stronger n-state machines.

'''Selected Results:'''

Following are some results for Domain-Restricted BBi, some of which follow logically from their definitions, and others which have been computed so far:

* BBi(i, -, -) = BBi(i). If we allow any number of states and symbols, then this is the standard BBi function.
* BBi(-, n, m) = BBi(nm, n, m) = BB(n, m) for n>=2, m>=2. If we allow any number of instructions, or if we specify nm instructions, then this is equivalent to the standard BB(n, m) function.
* BBi(nm-1, n, m) = BBi(nm, n, m) - 1 = BB(n, m) - 1 for n>=2, m>=2. With one undefined transition, the number of steps is one less than having a completely filled n x m transition table including a halting instruction, which adds one final step.
* For i>=1, BBi(i, -, 2) = 1, 3, 5, 16, 20, 37, 106, 793, …. BBi(9, -, 2) is at least 47176869, but it may be even larger if a more powerful 6-state, 2-symbol machine having three undefined transitions can be found. BBi(10, -, 2) will almost certainly be greater than the current lower bound of 47176870 = BB(5,2).
* For i>=1, BBi(i, <=2, -) = 1, 3, 5, 13, 37, 123, 3232963, …. The number of states is specified here as “<=2” rather than “2” so that we can have a valid value for one instruction. (A 1-instruction Turing machine has only one operating state.) BBi(8, <=2, -) will almost certainly be greater than the current lower bound of 3232964 = BB(2,4).

== Instruction-Limited Busy Beaver Variants ==
The concept of classifying a Turing machine's size by number of instructions, rather than by the number of states and symbols in its transition table, can also be utilized for Busy Beaver variants. The notation for the instruction-limited version of a variant can conveniently be denoted by appending an "i" to the already existing notation for the variant. Furthermore, the name of the Instruction-Limited version of the variant may be more conveniently started with "I-L" for brevity. Some known results for Instruction-Limited BB variants are as follows:

* I-L Greedy Busy Beaver. gBBi(n) is the largest number of steps taken by an n-instruction Turing machine, but where its first n-1 instructions (in the order encountered during running the machine) must also be a machine taking gBBi(n-1) steps. In other words, once the greedy n-instruction champion machines are identified, all machines having n+1 instructions must be extended from one of those n-instruction champions.
** gBBi(n) for n = 1 through n = 13 are: 1, 3, 5, 13, 19, 25, 41, 55, 238, 941, 1341, 10465, 10675. gBBi(14) must be at least 9,874,580 steps for current champion machine {{TM|0RB6RB1LB---3LA1RB7RB2LB_1LA2RB3LA4RB5RB1LB5LA---|halt}}.
* I-L Blanking Busy Beaver. BLBi(n) is the largest amount of steps taken by an n-instruction Turing machine when blanking the tape for the first time after having written a non-blank symbol to tape.
** BLBi(1) and BLBi(2) have no values because it is impossible to blank the tape with one or two instructions. BLBi(n) for n = 3 through n = 7 are: 4, 12, 30, 77, 808. BLBi(8) is at least 1,367,361,263,049 by the current champion {{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}.

== References ==
<references />
[[category:Functions]]

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-08-24T21:02:49Z

MrBrain: Changing n>=1 to n>=2 just to be safe for all small machines (which can get weird when you have only 1 state.)

An '''n-instruction Turing machine''' is a [[Turing machine]] having an arbitrary number of states and symbols, but having only ''n'' defined transitions/instructions in its transition table (all others are undefined).

The '''Instruction-Limited Busy Beaver function (BBi(n))''' is the maximum steps taken by any n-instruction Turing machines before eventually halting (when started on an initially blank tape). Similarly, the '''Instruction-Limited Symbols Busy Beaver function (Σi(n))''' is the maximum [[sigma score]] (number of non-blank symbols left on the tape at halting) for all halting n-instruction Turing machines (when started on an initially blank tape).

* The currently known values of BBi(n) for n>=0 are: 0, 1, 3, 5, 16, 37, 123, 3932963, .... BBi(8) is at least 6.889 x 101565, which is the number of steps taken by the current 8-instruction champion machine, discovered by Nick Drozd.
* The currently known values of Σi(n) for n>=0 are: 0, 1, 2, 4, 5, 9, 14, 2050, .... Σi(8) is at least 1.355 x 10783, which is the number of non-blank symbols written to tape by the same 8-instruction champion machine.

As with the traditional Busy Beaver sequences, BBi(n) and Σi(n) are both uncomputable, with each growing faster than any computable function.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') <ref>Tibor Radó (May 1962). "[https://computation4cognitivescientists.weebly.com/uploads/6/2/8/3/6283774/rado-on_non-computable_functions.pdf On non-computable functions]" (PDF). ''Bell System Technical Journal''. '''41''' (3): 877–884. https://doi.org/10.1002%2Fj.1538-7305.1962.tb00480.x</ref>. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n=1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m=1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? The current BBi(8) champion is a 3-state, 4-symbol machine, but this may still be surpassed by a 3-state, 3-symbol machine yet to be found.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!Step Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|0RH|halt}} {{TM|1RH---|halt}}
|BB(1,2)
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''3'''
|{{TM|0RB---_1LA---|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''5'''
|{{TM|1RB1LB_1LA---|halt}} (and 13 others)
|[[BB(2)|BB(2,2)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''16'''
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''37'''
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|[[BB(2,3)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''123'''
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''3,932,963'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|[[BB(2,4)]] - 1, and
A [[BB(3,3)]] high-ranking machine
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 6.889 \times 10^{1565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

{| class="wikitable"
!n
!'''Σ'''i(n)
!Sigma Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|1RH---|halt}}
|Σ(1,2)
|[[oeis:A384766|A384766]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''2'''
|{{TM|1RB---_1LA---|halt}} (and 7 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''4'''
|{{TM|1RB1LB_1LA---|halt}} (and 4 others)
|[[BB(2)|Σ(2,2)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''5'''
|{{TM|1RB0LB---_1LA2RA---|halt}} (and 40 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''9'''
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|Σ[[BB(2,3)|(2,3)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''14'''
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''2,050'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|Σ[[BB(2,4)|(2,4)]], and
A Σ[[BB(3,3)|(3,3)]] high-ranking machine
|[[oeis:A384766|A384766]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 1.355 \times 10^{783}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

Notes:
* Proven values for BBi(n) and Σi(n) are shown as bold in the above two tables.
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue. With the BBi(8) champion it appears that this trend has been broken (it is believed that <math>BB(3,3) < 10^{1565}</math>)
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5)-1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* The convention for writing TMs in standard text format here is to include as many states and symbols as are referenced. So for example, we write {{TM|1RH---|halt}} instead of {{TM|1RH|halt}} since this TM references 2 symbols (including the implicit blank symbol 0). If one state has no instructions leaving it, the convention is to treat it as the halt state <code>H</code>.

== Tree Normal Form (TNF) for BBi ==
Shown below are the total number of n-instruction Turing machines for each number of instructions n>=0. The numbers of halting and non-halting machines for n=8 are not yet known. The function Tree Normal Form for BBi is denoted BBiTNF(n)
{| class="wikitable"
!n
!All Machines
!All Cumulative
!Halting Machines
!Halting Cumulative
!Non-Halting Machines
!Non-Halting Cumulative
|-
| style="text-align: right;" |0
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |0
| style="text-align: right;" |0
|-
| style="text-align: right;" |1
| style="text-align: right;" |4
| style="text-align: right;" |5
| style="text-align: right;" |2
| style="text-align: right;" |3
| style="text-align: right;" |2
| style="text-align: right;" |2
|-
| style="text-align: right;" |2
| style="text-align: right;" |30
| style="text-align: right;" |35
| style="text-align: right;" |17
| style="text-align: right;" |20
| style="text-align: right;" |13
| style="text-align: right;" |15
|-
| style="text-align: right;" |3
| style="text-align: right;" |378
| style="text-align: right;" |413
| style="text-align: right;" |260
| style="text-align: right;" |280
| style="text-align: right;" |118
| style="text-align: right;" |133
|-
| style="text-align: right;" |4
| style="text-align: right;" |7,640
| style="text-align: right;" |8,053
| style="text-align: right;" |5,581
| style="text-align: right;" |5,861
| style="text-align: right;" |2,059
| style="text-align: right;" |2,192
|-
| style="text-align: right;" |5
| style="text-align: right;" |205,580
| style="text-align: right;" |213,633
| style="text-align: right;" |160,952
| style="text-align: right;" |166,813
| style="text-align: right;" |44,628
| style="text-align: right;" |46.820
|-
| style="text-align: right;" |6
| style="text-align: right;" |7,149,820
| style="text-align: right;" |7,363,453
| style="text-align: right;" |5,843,696
| style="text-align: right;" |6,010,509
| style="text-align: right;" |1,306,124
| style="text-align: right;" |1,352,944
|-
| style="text-align: right;" |7
| style="text-align: right;" |305,333,128
| style="text-align: right;" |312,696,581
| style="text-align: right;" |258,044,501
| style="text-align: right;" |264,055,010
| style="text-align: right;" |47,288,627
| style="text-align: right;" |48,641,571
|-
| style="text-align: right;" |8
| style="text-align: right;" |15,561,793,526
| style="text-align: right;" |15,874,490,107
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
|}
Enumeration of Turing machines in Tree Normal Form ([[TNF]]) for the Limited-Instruction Busy Beaver problem uses the following procedure:

* The "null Turing machine" with 0 instructions is the root of the tree. This "machine" halts after 0 steps, writing 0 non-blank symbols to the tape.
* The first instruction starts from A, the implied starting state, reads a 0 on the tape (which is blank at machine start) and must shift right (to avoid consideration of trivial mirror-image machines).
** This allows the following four possibilities: A0→0RA, A0→0RB, A0→1RA, and A0→1RB. The two machines transitioning to state A never halt, while the two machines transitioning to state B halt. The two halting machines are next expecting an instruction for B0, but since no such instruction exists, the machines halt after a single step.
* Following the enumeration of machines for each previous number of instructions, each of the halting machines is extended from the instruction at which it halted. For n=2, this instruction must be B0. (Non-halting machines cannot be extended since they can never reach a new instruction to be added.)
* As per normal TNF procedure, the new instruction can write any symbol to tape, shift left or right, and transition to any state as long as:
** The symbol written can be at most one greater than the highest symbol previously read or written, and
** The state to which the instruction transitions can be at most one state greater than the current maximum state.
* For the BBi problem, any instruction that transitions to a "new" state (one not yet containing any instructions) can be interpreted as a "halting instruction". (e.g. The one-instruction machine A0→1RB is exactly the same as A0→1RH.) Similarly, this new state can be considered a "halt state", as there is functionally no difference between a halt state and an operational state having no instructions. (B = H.)
** Unlike the traditional BB problem, where a halting instruction is normally used in the last remaining cell in an n x m transition table, it is necessary to consider all possible halting transitions for BBi, (not just those of the form →1RH), as this allows the most possible extensions of n-instruction machines to (n+1)-instruction machines.

== Domain-Restricted BBi ==
In addition to the standard Instruction-Limited Busy Beaver function BBi(n), where the numbers of states and symbols a Turing machine may have is unlimited, it may also be useful to consider the “Domain-Restricted BBi”, where the numbers of instructions, states, and/or symbols are specified values or ranges.

Following are some example three-argument functions within the Domain-Restricted BBi. For each function, the value is always the greatest number of steps taken by a Turing machine satisfying the corresponding restrictions:

* BBi(i, n, m): An n-state, m-symbol Turing machine having i instructions
* BBi(i, <=n, m): An i-instruction, m-symbol Turing machine having no greater than n states
* BBi(i, -, m): An i-instruction, m-symbol Turing machine having any number of states
* BBi(i, n, <=m): An i-instruction, n-state Turing machine having no greater than m instructions
* Etc.

Note that we have replaced the argument n from the standard BBi function with i in this three-argument version, primarily to avoid confusion with the n normally used to denote the number of states in a standard BB Turing machine.

For each of the three arguments, we may simply list a value, in which case the Turing machine must have exactly that number of instructions, states, or symbols. Another option is to specify a range, such as “<=n”, or “<=m”, in which case the Turing machine must have n or fewer states, or m or fewer symbols, respectively. And the final option is to give an argument as “-“, in which case, there is no restriction to the Turing machine’s number of instructions, states, or symbols.

'''Example Questions:'''

The main reason for having the flexibility in the way that the arguments are presented within the Domain-Restricted BBi function is that some problems relating to BB and BBi are best suited to examining Turing machines having specific numbers of states and symbols, while others are more practically examined within a range of state and symbol values instead. For example, consider the following two questions, along with two possible ways to interpret and answer each question:

'''Question 1:''' What are the maximum numbers of steps an i-instruction, 5-state, 2-symbol Turing machine can take before halting?

* Interpretation #1: For i=1 through 10, BBi(i, 5, 2) = -, -, -, -, 9, 31, 65, 793, 47176869, 47176870. In this interpretation, there are no values for 1<=i<=4, because there is no way to construct a 5-state Turing machine using fewer than 5 instructions. Also, the values for i=5, 6, and 7 are lower than those in the following second interpretation.
* Interpretation #2: For i=1 through 10, BBi(i, <=5, 2) = 1, 3, 5, 16, 20, 37, 106, 793, 47176869, 47176870. For this interpretation, we do not take the question’s wording of “5-state” so literally, but rather we allow the consideration of any Turing machine having no greater than 5 states. For example, BBi(7, <=5, 2) = 106 = BB(4,2) – 1.

The second interpretation, despite being less literally adherent to the question wording than the first, seems more useful for this particular question, specifically because we are probably interested in seeing how the number of steps grows for the best possible arrangement of i instructions within the 5 x 2 state table, and not just among the arrangements that specifically include defined transitions for all five states.

'''Question 2:''' For all n>1, will the 2n-1 instruction, n-state, 2-symbol BBi champion (one undefined transition) always exceed the 2n-1 instruction, n+1 state, 2-symbol BBi champion (3 undefined transitions)? (This is currently conjectured to be false.)

* Interpretation #1: Compare values of BBi(2n-1, n, 2) with those of BBi(2n-1, n+1, 2) for n=2 through 5:
** For n=2 through 5, BBi(2n-1, n, 2) = 5, 20, 106, 47176869.
** For n=2 through 5, BBi(2n-1, n+1, 2) = 4, 19, 65, ???.
* Interpretation #2: Compare values of BBi(2n-1, <=n, 2) with those of BBi(2n-1, <=n+1, 2) for n=2 through 5:
** For n=2 through 5, BBi(2n-1, <=n, 2) = 5, 20, 106, 47176869.
** For n=2 through 5, BBi(2n-1, <=n+1, 2) = 5, 20, 106, ???.

For this second question, the first interpretation clearly seems to be the more useful one, as we are probably interested in comparing the steps differences between machines having strictly n states (with one undefined transition) with machines having strictly n+1 states (with three undefined transitions). Using the second interpretation, information about these steps differences is lost since we have replaced the steps values of the machines with n+1 states with the values from the stronger n-state machines.

'''Selected Results:'''

Following are some results for Domain-Restricted BBi, some of which follow logically from their definitions, and others which have been computed so far:

* BBi(i, -, -) = BBi(i). If we allow any number of states and symbols, then this is the standard BBi function.
* BBi(-, n, m) = BBi(nm, n, m) = BB(n, m) for n>=2, m>=2. If we allow any number of instructions, or if we specify nm instructions, then this is equivalent to the standard BB(n, m) function.
* BBi(nm-1, n, m) = BBi(nm, n, m) - 1 = BB(n, m) - 1 for n>=2, m>=2. With one undefined transition, the number of steps is one less than having a completely filled n x m transition table including a halting instruction, which adds one final step.
* For i>=1, BBi(i, -, 2) = 1, 3, 5, 16, 20, 37, 106, 793, …. BBi(9, -, 2) is at least 47176869, but it may be even larger if a more powerful 6-state, 2-symbol machine having three undefined transitions can be found. BBi(10, -, 2) will almost certainly be greater than the current lower bound of 47176870 = BB(5,2).
* For i>=1, BBi(i, <=2, -) = 1, 3, 5, 13, 37, 123, 3232963, …. The number of states is specified here as “<=2” rather than “2” so that we can have a valid value for one instruction. (A 1-instruction Turing machine has only one operating state.) BBi(8, <=2, -) will almost certainly be greater than the current lower bound of 3232964 = BB(2,4).

== Instruction-Limited Busy Beaver Variants ==
The concept of classifying a Turing machine's size by number of instructions, rather than by the number of states and symbols in its transition table, can also be utilized for Busy Beaver variants. The notation for the instruction-limited version of a variant can conveniently be denoted by appending an "i" to the already existing notation for the variant. Furthermore, the name of the Instruction-Limited version of the variant may be more conveniently started with "I-L" for brevity. Some known results for Instruction-Limited BB variants are as follows:

* I-L Greedy Busy Beaver. gBBi(n) is the largest number of steps taken by an n-instruction Turing machine, but where its first n-1 instructions (in the order encountered during running the machine) must also be a machine taking gBBi(n-1) steps. In other words, once the greedy n-instruction champion machines are identified, all machines having n+1 instructions must be extended from one of those n-instruction champions.
** gBBi(n) for n=1 through n=13 are: 1, 3, 5, 13, 19, 25, 41, 55, 238, 941, 1341, 10465, 10675. gBBi(14) must be at least 9,874,580 steps for current champion machine {{TM|0RB6RB1LB---3LA1RB7RB2LB_1LA2RB3LA4RB5RB1LB5LA---|halt}}.
* I-L Blanking Busy Beaver. BLBi(n) is the largest amount of steps taken by an n-instruction Turing machine when blanking the tape for the first time after having written a non-blank symbol to tape.
** BLBi(1) and BLBi(2) have no values because it is impossible to blank the tape with one or two instructions. BLBi(n) for n=3 through n=7 are: 4, 12, 30, 77, 808. BLBi(8) is at least 1,367,361,263,049 by the current champion {{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}.

== References ==
<references />
[[category:Functions]]

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-08-24T20:39:39Z

MrBrain: Changed a faulty "=" to a correct "-".

An '''n-instruction Turing machine''' is a [[Turing machine]] having an arbitrary number of states and symbols, but having only ''n'' defined transitions/instructions in its transition table (all others are undefined).

The '''Instruction-Limited Busy Beaver function (BBi(n))''' is the maximum steps taken by any n-instruction Turing machines before eventually halting (when started on an initially blank tape). Similarly, the '''Instruction-Limited Symbols Busy Beaver function (Σi(n))''' is the maximum [[sigma score]] (number of non-blank symbols left on the tape at halting) for all halting n-instruction Turing machines (when started on an initially blank tape).

* The currently known values of BBi(n) for n>=0 are: 0, 1, 3, 5, 16, 37, 123, 3932963, .... BBi(8) is at least 6.889 x 101565, which is the number of steps taken by the current 8-instruction champion machine, discovered by Nick Drozd.
* The currently known values of Σi(n) for n>=0 are: 0, 1, 2, 4, 5, 9, 14, 2050, .... Σi(8) is at least 1.355 x 10783, which is the number of non-blank symbols written to tape by the same 8-instruction champion machine.

As with the traditional Busy Beaver sequences, BBi(n) and Σi(n) are both uncomputable, with each growing faster than any computable function.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') <ref>Tibor Radó (May 1962). "[https://computation4cognitivescientists.weebly.com/uploads/6/2/8/3/6283774/rado-on_non-computable_functions.pdf On non-computable functions]" (PDF). ''Bell System Technical Journal''. '''41''' (3): 877–884. https://doi.org/10.1002%2Fj.1538-7305.1962.tb00480.x</ref>. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n=1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m=1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? The current BBi(8) champion is a 3-state, 4-symbol machine, but this may still be surpassed by a 3-state, 3-symbol machine yet to be found.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!Step Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|0RH|halt}} {{TM|1RH---|halt}}
|BB(1,2)
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''3'''
|{{TM|0RB---_1LA---|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''5'''
|{{TM|1RB1LB_1LA---|halt}} (and 13 others)
|[[BB(2)|BB(2,2)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''16'''
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''37'''
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|[[BB(2,3)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''123'''
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''3,932,963'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|[[BB(2,4)]] - 1, and
A [[BB(3,3)]] high-ranking machine
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 6.889 \times 10^{1565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

{| class="wikitable"
!n
!'''Σ'''i(n)
!Sigma Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|1RH---|halt}}
|Σ(1,2)
|[[oeis:A384766|A384766]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''2'''
|{{TM|1RB---_1LA---|halt}} (and 7 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''4'''
|{{TM|1RB1LB_1LA---|halt}} (and 4 others)
|[[BB(2)|Σ(2,2)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''5'''
|{{TM|1RB0LB---_1LA2RA---|halt}} (and 40 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''9'''
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|Σ[[BB(2,3)|(2,3)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''14'''
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''2,050'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|Σ[[BB(2,4)|(2,4)]], and
A Σ[[BB(3,3)|(3,3)]] high-ranking machine
|[[oeis:A384766|A384766]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 1.355 \times 10^{783}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

Notes:
* Proven values for BBi(n) and Σi(n) are shown as bold in the above two tables.
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue. With the BBi(8) champion it appears that this trend has been broken (it is believed that <math>BB(3,3) < 10^{1565}</math>)
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5)-1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* The convention for writing TMs in standard text format here is to include as many states and symbols as are referenced. So for example, we write {{TM|1RH---|halt}} instead of {{TM|1RH|halt}} since this TM references 2 symbols (including the implicit blank symbol 0). If one state has no instructions leaving it, the convention is to treat it as the halt state <code>H</code>.

== Tree Normal Form (TNF) for BBi ==
Shown below are the total number of n-instruction Turing machines for each number of instructions n>=0. The numbers of halting and non-halting machines for n=8 are not yet known. The function Tree Normal Form for BBi is denoted BBiTNF(n)
{| class="wikitable"
!n
!All Machines
!All Cumulative
!Halting Machines
!Halting Cumulative
!Non-Halting Machines
!Non-Halting Cumulative
|-
| style="text-align: right;" |0
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |0
| style="text-align: right;" |0
|-
| style="text-align: right;" |1
| style="text-align: right;" |4
| style="text-align: right;" |5
| style="text-align: right;" |2
| style="text-align: right;" |3
| style="text-align: right;" |2
| style="text-align: right;" |2
|-
| style="text-align: right;" |2
| style="text-align: right;" |30
| style="text-align: right;" |35
| style="text-align: right;" |17
| style="text-align: right;" |20
| style="text-align: right;" |13
| style="text-align: right;" |15
|-
| style="text-align: right;" |3
| style="text-align: right;" |378
| style="text-align: right;" |413
| style="text-align: right;" |260
| style="text-align: right;" |280
| style="text-align: right;" |118
| style="text-align: right;" |133
|-
| style="text-align: right;" |4
| style="text-align: right;" |7,640
| style="text-align: right;" |8,053
| style="text-align: right;" |5,581
| style="text-align: right;" |5,861
| style="text-align: right;" |2,059
| style="text-align: right;" |2,192
|-
| style="text-align: right;" |5
| style="text-align: right;" |205,580
| style="text-align: right;" |213,633
| style="text-align: right;" |160,952
| style="text-align: right;" |166,813
| style="text-align: right;" |44,628
| style="text-align: right;" |46.820
|-
| style="text-align: right;" |6
| style="text-align: right;" |7,149,820
| style="text-align: right;" |7,363,453
| style="text-align: right;" |5,843,696
| style="text-align: right;" |6,010,509
| style="text-align: right;" |1,306,124
| style="text-align: right;" |1,352,944
|-
| style="text-align: right;" |7
| style="text-align: right;" |305,333,128
| style="text-align: right;" |312,696,581
| style="text-align: right;" |258,044,501
| style="text-align: right;" |264,055,010
| style="text-align: right;" |47,288,627
| style="text-align: right;" |48,641,571
|-
| style="text-align: right;" |8
| style="text-align: right;" |15,561,793,526
| style="text-align: right;" |15,874,490,107
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
|}
Enumeration of Turing machines in Tree Normal Form ([[TNF]]) for the Limited-Instruction Busy Beaver problem uses the following procedure:

* The "null Turing machine" with 0 instructions is the root of the tree. This "machine" halts after 0 steps, writing 0 non-blank symbols to the tape.
* The first instruction starts from A, the implied starting state, reads a 0 on the tape (which is blank at machine start) and must shift right (to avoid consideration of trivial mirror-image machines).
** This allows the following four possibilities: A0→0RA, A0→0RB, A0→1RA, and A0→1RB. The two machines transitioning to state A never halt, while the two machines transitioning to state B halt. The two halting machines are next expecting an instruction for B0, but since no such instruction exists, the machines halt after a single step.
* Following the enumeration of machines for each previous number of instructions, each of the halting machines is extended from the instruction at which it halted. For n=2, this instruction must be B0. (Non-halting machines cannot be extended since they can never reach a new instruction to be added.)
* As per normal TNF procedure, the new instruction can write any symbol to tape, shift left or right, and transition to any state as long as:
** The symbol written can be at most one greater than the highest symbol previously read or written, and
** The state to which the instruction transitions can be at most one state greater than the current maximum state.
* For the BBi problem, any instruction that transitions to a "new" state (one not yet containing any instructions) can be interpreted as a "halting instruction". (e.g. The one-instruction machine A0→1RB is exactly the same as A0→1RH.) Similarly, this new state can be considered a "halt state", as there is functionally no difference between a halt state and an operational state having no instructions. (B = H.)
** Unlike the traditional BB problem, where a halting instruction is normally used in the last remaining cell in an n x m transition table, it is necessary to consider all possible halting transitions for BBi, (not just those of the form →1RH), as this allows the most possible extensions of n-instruction machines to (n+1)-instruction machines.

== Domain-Restricted BBi ==
In addition to the standard Instruction-Limited Busy Beaver function BBi(n), where the numbers of states and symbols a Turing machine may have is unlimited, it may also be useful to consider the “Domain-Restricted BBi”, where the numbers of instructions, states, and/or symbols are specified values or ranges.

Following are some example three-argument functions within the Domain-Restricted BBi. For each function, the value is always the greatest number of steps taken by a Turing machine satisfying the corresponding restrictions:

* BBi(i, n, m): An n-state, m-symbol Turing machine having i instructions
* BBi(i, <=n, m): An i-instruction, m-symbol Turing machine having no greater than n states
* BBi(i, -, m): An i-instruction, m-symbol Turing machine having any number of states
* BBi(i, n, <=m): An i-instruction, n-state Turing machine having no greater than m instructions
* Etc.

Note that we have replaced the argument n from the standard BBi function with i in this three-argument version, primarily to avoid confusion with the n normally used to denote the number of states in a standard BB Turing machine.

For each of the three arguments, we may simply list a value, in which case the Turing machine must have exactly that number of instructions, states, or symbols. Another option is to specify a range, such as “<=n”, or “<=m”, in which case the Turing machine must have n or fewer states, or m or fewer symbols, respectively. And the final option is to give an argument as “-“, in which case, there is no restriction to the Turing machine’s number of instructions, states, or symbols.

'''Example Questions:'''

The main reason for having the flexibility in the way that the arguments are presented within the Domain-Restricted BBi function is that some problems relating to BB and BBi are best suited to examining Turing machines having specific numbers of states and symbols, while others are more practically examined within a range of state and symbol values instead. For example, consider the following two questions, along with two possible ways to interpret and answer each question:

'''Question 1:''' What are the maximum numbers of steps an i-instruction, 5-state, 2-symbol Turing machine can take before halting?

* Interpretation #1: For i=1 through 10, BBi(i, 5, 2) = -, -, -, -, 9, 31, 65, 793, 47176869, 47176870. In this interpretation, there are no values for 1<=i<=4, because there is no way to construct a 5-state Turing machine using fewer than 5 instructions. Also, the values for i=5, 6, and 7 are lower than those in the following second interpretation.
* Interpretation #2: For i=1 through 10, BBi(i, <=5, 2) = 1, 3, 5, 16, 20, 37, 106, 793, 47176869, 47176870. For this interpretation, we do not take the question’s wording of “5-state” so literally, but rather we allow the consideration of any Turing machine having no greater than 5 states. For example, BBi(7, <=5, 2) = 106 = BB(4,2) – 1.

The second interpretation, despite being less literally adherent to the question wording than the first, seems more useful for this particular question, specifically because we are probably interested in seeing how the number of steps grows for the best possible arrangement of i instructions within the 5 x 2 state table, and not just among the arrangements that specifically include defined transitions for all five states.

'''Question 2:''' For all n>1, will the 2n-1 instruction, n-state, 2-symbol BBi champion (one undefined transition) always exceed the 2n-1 instruction, n+1 state, 2-symbol BBi champion (3 undefined transitions)? (This is currently conjectured to be false.)

* Interpretation #1: Compare values of BBi(2n-1, n, 2) with those of BBi(2n-1, n+1, 2) for n=2 through 5:
** For n=2 through 5, BBi(2n-1, n, 2) = 5, 20, 106, 47176869.
** For n=2 through 5, BBi(2n-1, n+1, 2) = 4, 19, 65, ???.
* Interpretation #2: Compare values of BBi(2n-1, <=n, 2) with those of BBi(2n-1, <=n+1, 2) for n=2 through 5:
** For n=2 through 5, BBi(2n-1, <=n, 2) = 5, 20, 106, 47176869.
** For n=2 through 5, BBi(2n-1, <=n+1, 2) = 5, 20, 106, ???.

For this second question, the first interpretation clearly seems to be the more useful one, as we are probably interested in comparing the steps differences between machines having strictly n states (with one undefined transition) with machines having strictly n+1 states (with three undefined transitions). Using the second interpretation, information about these steps differences is lost since we have replaced the steps values of the machines with n+1 states with the values from the stronger n-state machines.

'''Selected Results:'''

Following are some results for Domain-Restricted BBi, some of which follow logically from their definitions, and others which have been computed so far:

* BBi(i, -, -) = BBi(i). If we allow any number of states and symbols, then this is the standard BBi function.
* BBi(-, n, m) = BBi(nm, n, m) = BB(n, m) for n>=1, m>=2. If we allow any number of instructions, or if we specify nm instructions, then this is equivalent to the standard BB(n, m) function.
* BBi(nm-1, n, m) = BBi(nm, n, m) - 1 = BB(n, m) - 1 for n>=2, m>=2. With one undefined transition, the number of steps is one less than having a completely filled n x m transition table including a halting instruction, which adds one final step.
* For i>=1, BBi(i, -, 2) = 1, 3, 5, 16, 20, 37, 106, 793, …. BBi(9, -, 2) is at least 47176869, but it may be even larger if a more powerful 6-state, 2-symbol machine having three undefined transitions can be found. BBi(10, -, 2) will almost certainly be greater than the current lower bound of 47176870 = BB(5,2).
* For i>=1, BBi(i, <=2, -) = 1, 3, 5, 13, 37, 123, 3232963, …. The number of states is specified here as “<=2” rather than “2” so that we can have a valid value for one instruction. (A 1-instruction Turing machine has only one operating state.) BBi(8, <=2, -) will almost certainly be greater than the current lower bound of 3232964 = BB(2,4).

== Instruction-Limited Busy Beaver Variants ==
The concept of classifying a Turing machine's size by number of instructions, rather than by the number of states and symbols in its transition table, can also be utilized for Busy Beaver variants. The notation for the instruction-limited version of a variant can conveniently be denoted by appending an "i" to the already existing notation for the variant. Furthermore, the name of the Instruction-Limited version of the variant may be more conveniently started with "I-L" for brevity. Some known results for Instruction-Limited BB variants are as follows:

* I-L Greedy Busy Beaver. gBBi(n) is the largest number of steps taken by an n-instruction Turing machine, but where its first n-1 instructions (in the order encountered during running the machine) must also be a machine taking gBBi(n-1) steps. In other words, once the greedy n-instruction champion machines are identified, all machines having n+1 instructions must be extended from one of those n-instruction champions.
** gBBi(n) for n=1 through n=13 are: 1, 3, 5, 13, 19, 25, 41, 55, 238, 941, 1341, 10465, 10675. gBBi(14) must be at least 9,874,580 steps for current champion machine {{TM|0RB6RB1LB---3LA1RB7RB2LB_1LA2RB3LA4RB5RB1LB5LA---|halt}}.
* I-L Blanking Busy Beaver. BLBi(n) is the largest amount of steps taken by an n-instruction Turing machine when blanking the tape for the first time after having written a non-blank symbol to tape.
** BLBi(1) and BLBi(2) have no values because it is impossible to blank the tape with one or two instructions. BLBi(n) for n=3 through n=7 are: 4, 12, 30, 77, 808. BLBi(8) is at least 1,367,361,263,049 by the current champion {{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}.

== References ==
<references />
[[category:Functions]]

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-08-24T20:31:04Z

MrBrain: For the third Domain-Restricted BBi result, n must be at least 2.

An '''n-instruction Turing machine''' is a [[Turing machine]] having an arbitrary number of states and symbols, but having only ''n'' defined transitions/instructions in its transition table (all others are undefined).

The '''Instruction-Limited Busy Beaver function (BBi(n))''' is the maximum steps taken by any n-instruction Turing machines before eventually halting (when started on an initially blank tape). Similarly, the '''Instruction-Limited Symbols Busy Beaver function (Σi(n))''' is the maximum [[sigma score]] (number of non-blank symbols left on the tape at halting) for all halting n-instruction Turing machines (when started on an initially blank tape).

* The currently known values of BBi(n) for n>=0 are: 0, 1, 3, 5, 16, 37, 123, 3932963, .... BBi(8) is at least 6.889 x 101565, which is the number of steps taken by the current 8-instruction champion machine, discovered by Nick Drozd.
* The currently known values of Σi(n) for n>=0 are: 0, 1, 2, 4, 5, 9, 14, 2050, .... Σi(8) is at least 1.355 x 10783, which is the number of non-blank symbols written to tape by the same 8-instruction champion machine.

As with the traditional Busy Beaver sequences, BBi(n) and Σi(n) are both uncomputable, with each growing faster than any computable function.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') <ref>Tibor Radó (May 1962). "[https://computation4cognitivescientists.weebly.com/uploads/6/2/8/3/6283774/rado-on_non-computable_functions.pdf On non-computable functions]" (PDF). ''Bell System Technical Journal''. '''41''' (3): 877–884. https://doi.org/10.1002%2Fj.1538-7305.1962.tb00480.x</ref>. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n=1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m=1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? The current BBi(8) champion is a 3-state, 4-symbol machine, but this may still be surpassed by a 3-state, 3-symbol machine yet to be found.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!Step Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|0RH|halt}} {{TM|1RH---|halt}}
|BB(1,2)
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''3'''
|{{TM|0RB---_1LA---|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''5'''
|{{TM|1RB1LB_1LA---|halt}} (and 13 others)
|[[BB(2)|BB(2,2)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''16'''
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''37'''
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|[[BB(2,3)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''123'''
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''3,932,963'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|[[BB(2,4)]] - 1, and
A [[BB(3,3)]] high-ranking machine
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 6.889 \times 10^{1565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

{| class="wikitable"
!n
!'''Σ'''i(n)
!Sigma Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|1RH---|halt}}
|Σ(1,2)
|[[oeis:A384766|A384766]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''2'''
|{{TM|1RB---_1LA---|halt}} (and 7 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''4'''
|{{TM|1RB1LB_1LA---|halt}} (and 4 others)
|[[BB(2)|Σ(2,2)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''5'''
|{{TM|1RB0LB---_1LA2RA---|halt}} (and 40 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''9'''
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|Σ[[BB(2,3)|(2,3)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''14'''
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''2,050'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|Σ[[BB(2,4)|(2,4)]], and
A Σ[[BB(3,3)|(3,3)]] high-ranking machine
|[[oeis:A384766|A384766]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 1.355 \times 10^{783}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

Notes:
* Proven values for BBi(n) and Σi(n) are shown as bold in the above two tables.
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue. With the BBi(8) champion it appears that this trend has been broken (it is believed that <math>BB(3,3) < 10^{1565}</math>)
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5)-1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* The convention for writing TMs in standard text format here is to include as many states and symbols as are referenced. So for example, we write {{TM|1RH---|halt}} instead of {{TM|1RH|halt}} since this TM references 2 symbols (including the implicit blank symbol 0). If one state has no instructions leaving it, the convention is to treat it as the halt state <code>H</code>.

== Tree Normal Form (TNF) for BBi ==
Shown below are the total number of n-instruction Turing machines for each number of instructions n>=0. The numbers of halting and non-halting machines for n=8 are not yet known. The function Tree Normal Form for BBi is denoted BBiTNF(n)
{| class="wikitable"
!n
!All Machines
!All Cumulative
!Halting Machines
!Halting Cumulative
!Non-Halting Machines
!Non-Halting Cumulative
|-
| style="text-align: right;" |0
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |0
| style="text-align: right;" |0
|-
| style="text-align: right;" |1
| style="text-align: right;" |4
| style="text-align: right;" |5
| style="text-align: right;" |2
| style="text-align: right;" |3
| style="text-align: right;" |2
| style="text-align: right;" |2
|-
| style="text-align: right;" |2
| style="text-align: right;" |30
| style="text-align: right;" |35
| style="text-align: right;" |17
| style="text-align: right;" |20
| style="text-align: right;" |13
| style="text-align: right;" |15
|-
| style="text-align: right;" |3
| style="text-align: right;" |378
| style="text-align: right;" |413
| style="text-align: right;" |260
| style="text-align: right;" |280
| style="text-align: right;" |118
| style="text-align: right;" |133
|-
| style="text-align: right;" |4
| style="text-align: right;" |7,640
| style="text-align: right;" |8,053
| style="text-align: right;" |5,581
| style="text-align: right;" |5,861
| style="text-align: right;" |2,059
| style="text-align: right;" |2,192
|-
| style="text-align: right;" |5
| style="text-align: right;" |205,580
| style="text-align: right;" |213,633
| style="text-align: right;" |160,952
| style="text-align: right;" |166,813
| style="text-align: right;" |44,628
| style="text-align: right;" |46.820
|-
| style="text-align: right;" |6
| style="text-align: right;" |7,149,820
| style="text-align: right;" |7,363,453
| style="text-align: right;" |5,843,696
| style="text-align: right;" |6,010,509
| style="text-align: right;" |1,306,124
| style="text-align: right;" |1,352,944
|-
| style="text-align: right;" |7
| style="text-align: right;" |305,333,128
| style="text-align: right;" |312,696,581
| style="text-align: right;" |258,044,501
| style="text-align: right;" |264,055,010
| style="text-align: right;" |47,288,627
| style="text-align: right;" |48,641,571
|-
| style="text-align: right;" |8
| style="text-align: right;" |15,561,793,526
| style="text-align: right;" |15,874,490,107
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
|}
Enumeration of Turing machines in Tree Normal Form ([[TNF]]) for the Limited-Instruction Busy Beaver problem uses the following procedure:

* The "null Turing machine" with 0 instructions is the root of the tree. This "machine" halts after 0 steps, writing 0 non-blank symbols to the tape.
* The first instruction starts from A, the implied starting state, reads a 0 on the tape (which is blank at machine start) and must shift right (to avoid consideration of trivial mirror-image machines).
** This allows the following four possibilities: A0→0RA, A0→0RB, A0→1RA, and A0→1RB. The two machines transitioning to state A never halt, while the two machines transitioning to state B halt. The two halting machines are next expecting an instruction for B0, but since no such instruction exists, the machines halt after a single step.
* Following the enumeration of machines for each previous number of instructions, each of the halting machines is extended from the instruction at which it halted. For n=2, this instruction must be B0. (Non-halting machines cannot be extended since they can never reach a new instruction to be added.)
* As per normal TNF procedure, the new instruction can write any symbol to tape, shift left or right, and transition to any state as long as:
** The symbol written can be at most one greater than the highest symbol previously read or written, and
** The state to which the instruction transitions can be at most one state greater than the current maximum state.
* For the BBi problem, any instruction that transitions to a "new" state (one not yet containing any instructions) can be interpreted as a "halting instruction". (e.g. The one-instruction machine A0→1RB is exactly the same as A0→1RH.) Similarly, this new state can be considered a "halt state", as there is functionally no difference between a halt state and an operational state having no instructions. (B = H.)
** Unlike the traditional BB problem, where a halting instruction is normally used in the last remaining cell in an n x m transition table, it is necessary to consider all possible halting transitions for BBi, (not just those of the form →1RH), as this allows the most possible extensions of n-instruction machines to (n+1)-instruction machines.

== Domain-Restricted BBi ==
In addition to the standard Instruction-Limited Busy Beaver function BBi(n), where the numbers of states and symbols a Turing machine may have is unlimited, it may also be useful to consider the “Domain-Restricted BBi”, where the numbers of instructions, states, and/or symbols are specified values or ranges.

Following are some example three-argument functions within the Domain-Restricted BBi. For each function, the value is always the greatest number of steps taken by a Turing machine satisfying the corresponding restrictions:

* BBi(i, n, m): An n-state, m-symbol Turing machine having i instructions
* BBi(i, <=n, m): An i-instruction, m-symbol Turing machine having no greater than n states
* BBi(i, -, m): An i-instruction, m-symbol Turing machine having any number of states
* BBi(i, n, <=m): An i-instruction, n-state Turing machine having no greater than m instructions
* Etc.

Note that we have replaced the argument n from the standard BBi function with i in this three-argument version, primarily to avoid confusion with the n normally used to denote the number of states in a standard BB Turing machine.

For each of the three arguments, we may simply list a value, in which case the Turing machine must have exactly that number of instructions, states, or symbols. Another option is to specify a range, such as “<=n”, or “<=m”, in which case the Turing machine must have n or fewer states, or m or fewer symbols, respectively. And the final option is to give an argument as “-“, in which case, there is no restriction to the Turing machine’s number of instructions, states, or symbols.

'''Example Questions:'''

The main reason for having the flexibility in the way that the arguments are presented within the Domain-Restricted BBi function is that some problems relating to BB and BBi are best suited to examining Turing machines having specific numbers of states and symbols, while others are more practically examined within a range of state and symbol values instead. For example, consider the following two questions, along with two possible ways to interpret and answer each question:

'''Question 1:''' What are the maximum numbers of steps an i-instruction, 5-state, 2-symbol Turing machine can take before halting?

* Interpretation #1: For i=1 through 10, BBi(i, 5, 2) = -, -, -, -, 9, 31, 65, 793, 47176869, 47176870. In this interpretation, there are no values for 1<=i<=4, because there is no way to construct a 5-state Turing machine using fewer than 5 instructions. Also, the values for i=5, 6, and 7 are lower than those in the following second interpretation.
* Interpretation #2: For i=1 through 10, BBi(i, <=5, 2) = 1, 3, 5, 16, 20, 37, 106, 793, 47176869, 47176870. For this interpretation, we do not take the question’s wording of “5-state” so literally, but rather we allow the consideration of any Turing machine having no greater than 5 states. For example, BBi(7, <=5, 2) = 106 = BB(4,2) – 1.

The second interpretation, despite being less literally adherent to the question wording than the first, seems more useful for this particular question, specifically because we are probably interested in seeing how the number of steps grows for the best possible arrangement of i instructions within the 5 x 2 state table, and not just among the arrangements that specifically include defined transitions for all five states.

'''Question 2:''' For all n>1, will the 2n-1 instruction, n-state, 2-symbol BBi champion (one undefined transition) always exceed the 2n-1 instruction, n+1 state, 2-symbol BBi champion (3 undefined transitions)? (This is currently conjectured to be false.)

* Interpretation #1: Compare values of BBi(2n-1, n, 2) with those of BBi(2n-1, n+1, 2) for n=2 through 5:
** For n=2 through 5, BBi(2n-1, n, 2) = 5, 20, 106, 47176869.
** For n=2 through 5, BBi(2n-1, n+1, 2) = 4, 19, 65, ???.
* Interpretation #2: Compare values of BBi(2n-1, <=n, 2) with those of BBi(2n-1, <=n+1, 2) for n=2 through 5:
** For n=2 through 5, BBi(2n-1, <=n, 2) = 5, 20, 106, 47176869.
** For n=2 through 5, BBi(2n-1, <=n+1, 2) = 5, 20, 106, ???.

For this second question, the first interpretation clearly seems to be the more useful one, as we are probably interested in comparing the steps differences between machines having strictly n states (with one undefined transition) with machines having strictly n+1 states (with three undefined transitions). Using the second interpretation, information about these steps differences is lost since we have replaced the steps values of the machines with n+1 states with the values from the stronger n-state machines.

'''Selected Results:'''

Following are some results for Domain-Restricted BBi, some of which follow logically from their definitions, and others which have been computed so far:

* BBi(i, -, -) = BBi(i). If we allow any number of states and symbols, then this is the standard BBi function.
* BBi(-, n, m) = BBi(nm, n, m) = BB(n, m) for n>=1, m>=2. If we allow any number of instructions, or if we specify nm instructions, then this is equivalent to the standard BB(n, m) function.
* BBi(nm-1, n, m) = BBi(nm, n, m) - 1 = BB(n, m) = 1 for n>=2, m>=2. With one undefined transition, the number of steps is one less than having a completely filled n x m transition table including a halting instruction, which adds one final step.
* For i>=1, BBi(i, -, 2) = 1, 3, 5, 16, 20, 37, 106, 793, …. BBi(9, -, 2) is at least 47176869, but it may be even larger if a more powerful 6-state, 2-symbol machine having three undefined transitions can be found. BBi(10, -, 2) will almost certainly be greater than the current lower bound of 47176870 = BB(5,2).
* For i>=1, BBi(i, <=2, -) = 1, 3, 5, 13, 37, 123, 3232963, …. The number of states is specified here as “<=2” rather than “2” so that we can have a valid value for one instruction. (A 1-instruction Turing machine has only one operating state.) BBi(8, <=2, -) will almost certainly be greater than the current lower bound of 3232964 = BB(2,4).

== Instruction-Limited Busy Beaver Variants ==
The concept of classifying a Turing machine's size by number of instructions, rather than by the number of states and symbols in its transition table, can also be utilized for Busy Beaver variants. The notation for the instruction-limited version of a variant can conveniently be denoted by appending an "i" to the already existing notation for the variant. Furthermore, the name of the Instruction-Limited version of the variant may be more conveniently started with "I-L" for brevity. Some known results for Instruction-Limited BB variants are as follows:

* I-L Greedy Busy Beaver. gBBi(n) is the largest number of steps taken by an n-instruction Turing machine, but where its first n-1 instructions (in the order encountered during running the machine) must also be a machine taking gBBi(n-1) steps. In other words, once the greedy n-instruction champion machines are identified, all machines having n+1 instructions must be extended from one of those n-instruction champions.
** gBBi(n) for n=1 through n=13 are: 1, 3, 5, 13, 19, 25, 41, 55, 238, 941, 1341, 10465, 10675. gBBi(14) must be at least 9,874,580 steps for current champion machine {{TM|0RB6RB1LB---3LA1RB7RB2LB_1LA2RB3LA4RB5RB1LB5LA---|halt}}.
* I-L Blanking Busy Beaver. BLBi(n) is the largest amount of steps taken by an n-instruction Turing machine when blanking the tape for the first time after having written a non-blank symbol to tape.
** BLBi(1) and BLBi(2) have no values because it is impossible to blank the tape with one or two instructions. BLBi(n) for n=3 through n=7 are: 4, 12, 30, 77, 808. BLBi(8) is at least 1,367,361,263,049 by the current champion {{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}.

== References ==
<references />
[[category:Functions]]

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-08-24T20:14:31Z

MrBrain: Added a new section for Domain-Restricted BBi. This section includes examples of the flexible notation, which allows for exact values for i, n, and m, or ranges. Illustrates the value of having both versions, which might be better suited to different questions. Also gives a few selected logical and computed results.

An '''n-instruction Turing machine''' is a [[Turing machine]] having an arbitrary number of states and symbols, but having only ''n'' defined transitions/instructions in its transition table (all others are undefined).

The '''Instruction-Limited Busy Beaver function (BBi(n))''' is the maximum steps taken by any n-instruction Turing machines before eventually halting (when started on an initially blank tape). Similarly, the '''Instruction-Limited Symbols Busy Beaver function (Σi(n))''' is the maximum [[sigma score]] (number of non-blank symbols left on the tape at halting) for all halting n-instruction Turing machines (when started on an initially blank tape).

* The currently known values of BBi(n) for n>=0 are: 0, 1, 3, 5, 16, 37, 123, 3932963, .... BBi(8) is at least 6.889 x 101565, which is the number of steps taken by the current 8-instruction champion machine, discovered by Nick Drozd.
* The currently known values of Σi(n) for n>=0 are: 0, 1, 2, 4, 5, 9, 14, 2050, .... Σi(8) is at least 1.355 x 10783, which is the number of non-blank symbols written to tape by the same 8-instruction champion machine.

As with the traditional Busy Beaver sequences, BBi(n) and Σi(n) are both uncomputable, with each growing faster than any computable function.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') <ref>Tibor Radó (May 1962). "[https://computation4cognitivescientists.weebly.com/uploads/6/2/8/3/6283774/rado-on_non-computable_functions.pdf On non-computable functions]" (PDF). ''Bell System Technical Journal''. '''41''' (3): 877–884. https://doi.org/10.1002%2Fj.1538-7305.1962.tb00480.x</ref>. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n=1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m=1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? The current BBi(8) champion is a 3-state, 4-symbol machine, but this may still be surpassed by a 3-state, 3-symbol machine yet to be found.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!Step Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|0RH|halt}} {{TM|1RH---|halt}}
|BB(1,2)
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''3'''
|{{TM|0RB---_1LA---|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''5'''
|{{TM|1RB1LB_1LA---|halt}} (and 13 others)
|[[BB(2)|BB(2,2)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''16'''
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''37'''
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|[[BB(2,3)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''123'''
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''3,932,963'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|[[BB(2,4)]] - 1, and
A [[BB(3,3)]] high-ranking machine
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 6.889 \times 10^{1565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

{| class="wikitable"
!n
!'''Σ'''i(n)
!Sigma Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|1RH---|halt}}
|Σ(1,2)
|[[oeis:A384766|A384766]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''2'''
|{{TM|1RB---_1LA---|halt}} (and 7 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''4'''
|{{TM|1RB1LB_1LA---|halt}} (and 4 others)
|[[BB(2)|Σ(2,2)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''5'''
|{{TM|1RB0LB---_1LA2RA---|halt}} (and 40 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''9'''
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|Σ[[BB(2,3)|(2,3)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''14'''
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''2,050'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|Σ[[BB(2,4)|(2,4)]], and
A Σ[[BB(3,3)|(3,3)]] high-ranking machine
|[[oeis:A384766|A384766]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 1.355 \times 10^{783}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

Notes:
* Proven values for BBi(n) and Σi(n) are shown as bold in the above two tables.
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue. With the BBi(8) champion it appears that this trend has been broken (it is believed that <math>BB(3,3) < 10^{1565}</math>)
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5)-1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* The convention for writing TMs in standard text format here is to include as many states and symbols as are referenced. So for example, we write {{TM|1RH---|halt}} instead of {{TM|1RH|halt}} since this TM references 2 symbols (including the implicit blank symbol 0). If one state has no instructions leaving it, the convention is to treat it as the halt state <code>H</code>.

== Tree Normal Form (TNF) for BBi ==
Shown below are the total number of n-instruction Turing machines for each number of instructions n>=0. The numbers of halting and non-halting machines for n=8 are not yet known. The function Tree Normal Form for BBi is denoted BBiTNF(n)
{| class="wikitable"
!n
!All Machines
!All Cumulative
!Halting Machines
!Halting Cumulative
!Non-Halting Machines
!Non-Halting Cumulative
|-
| style="text-align: right;" |0
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |0
| style="text-align: right;" |0
|-
| style="text-align: right;" |1
| style="text-align: right;" |4
| style="text-align: right;" |5
| style="text-align: right;" |2
| style="text-align: right;" |3
| style="text-align: right;" |2
| style="text-align: right;" |2
|-
| style="text-align: right;" |2
| style="text-align: right;" |30
| style="text-align: right;" |35
| style="text-align: right;" |17
| style="text-align: right;" |20
| style="text-align: right;" |13
| style="text-align: right;" |15
|-
| style="text-align: right;" |3
| style="text-align: right;" |378
| style="text-align: right;" |413
| style="text-align: right;" |260
| style="text-align: right;" |280
| style="text-align: right;" |118
| style="text-align: right;" |133
|-
| style="text-align: right;" |4
| style="text-align: right;" |7,640
| style="text-align: right;" |8,053
| style="text-align: right;" |5,581
| style="text-align: right;" |5,861
| style="text-align: right;" |2,059
| style="text-align: right;" |2,192
|-
| style="text-align: right;" |5
| style="text-align: right;" |205,580
| style="text-align: right;" |213,633
| style="text-align: right;" |160,952
| style="text-align: right;" |166,813
| style="text-align: right;" |44,628
| style="text-align: right;" |46.820
|-
| style="text-align: right;" |6
| style="text-align: right;" |7,149,820
| style="text-align: right;" |7,363,453
| style="text-align: right;" |5,843,696
| style="text-align: right;" |6,010,509
| style="text-align: right;" |1,306,124
| style="text-align: right;" |1,352,944
|-
| style="text-align: right;" |7
| style="text-align: right;" |305,333,128
| style="text-align: right;" |312,696,581
| style="text-align: right;" |258,044,501
| style="text-align: right;" |264,055,010
| style="text-align: right;" |47,288,627
| style="text-align: right;" |48,641,571
|-
| style="text-align: right;" |8
| style="text-align: right;" |15,561,793,526
| style="text-align: right;" |15,874,490,107
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
|}
Enumeration of Turing machines in Tree Normal Form ([[TNF]]) for the Limited-Instruction Busy Beaver problem uses the following procedure:

* The "null Turing machine" with 0 instructions is the root of the tree. This "machine" halts after 0 steps, writing 0 non-blank symbols to the tape.
* The first instruction starts from A, the implied starting state, reads a 0 on the tape (which is blank at machine start) and must shift right (to avoid consideration of trivial mirror-image machines).
** This allows the following four possibilities: A0→0RA, A0→0RB, A0→1RA, and A0→1RB. The two machines transitioning to state A never halt, while the two machines transitioning to state B halt. The two halting machines are next expecting an instruction for B0, but since no such instruction exists, the machines halt after a single step.
* Following the enumeration of machines for each previous number of instructions, each of the halting machines is extended from the instruction at which it halted. For n=2, this instruction must be B0. (Non-halting machines cannot be extended since they can never reach a new instruction to be added.)
* As per normal TNF procedure, the new instruction can write any symbol to tape, shift left or right, and transition to any state as long as:
** The symbol written can be at most one greater than the highest symbol previously read or written, and
** The state to which the instruction transitions can be at most one state greater than the current maximum state.
* For the BBi problem, any instruction that transitions to a "new" state (one not yet containing any instructions) can be interpreted as a "halting instruction". (e.g. The one-instruction machine A0→1RB is exactly the same as A0→1RH.) Similarly, this new state can be considered a "halt state", as there is functionally no difference between a halt state and an operational state having no instructions. (B = H.)
** Unlike the traditional BB problem, where a halting instruction is normally used in the last remaining cell in an n x m transition table, it is necessary to consider all possible halting transitions for BBi, (not just those of the form →1RH), as this allows the most possible extensions of n-instruction machines to (n+1)-instruction machines.

== Domain-Restricted BBi ==
In addition to the standard Instruction-Limited Busy Beaver function BBi(n), where the numbers of states and symbols a Turing machine may have is unlimited, it may also be useful to consider the “Domain-Restricted BBi”, where the numbers of instructions, states, and/or symbols are specified values or ranges.

Following are some example three-argument functions within the Domain-Restricted BBi. For each function, the value is always the greatest number of steps taken by a Turing machine satisfying the corresponding restrictions:

* BBi(i, n, m): An n-state, m-symbol Turing machine having i instructions
* BBi(i, <=n, m): An i-instruction, m-symbol Turing machine having no greater than n states
* BBi(i, -, m): An i-instruction, m-symbol Turing machine having any number of states
* BBi(i, n, <=m): An i-instruction, n-state Turing machine having no greater than m instructions
* Etc.

Note that we have replaced the argument n from the standard BBi function with i in this three-argument version, primarily to avoid confusion with the n normally used to denote the number of states in a standard BB Turing machine.

For each of the three arguments, we may simply list a value, in which case the Turing machine must have exactly that number of instructions, states, or symbols. Another option is to specify a range, such as “<=n”, or “<=m”, in which case the Turing machine must have n or fewer states, or m or fewer symbols, respectively. And the final option is to give an argument as “-“, in which case, there is no restriction to the Turing machine’s number of instructions, states, or symbols.

'''Example Questions:'''

The main reason for having the flexibility in the way that the arguments are presented within the Domain-Restricted BBi function is that some problems relating to BB and BBi are best suited to examining Turing machines having specific numbers of states and symbols, while others are more practically examined within a range of state and symbol values instead. For example, consider the following two questions, along with two possible ways to interpret and answer each question:

'''Question 1:''' What are the maximum numbers of steps an i-instruction, 5-state, 2-symbol Turing machine can take before halting?

* Interpretation #1: For i=1 through 10, BBi(i, 5, 2) = -, -, -, -, 9, 31, 65, 793, 47176869, 47176870. In this interpretation, there are no values for 1<=i<=4, because there is no way to construct a 5-state Turing machine using fewer than 5 instructions. Also, the values for i=5, 6, and 7 are lower than those in the following second interpretation.
* Interpretation #2: For i=1 through 10, BBi(i, <=5, 2) = 1, 3, 5, 16, 20, 37, 106, 793, 47176869, 47176870. For this interpretation, we do not take the question’s wording of “5-state” so literally, but rather we allow the consideration of any Turing machine having no greater than 5 states. For example, BBi(7, <=5, 2) = 106 = BB(4,2) – 1.

The second interpretation, despite being less literally adherent to the question wording than the first, seems more useful for this particular question, specifically because we are probably interested in seeing how the number of steps grows for the best possible arrangement of i instructions within the 5 x 2 state table, and not just among the arrangements that specifically include defined transitions for all five states.

'''Question 2:''' For all n>1, will the 2n-1 instruction, n-state, 2-symbol BBi champion (one undefined transition) always exceed the 2n-1 instruction, n+1 state, 2-symbol BBi champion (3 undefined transitions)? (This is currently conjectured to be false.)

* Interpretation #1: Compare values of BBi(2n-1, n, 2) with those of BBi(2n-1, n+1, 2) for n=2 through 5:
** For n=2 through 5, BBi(2n-1, n, 2) = 5, 20, 106, 47176869.
** For n=2 through 5, BBi(2n-1, n+1, 2) = 4, 19, 65, ???.
* Interpretation #2: Compare values of BBi(2n-1, <=n, 2) with those of BBi(2n-1, <=n+1, 2) for n=2 through 5:
** For n=2 through 5, BBi(2n-1, <=n, 2) = 5, 20, 106, 47176869.
** For n=2 through 5, BBi(2n-1, <=n+1, 2) = 5, 20, 106, ???.

For this second question, the first interpretation clearly seems to be the more useful one, as we are probably interested in comparing the steps differences between machines having strictly n states (with one undefined transition) with machines having strictly n+1 states (with three undefined transitions). Using the second interpretation, information about these steps differences is lost since we have replaced the steps values of the machines with n+1 states with the values from the stronger n-state machines.

'''Selected Results:'''

Following are some results for Domain-Restricted BBi, some of which follow logically from their definitions, and others which have been computed so far:

* BBi(i, -, -) = BBi(i). If we allow any number of states and symbols, then this is the standard BBi function.
* BBi(-, n, m) = BBi(nm, n, m) = BB(n, m) for n>=1, m>=2. If we allow any number of instructions, or if we specify nm instructions, then this is equivalent to the standard BB(n, m) function.
* BBi(nm-1, n, m) = BBi(nm, n, m) - 1 = BB(n, m) = 1 for n>=1, m>=2. With one undefined transition, the number of steps is one less than having a completely filled n x m transition table including a halting instruction, which adds one final step.
* For i>=1, BBi(i, -, 2) = 1, 3, 5, 16, 20, 37, 106, 793, …. BBi(9, -, 2) is at least 47176869, but it may be even larger if a more powerful 6-state, 2-symbol machine having three undefined transitions can be found. BBi(10, -, 2) will almost certainly be greater than the current lower bound of 47176870 = BB(5,2).
* For i>=1, BBi(i, <=2, -) = 1, 3, 5, 13, 37, 123, 3232963, …. The number of states is specified here as “<=2” rather than “2” so that we can have a valid value for one instruction. (A 1-instruction Turing machine has only one operating state.) BBi(8, <=2, -) will almost certainly be greater than the current lower bound of 3232964 = BB(2,4).

== Instruction-Limited Busy Beaver Variants ==
The concept of classifying a Turing machine's size by number of instructions, rather than by the number of states and symbols in its transition table, can also be utilized for Busy Beaver variants. The notation for the instruction-limited version of a variant can conveniently be denoted by appending an "i" to the already existing notation for the variant. Furthermore, the name of the Instruction-Limited version of the variant may be more conveniently started with "I-L" for brevity. Some known results for Instruction-Limited BB variants are as follows:

* I-L Greedy Busy Beaver. gBBi(n) is the largest number of steps taken by an n-instruction Turing machine, but where its first n-1 instructions (in the order encountered during running the machine) must also be a machine taking gBBi(n-1) steps. In other words, once the greedy n-instruction champion machines are identified, all machines having n+1 instructions must be extended from one of those n-instruction champions.
** gBBi(n) for n=1 through n=13 are: 1, 3, 5, 13, 19, 25, 41, 55, 238, 941, 1341, 10465, 10675. gBBi(14) must be at least 9,874,580 steps for current champion machine {{TM|0RB6RB1LB---3LA1RB7RB2LB_1LA2RB3LA4RB5RB1LB5LA---|halt}}.
* I-L Blanking Busy Beaver. BLBi(n) is the largest amount of steps taken by an n-instruction Turing machine when blanking the tape for the first time after having written a non-blank symbol to tape.
** BLBi(1) and BLBi(2) have no values because it is impossible to blank the tape with one or two instructions. BLBi(n) for n=3 through n=7 are: 4, 12, 30, 77, 808. BLBi(8) is at least 1,367,361,263,049 by the current champion {{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}.

== References ==
<references />
[[category:Functions]]

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-08-15T22:55:04Z

MrBrain: Updated description and values for BLBi, the I-L Blanking Busy Beaver.

An '''n-instruction Turing machine''' is a [[Turing machine]] having an arbitrary number of states and symbols, but having only ''n'' defined transitions/instructions in its transition table (all others are undefined).

The '''Instruction-Limited Busy Beaver function (BBi(n))''' is the maximum steps taken by any n-instruction Turing machines before eventually halting (when started on an initially blank tape). Similarly, the '''Instruction-Limited Symbols Busy Beaver function (Σi(n))''' is the maximum [[sigma score]] (number of non-blank symbols left on the tape at halting) for all halting n-instruction Turing machines (when started on an initially blank tape).

* The currently known values of BBi(n) for n>=0 are: 0, 1, 3, 5, 16, 37, 123, 3932963, .... BBi(8) is at least 6.889 x 101565, which is the number of steps taken by the current 8-instruction champion machine, discovered by Nick Drozd.
* The currently known values of Σi(n) for n>=0 are: 0, 1, 2, 4, 5, 9, 14, 2050, .... Σi(8) is at least 1.355 x 10783, which is the number of non-blank symbols written to tape by the same 8-instruction champion machine.

As with the traditional Busy Beaver sequences, BBi(n) and Σi(n) are both uncomputable, with each growing faster than any computable function.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') <ref>Tibor Radó (May 1962). "[https://computation4cognitivescientists.weebly.com/uploads/6/2/8/3/6283774/rado-on_non-computable_functions.pdf On non-computable functions]" (PDF). ''Bell System Technical Journal''. '''41''' (3): 877–884. https://doi.org/10.1002%2Fj.1538-7305.1962.tb00480.x</ref>. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n=1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m=1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? The current BBi(8) champion is a 3-state, 4-symbol machine, but this may still be surpassed by a 3-state, 3-symbol machine yet to be found.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!Step Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|0RH}} {{TM|1RH---}}
|BB(1,2)
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''3'''
|{{TM|0RB---_1LA---}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''5'''
|{{TM|1RB1LB_1LA---}} (and 13 others)
|[[BB(2)|BB(2,2)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''16'''
|{{TM|1RB---_0RC---_1LC0LA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''37'''
|{{TM|1RB2LB---_2LA2RB1LB}}
|[[BB(2,3)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''123'''
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''3,932,963'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|[[BB(2,4)]] - 1, and
A [[BB(3,3)]] high-ranking machine
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 6.889 \times 10^{1565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

{| class="wikitable"
!n
!'''Σ'''i(n)
!Sigma Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|1RH---}}
|Σ(1,2)
|[[oeis:A384766|A384766]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''2'''
|{{TM|1RB---_1LA---}} (and 7 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''4'''
|{{TM|1RB1LB_1LA---}} (and 4 others)
|[[BB(2)|Σ(2,2)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''5'''
|{{TM|1RB0LB---_1LA2RA---}} (and 40 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''9'''
|{{TM|1RB2LB---_2LA2RB1LB}}
|Σ[[BB(2,3)|(2,3)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''14'''
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''2,050'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|Σ[[BB(2,4)|(2,4)]], and
A Σ[[BB(3,3)|(3,3)]] high-ranking machine
|[[oeis:A384766|A384766]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 1.355 \times 10^{783}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

Notes:
* Proven values for BBi(n) and Σi(n) are shown as bold in the above two tables.
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue. With the BBi(8) champion it appears that this trend has been broken (it is believed that <math>BB(3,3) < 10^{1565}</math>)
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5)-1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* The convention for writing TMs in standard text format here is to include as many states and symbols as are referenced. So for example, we write {{TM|1RH---}} instead of {{TM|1RH}} since this TM references 2 symbols (including the implicit blank symbol 0). If one state has no instructions leaving it, the convention is to treat it as the halt state <code>H</code>.

== Tree Normal Form (TNF) for BBi ==
Shown below are the total number of n-instruction Turing machines for each number of instructions n>=0. The numbers of halting and non-halting machines for n=8 are not yet known. The function Tree Normal Form for BBi is denoted BBiTNF(n)
{| class="wikitable"
!n
!All Machines
!All Cumulative
!Halting Machines
!Halting Cumulative
!Non-Halting Machines
!Non-Halting Cumulative
|-
| style="text-align: right;" |0
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |0
| style="text-align: right;" |0
|-
| style="text-align: right;" |1
| style="text-align: right;" |4
| style="text-align: right;" |5
| style="text-align: right;" |2
| style="text-align: right;" |3
| style="text-align: right;" |2
| style="text-align: right;" |2
|-
| style="text-align: right;" |2
| style="text-align: right;" |30
| style="text-align: right;" |35
| style="text-align: right;" |17
| style="text-align: right;" |20
| style="text-align: right;" |13
| style="text-align: right;" |15
|-
| style="text-align: right;" |3
| style="text-align: right;" |378
| style="text-align: right;" |413
| style="text-align: right;" |260
| style="text-align: right;" |280
| style="text-align: right;" |118
| style="text-align: right;" |133
|-
| style="text-align: right;" |4
| style="text-align: right;" |7,640
| style="text-align: right;" |8,053
| style="text-align: right;" |5,581
| style="text-align: right;" |5,861
| style="text-align: right;" |2,059
| style="text-align: right;" |2,192
|-
| style="text-align: right;" |5
| style="text-align: right;" |205,580
| style="text-align: right;" |213,633
| style="text-align: right;" |160,952
| style="text-align: right;" |166,813
| style="text-align: right;" |44,628
| style="text-align: right;" |46.820
|-
| style="text-align: right;" |6
| style="text-align: right;" |7,149,820
| style="text-align: right;" |7,363,453
| style="text-align: right;" |5,843,696
| style="text-align: right;" |6,010,509
| style="text-align: right;" |1,306,124
| style="text-align: right;" |1,352,944
|-
| style="text-align: right;" |7
| style="text-align: right;" |305,333,128
| style="text-align: right;" |312,696,581
| style="text-align: right;" |258,044,501
| style="text-align: right;" |264,055,010
| style="text-align: right;" |47,288,627
| style="text-align: right;" |48,641,571
|-
| style="text-align: right;" |8
| style="text-align: right;" |15,561,793,526
| style="text-align: right;" |15,874,490,107
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
|}
Enumeration of Turing machines in Tree Normal Form ([[TNF]]) for the Limited-Instruction Busy Beaver problem uses the following procedure:

* The "null Turing machine" with 0 instructions is the root of the tree. This "machine" halts after 0 steps, writing 0 non-blank symbols to the tape.
* The first instruction starts from A, the implied starting state, reads a 0 on the tape (which is blank at machine start) and must shift right (to avoid consideration of trivial mirror-image machines).
** This allows the following four possibilities: A0→0RA, A0→0RB, A0→1RA, and A0→1RB. The two machines transitioning to state A never halt, while the two machines transitioning to state B halt. The two halting machines are next expecting an instruction for B0, but since no such instruction exists, the machines halt after a single step.
* Following the enumeration of machines for each previous number of instructions, each of the halting machines is extended from the instruction at which it halted. For n=2, this instruction must be B0. (Non-halting machines cannot be extended since they can never reach a new instruction to be added.)
* As per normal TNF procedure, the new instruction can write any symbol to tape, shift left or right, and transition to any state as long as:
** The symbol written can be at most one greater than the highest symbol previously read or written, and
** The state to which the instruction transitions can be at most one state greater than the current maximum state.
* For the BBi problem, any instruction that transitions to a "new" state (one not yet containing any instructions) can be interpreted as a "halting instruction". (e.g. The one-instruction machine A0→1RB is exactly the same as A0→1RH.) Similarly, this new state can be considered a "halt state", as there is functionally no difference between a halt state and an operational state having no instructions. (B = H.)
** Unlike the traditional BB problem, where a halting instruction is normally used in the last remaining cell in an n x m transition table, it is necessary to consider all possible halting transitions for BBi, (not just those of the form →1RH), as this allows the most possible extensions of n-instruction machines to (n+1)-instruction machines.

== Instruction-Limited Busy Beaver Variants ==
The concept of classifying a Turing machine's size by number of instructions, rather than by the number of states and symbols in its transition table, can also be utilized for Busy Beaver variants. The notation for the instruction-limited version of a variant can conveniently be denoted by appending an "i" to the already existing notation for the variant. Furthermore, the name of the Instruction-Limited version of the variant may be more conveniently started with "I-L" for brevity. Some known results for Instruction-Limited BB variants are as follows:

* I-L Greedy Busy Beaver. gBBi(n) is the largest number of steps taken by an n-instruction Turing machine, but where its first n-1 instructions (in the order encountered during running the machine) must also be a machine taking gBBi(n-1) steps. In other words, once the greedy n-instruction champion machines are identified, all machines having n+1 instructions must be extended from one of those n-instruction champions.
** gBBi(n) for n=1 through n=13 are: 1, 3, 5, 13, 19, 25, 41, 55, 238, 941, 1341, 10465, 10675. gBBi(14) must be at least 9,874,580 steps for current champion machine {{TM|0RB6RB1LB---3LA1RB7RB2LB_1LA2RB3LA4RB5RB1LB5LA---}}.
* I-L Blanking Busy Beaver. BLBi(n) is the largest amount of steps taken by an n-instruction Turing machine when blanking the tape for the first time after having written a non-blank symbol to tape.
** BLBi(1) and BLBi(2) have no values because it is impossible to blank the tape with one or two instructions. BLBi(n) for n=3 through n=7 are: 4, 12, 30, 77, 808. BLBi(8) is at least 1,367,361,263,049.

== References ==
<references />
[[category:Functions]]

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-08-15T17:59:29Z

MrBrain:

An '''n-instruction Turing machine''' is a [[Turing machine]] having an arbitrary number of states and symbols, but having only ''n'' defined transitions/instructions in its transition table (all others are undefined).

The '''Instruction-Limited Busy Beaver function (BBi(n))''' is the maximum steps taken by any n-instruction Turing machines before eventually halting (when started on an initially blank tape). Similarly, the '''Instruction-Limited Symbols Busy Beaver function (Σi(n))''' is the maximum [[sigma score]] (number of non-blank symbols left on the tape at halting) for all halting n-instruction Turing machines (when started on an initially blank tape).

* The currently known values of BBi(n) for n>=0 are: 0, 1, 3, 5, 16, 37, 123, 3932963, .... BBi(8) is at least 6.889 x 101565, which is the number of steps taken by the current 8-instruction champion machine, discovered by Nick Drozd.
* The currently known values of Σi(n) for n>=0 are: 0, 1, 2, 4, 5, 9, 14, 2050, .... Σi(8) is at least 1.355 x 10783, which is the number of non-blank symbols written to tape by the same 8-instruction champion machine.

As with the traditional Busy Beaver sequences, BBi(n) and Σi(n) are both uncomputable, with each growing faster than any computable function.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') <ref>Tibor Radó (May 1962). "[https://computation4cognitivescientists.weebly.com/uploads/6/2/8/3/6283774/rado-on_non-computable_functions.pdf On non-computable functions]" (PDF). ''Bell System Technical Journal''. '''41''' (3): 877–884. https://doi.org/10.1002%2Fj.1538-7305.1962.tb00480.x</ref>. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n=1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m=1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? The current BBi(8) champion is a 3-state, 4-symbol machine, but this may still be surpassed by a 3-state, 3-symbol machine yet to be found.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!Step Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|0RH}} {{TM|1RH---}}
|BB(1,2)
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''3'''
|{{TM|0RB---_1LA---}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''5'''
|{{TM|1RB1LB_1LA---}} (and 13 others)
|[[BB(2)|BB(2,2)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''16'''
|{{TM|1RB---_0RC---_1LC0LA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''37'''
|{{TM|1RB2LB---_2LA2RB1LB}}
|[[BB(2,3)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''123'''
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''3,932,963'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|[[BB(2,4)]] - 1, and
A [[BB(3,3)]] high-ranking machine
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 6.889 \times 10^{1565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

{| class="wikitable"
!n
!'''Σ'''i(n)
!Sigma Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|1RH---}}
|Σ(1,2)
|[[oeis:A384766|A384766]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''2'''
|{{TM|1RB---_1LA---}} (and 7 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''4'''
|{{TM|1RB1LB_1LA---}} (and 4 others)
|[[BB(2)|Σ(2,2)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''5'''
|{{TM|1RB0LB---_1LA2RA---}} (and 40 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''9'''
|{{TM|1RB2LB---_2LA2RB1LB}}
|Σ[[BB(2,3)|(2,3)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''14'''
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''2,050'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|Σ[[BB(2,4)|(2,4)]], and
A Σ[[BB(3,3)|(3,3)]] high-ranking machine
|[[oeis:A384766|A384766]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 1.355 \times 10^{783}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

Notes:
* Proven values for BBi(n) and Σi(n) are shown as bold in the above two tables.
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue. With the BBi(8) champion it appears that this trend has been broken (it is believed that <math>BB(3,3) < 10^{1565}</math>)
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5)-1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* The convention for writing TMs in standard text format here is to include as many states and symbols as are referenced. So for example, we write {{TM|1RH---}} instead of {{TM|1RH}} since this TM references 2 symbols (including the implicit blank symbol 0). If one state has no instructions leaving it, the convention is to treat it as the halt state <code>H</code>.

== Tree Normal Form (TNF) for BBi ==
Shown below are the total number of n-instruction Turing machines for each number of instructions n>=0. The numbers of halting and non-halting machines for n=8 are not yet known. The function Tree Normal Form for BBi is denoted BBiTNF(n)
{| class="wikitable"
!n
!All Machines
!All Cumulative
!Halting Machines
!Halting Cumulative
!Non-Halting Machines
!Non-Halting Cumulative
|-
| style="text-align: right;" |0
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |0
| style="text-align: right;" |0
|-
| style="text-align: right;" |1
| style="text-align: right;" |4
| style="text-align: right;" |5
| style="text-align: right;" |2
| style="text-align: right;" |3
| style="text-align: right;" |2
| style="text-align: right;" |2
|-
| style="text-align: right;" |2
| style="text-align: right;" |30
| style="text-align: right;" |35
| style="text-align: right;" |17
| style="text-align: right;" |20
| style="text-align: right;" |13
| style="text-align: right;" |15
|-
| style="text-align: right;" |3
| style="text-align: right;" |378
| style="text-align: right;" |413
| style="text-align: right;" |260
| style="text-align: right;" |280
| style="text-align: right;" |118
| style="text-align: right;" |133
|-
| style="text-align: right;" |4
| style="text-align: right;" |7,640
| style="text-align: right;" |8,053
| style="text-align: right;" |5,581
| style="text-align: right;" |5,861
| style="text-align: right;" |2,059
| style="text-align: right;" |2,192
|-
| style="text-align: right;" |5
| style="text-align: right;" |205,580
| style="text-align: right;" |213,633
| style="text-align: right;" |160,952
| style="text-align: right;" |166,813
| style="text-align: right;" |44,628
| style="text-align: right;" |46.820
|-
| style="text-align: right;" |6
| style="text-align: right;" |7,149,820
| style="text-align: right;" |7,363,453
| style="text-align: right;" |5,843,696
| style="text-align: right;" |6,010,509
| style="text-align: right;" |1,306,124
| style="text-align: right;" |1,352,944
|-
| style="text-align: right;" |7
| style="text-align: right;" |305,333,128
| style="text-align: right;" |312,696,581
| style="text-align: right;" |258,044,501
| style="text-align: right;" |264,055,010
| style="text-align: right;" |47,288,627
| style="text-align: right;" |48,641,571
|-
| style="text-align: right;" |8
| style="text-align: right;" |15,561,793,526
| style="text-align: right;" |15,874,490,107
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
|}
Enumeration of Turing machines in Tree Normal Form ([[TNF]]) for the Limited-Instruction Busy Beaver problem uses the following procedure:

* The "null Turing machine" with 0 instructions is the root of the tree. This "machine" halts after 0 steps, writing 0 non-blank symbols to the tape.
* The first instruction starts from A, the implied starting state, reads a 0 on the tape (which is blank at machine start) and must shift right (to avoid consideration of trivial mirror-image machines).
** This allows the following four possibilities: A0→0RA, A0→0RB, A0→1RA, and A0→1RB. The two machines transitioning to state A never halt, while the two machines transitioning to state B halt. The two halting machines are next expecting an instruction for B0, but since no such instruction exists, the machines halt after a single step.
* Following the enumeration of machines for each previous number of instructions, each of the halting machines is extended from the instruction at which it halted. For n=2, this instruction must be B0. (Non-halting machines cannot be extended since they can never reach a new instruction to be added.)
* As per normal TNF procedure, the new instruction can write any symbol to tape, shift left or right, and transition to any state as long as:
** The symbol written can be at most one greater than the highest symbol previously read or written, and
** The state to which the instruction transitions can be at most one state greater than the current maximum state.
* For the BBi problem, any instruction that transitions to a "new" state (one not yet containing any instructions) can be interpreted as a "halting instruction". (e.g. The one-instruction machine A0→1RB is exactly the same as A0→1RH.) Similarly, this new state can be considered a "halt state", as there is functionally no difference between a halt state and an operational state having no instructions. (B = H.)
** Unlike the traditional BB problem, where a halting instruction is normally used in the last remaining cell in an n x m transition table, it is necessary to consider all possible halting transitions for BBi, (not just those of the form →1RH), as this allows the most possible extensions of n-instruction machines to (n+1)-instruction machines.

== Instruction-Limited Busy Beaver Variants ==
The concept of classifying a Turing machine's size by number of instructions, rather than by the number of states and symbols in its transition table, can also be utilized for Busy Beaver variants. The notation for the instruction-limited version of a variant can conveniently be denoted by appending an "i" to the already existing notation for the variant. Furthermore, the name of the Instruction-Limited version of the variant may be more conveniently started with "I-L" for brevity. Some known results for Instruction-Limited BB variants are as follows:

* I-L Greedy Busy Beaver. gBBi(n) is the largest number of steps taken by an n-instruction Turing machine, but where its first n-1 instructions (in the order encountered during running the machine) must also be a machine taking gBBi(n-1) steps. In other words, once the greedy n-instruction champion machines are identified, all machines having n+1 instructions must be extended from one of those n-instruction champions.
** gBBi(n) for n=1 through n=13 are: 1, 3, 5, 13, 19, 25, 41, 55, 238, 941, 1341, 10465, 10675. gBBi(14) must be at least 9,874,580 steps for current champion machine {{TM|0RB6RB1LB---3LA1RB7RB2LB_1LA2RB3LA4RB5RB1LB5LA---}}.
* I-L Blanking Busy Beaver. BLBi(n) is the largest amount of steps taken by an n-instruction Turing machine before blanking the tape.
** BLBi(8) is at least 1,367,361,263,049.

== References ==
<references />
[[category:Functions]]

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-08-15T17:29:21Z

MrBrain: Proposed new "I-L" start for Instruction-Limited BB variants, for the purpose of brevity.

An '''n-instruction Turing machine''' is a [[Turing machine]] having an arbitrary number of states and symbols, but having only ''n'' defined transitions/instructions in its transition table (all others are undefined).

The '''Instruction-Limited Busy Beaver function (BBi(n))''' is the maximum steps taken by any n-instruction Turing machines before eventually halting (when started on an initially blank tape). Similarly, the '''Instruction-Limited Symbols Busy Beaver function (Σi(n))''' is the maximum [[sigma score]] (number of non-blank symbols left on the tape at halting) for all halting n-instruction Turing machines (when started on an initially blank tape).

* The currently known values of BBi(n) for n>=0 are: 0, 1, 3, 5, 16, 37, 123, 3932963, .... BBi(8) is at least 6.889 x 101565, which is the number of steps taken by the current 8-instruction champion machine, discovered by Nick Drozd.
* The currently known values of Σi(n) for n>=0 are: 0, 1, 2, 4, 5, 9, 14, 2050, .... Σi(8) is at least 1.355 x 10783, which is the number of non-blank symbols written to tape by the same 8-instruction champion machine.

As with the traditional Busy Beaver sequences, BBi(n) and Σi(n) are both uncomputable, with each growing faster than any computable function.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') <ref>Tibor Radó (May 1962). "[https://computation4cognitivescientists.weebly.com/uploads/6/2/8/3/6283774/rado-on_non-computable_functions.pdf On non-computable functions]" (PDF). ''Bell System Technical Journal''. '''41''' (3): 877–884. https://doi.org/10.1002%2Fj.1538-7305.1962.tb00480.x</ref>. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n=1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m=1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? The current BBi(8) champion is a 3-state, 4-symbol machine, but this may still be surpassed by a 3-state, 3-symbol machine yet to be found.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!Step Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|0RH}} {{TM|1RH---}}
|BB(1,2)
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''3'''
|{{TM|0RB---_1LA---}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''5'''
|{{TM|1RB1LB_1LA---}} (and 13 others)
|[[BB(2)|BB(2,2)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''16'''
|{{TM|1RB---_0RC---_1LC0LA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''37'''
|{{TM|1RB2LB---_2LA2RB1LB}}
|[[BB(2,3)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''123'''
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''3,932,963'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|[[BB(2,4)]] - 1, and
A [[BB(3,3)]] high-ranking machine
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 6.889 \times 10^{1565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

{| class="wikitable"
!n
!'''Σ'''i(n)
!Sigma Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|1RH---}}
|Σ(1,2)
|[[oeis:A384766|A384766]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''2'''
|{{TM|1RB---_1LA---}} (and 7 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''4'''
|{{TM|1RB1LB_1LA---}} (and 4 others)
|[[BB(2)|Σ(2,2)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''5'''
|{{TM|1RB0LB---_1LA2RA---}} (and 40 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''9'''
|{{TM|1RB2LB---_2LA2RB1LB}}
|Σ[[BB(2,3)|(2,3)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''14'''
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''2,050'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|Σ[[BB(2,4)|(2,4)]], and
A Σ[[BB(3,3)|(3,3)]] high-ranking machine
|[[oeis:A384766|A384766]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 1.355 \times 10^{783}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

Notes:
* Proven values for BBi(n) and Σi(n) are shown as bold in the above two tables.
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue. With the BBi(8) champion it appears that this trend has been broken (it is believed that <math>BB(3,3) < 10^{1565}</math>)
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5)-1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* The convention for writing TMs in standard text format here is to include as many states and symbols as are referenced. So for example, we write {{TM|1RH---}} instead of {{TM|1RH}} since this TM references 2 symbols (including the implicit blank symbol 0). If one state has no instructions leaving it, the convention is to treat it as the halt state <code>H</code>.

== Tree Normal Form (TNF) for BBi ==
Shown below are the total number of n-instruction Turing machines for each number of instructions n>=0. The numbers of halting and non-halting machines for n=8 are not yet known. The function Tree Normal Form for BBi is denoted BBiTNF(n)
{| class="wikitable"
!n
!All Machines
!All Cumulative
!Halting Machines
!Halting Cumulative
!Non-Halting Machines
!Non-Halting Cumulative
|-
| style="text-align: right;" |0
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |0
| style="text-align: right;" |0
|-
| style="text-align: right;" |1
| style="text-align: right;" |4
| style="text-align: right;" |5
| style="text-align: right;" |2
| style="text-align: right;" |3
| style="text-align: right;" |2
| style="text-align: right;" |2
|-
| style="text-align: right;" |2
| style="text-align: right;" |30
| style="text-align: right;" |35
| style="text-align: right;" |17
| style="text-align: right;" |20
| style="text-align: right;" |13
| style="text-align: right;" |15
|-
| style="text-align: right;" |3
| style="text-align: right;" |378
| style="text-align: right;" |413
| style="text-align: right;" |260
| style="text-align: right;" |280
| style="text-align: right;" |118
| style="text-align: right;" |133
|-
| style="text-align: right;" |4
| style="text-align: right;" |7,640
| style="text-align: right;" |8,053
| style="text-align: right;" |5,581
| style="text-align: right;" |5,861
| style="text-align: right;" |2,059
| style="text-align: right;" |2,192
|-
| style="text-align: right;" |5
| style="text-align: right;" |205,580
| style="text-align: right;" |213,633
| style="text-align: right;" |160,952
| style="text-align: right;" |166,813
| style="text-align: right;" |44,628
| style="text-align: right;" |46.820
|-
| style="text-align: right;" |6
| style="text-align: right;" |7,149,820
| style="text-align: right;" |7,363,453
| style="text-align: right;" |5,843,696
| style="text-align: right;" |6,010,509
| style="text-align: right;" |1,306,124
| style="text-align: right;" |1,352,944
|-
| style="text-align: right;" |7
| style="text-align: right;" |305,333,128
| style="text-align: right;" |312,696,581
| style="text-align: right;" |258,044,501
| style="text-align: right;" |264,055,010
| style="text-align: right;" |47,288,627
| style="text-align: right;" |48,641,571
|-
| style="text-align: right;" |8
| style="text-align: right;" |15,561,793,526
| style="text-align: right;" |15,874,490,107
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
|}
Enumeration of Turing machines in Tree Normal Form ([[TNF]]) for the Limited-Instruction Busy Beaver problem uses the following procedure:

* The "null Turing machine" with 0 instructions is the root of the tree. This "machine" halts after 0 steps, writing 0 non-blank symbols to the tape.
* The first instruction starts from A, the implied starting state, reads a 0 on the tape (which is blank at machine start) and must shift right (to avoid consideration of trivial mirror-image machines).
** This allows the following four possibilities: A0→0RA, A0→0RB, A0→1RA, and A0→1RB. The two machines transitioning to state A never halt, while the two machines transitioning to state B halt. The two halting machines are next expecting an instruction for B0, but since no such instruction exists, the machines halt after a single step.
* Following the enumeration of machines for each previous number of instructions, each of the halting machines is extended from the instruction at which it halted. For n=2, this instruction must be B0. (Non-halting machines cannot be extended since they can never reach a new instruction to be added.)
* As per normal TNF procedure, the new instruction can write any symbol to tape, shift left or right, and transition to any state as long as:
** The symbol written can be at most one greater than the highest symbol previously read or written, and
** The state to which the instruction transitions can be at most one state greater than the current maximum state.
* For the BBi problem, any instruction that transitions to a "new" state (one not yet containing any instructions) can be interpreted as a "halting instruction". (e.g. The one-instruction machine A0→1RB is exactly the same as A0→1RH.) Similarly, this new state can be considered a "halt state", as there is functionally no difference between a halt state and an operational state having no instructions. (B = H.)
** Unlike the traditional BB problem, where a halting instruction is normally used in the last remaining cell in an n x m transition table, it is necessary to consider all possible halting transitions for BBi, (not just those of the form →1RH), as this allows the most possible extensions of n-instruction machines to (n+1)-instruction machines.

== Instruction-Limited Busy Beaver Variants ==
The concept of classifying a Turing machine's size by number of instructions, rather than by the number of states and symbols in its transition table, can also be utilized for Busy Beaver variants. The notation for the instruction-limited version of a variant can conveniently be denoted by appending an "i" to the already existing notation for the variant. Furthermore, the name of the Instruction-Limited version of the variant may be more conveniently started with "I-L" for brevity. Some known results for Instruction-Limited BB variants are as follows:

* I-L Greedy Busy Beaver. gBBi(n) is the largest number of steps taken by an n-instruction Turing machine, but where its first n-1 instructions (in the order encountered during running the machine) must also be a machine taking gBBi(n-1) steps. In other words, once the greedy n-instruction champion machines are identified, all machines having n+1 instructions must be extended from one of those n-instruction champions.
** gBBi(n) for n=1 through n=13 are: 1, 3, 5, 13, 19, 25, 41, 55, 238, 941, 1341, 10465, 10675. gBBi(14) must be at least 9,874,580 steps for current champion machine {{TM|0RB6RB1LB---3LA1RB7RB2LB_1LA2RB3LA4RB5RB1LB5LA---}}.
* I-L Blanking Busy Beaver. BLBi(n) is the largest amount of steps done by an n-instruction Turing machine before blanking the tape.
** BLBi(8) is at least 1,367,361,263,049.

== References ==
<references />
[[category:Functions]]

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-08-15T17:14:25Z

MrBrain: Corrected the champion 14-instruction machine for gBBi. Previously listed 2 holdouts. New one is the actual current champion.

An '''n-instruction Turing machine''' is a [[Turing machine]] having an arbitrary number of states and symbols, but having only ''n'' defined transitions/instructions in its transition table (all others are undefined).

The '''Instruction-Limited Busy Beaver function (BBi(n))''' is the maximum steps taken by any n-instruction Turing machines before eventually halting (when started on an initially blank tape). Similarly, the '''Instruction-Limited Symbols Busy Beaver function (Σi(n))''' is the maximum [[sigma score]] (number of non-blank symbols left on the tape at halting) for all halting n-instruction Turing machines (when started on an initially blank tape).

* The currently known values of BBi(n) for n>=0 are: 0, 1, 3, 5, 16, 37, 123, 3932963, .... BBi(8) is at least 6.889 x 101565, which is the number of steps taken by the current 8-instruction champion machine, discovered by Nick Drozd.
* The currently known values of Σi(n) for n>=0 are: 0, 1, 2, 4, 5, 9, 14, 2050, .... Σi(8) is at least 1.355 x 10783, which is the number of non-blank symbols written to tape by the same 8-instruction champion machine.

As with the traditional Busy Beaver sequences, BBi(n) and Σi(n) are both uncomputable, with each growing faster than any computable function.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') <ref>Tibor Radó (May 1962). "[https://computation4cognitivescientists.weebly.com/uploads/6/2/8/3/6283774/rado-on_non-computable_functions.pdf On non-computable functions]" (PDF). ''Bell System Technical Journal''. '''41''' (3): 877–884. https://doi.org/10.1002%2Fj.1538-7305.1962.tb00480.x</ref>. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n=1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m=1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? The current BBi(8) champion is a 3-state, 4-symbol machine, but this may still be surpassed by a 3-state, 3-symbol machine yet to be found.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!Step Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|0RH}} {{TM|1RH---}}
|BB(1,2)
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''3'''
|{{TM|0RB---_1LA---}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''5'''
|{{TM|1RB1LB_1LA---}} (and 13 others)
|[[BB(2)|BB(2,2)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''16'''
|{{TM|1RB---_0RC---_1LC0LA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''37'''
|{{TM|1RB2LB---_2LA2RB1LB}}
|[[BB(2,3)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''123'''
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''3,932,963'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|[[BB(2,4)]] - 1, and
A [[BB(3,3)]] high-ranking machine
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 6.889 \times 10^{1565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

{| class="wikitable"
!n
!'''Σ'''i(n)
!Sigma Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|1RH---}}
|Σ(1,2)
|[[oeis:A384766|A384766]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''2'''
|{{TM|1RB---_1LA---}} (and 7 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''4'''
|{{TM|1RB1LB_1LA---}} (and 4 others)
|[[BB(2)|Σ(2,2)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''5'''
|{{TM|1RB0LB---_1LA2RA---}} (and 40 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''9'''
|{{TM|1RB2LB---_2LA2RB1LB}}
|Σ[[BB(2,3)|(2,3)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''14'''
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''2,050'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|Σ[[BB(2,4)|(2,4)]], and
A Σ[[BB(3,3)|(3,3)]] high-ranking machine
|[[oeis:A384766|A384766]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 1.355 \times 10^{783}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

Notes:
* Proven values for BBi(n) and Σi(n) are shown as bold in the above two tables.
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue. With the BBi(8) champion it appears that this trend has been broken (it is believed that <math>BB(3,3) < 10^{1565}</math>)
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5)-1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* The convention for writing TMs in standard text format here is to include as many states and symbols as are referenced. So for example, we write {{TM|1RH---}} instead of {{TM|1RH}} since this TM references 2 symbols (including the implicit blank symbol 0). If one state has no instructions leaving it, the convention is to treat it as the halt state <code>H</code>.

== Tree Normal Form (TNF) for BBi ==
Shown below are the total number of n-instruction Turing machines for each number of instructions n>=0. The numbers of halting and non-halting machines for n=8 are not yet known. The function Tree Normal Form for BBi is denoted BBiTNF(n)
{| class="wikitable"
!n
!All Machines
!All Cumulative
!Halting Machines
!Halting Cumulative
!Non-Halting Machines
!Non-Halting Cumulative
|-
| style="text-align: right;" |0
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |0
| style="text-align: right;" |0
|-
| style="text-align: right;" |1
| style="text-align: right;" |4
| style="text-align: right;" |5
| style="text-align: right;" |2
| style="text-align: right;" |3
| style="text-align: right;" |2
| style="text-align: right;" |2
|-
| style="text-align: right;" |2
| style="text-align: right;" |30
| style="text-align: right;" |35
| style="text-align: right;" |17
| style="text-align: right;" |20
| style="text-align: right;" |13
| style="text-align: right;" |15
|-
| style="text-align: right;" |3
| style="text-align: right;" |378
| style="text-align: right;" |413
| style="text-align: right;" |260
| style="text-align: right;" |280
| style="text-align: right;" |118
| style="text-align: right;" |133
|-
| style="text-align: right;" |4
| style="text-align: right;" |7,640
| style="text-align: right;" |8,053
| style="text-align: right;" |5,581
| style="text-align: right;" |5,861
| style="text-align: right;" |2,059
| style="text-align: right;" |2,192
|-
| style="text-align: right;" |5
| style="text-align: right;" |205,580
| style="text-align: right;" |213,633
| style="text-align: right;" |160,952
| style="text-align: right;" |166,813
| style="text-align: right;" |44,628
| style="text-align: right;" |46.820
|-
| style="text-align: right;" |6
| style="text-align: right;" |7,149,820
| style="text-align: right;" |7,363,453
| style="text-align: right;" |5,843,696
| style="text-align: right;" |6,010,509
| style="text-align: right;" |1,306,124
| style="text-align: right;" |1,352,944
|-
| style="text-align: right;" |7
| style="text-align: right;" |305,333,128
| style="text-align: right;" |312,696,581
| style="text-align: right;" |258,044,501
| style="text-align: right;" |264,055,010
| style="text-align: right;" |47,288,627
| style="text-align: right;" |48,641,571
|-
| style="text-align: right;" |8
| style="text-align: right;" |15,561,793,526
| style="text-align: right;" |15,874,490,107
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
|}
Enumeration of Turing machines in Tree Normal Form ([[TNF]]) for the Limited-Instruction Busy Beaver problem uses the following procedure:

* The "null Turing machine" with 0 instructions is the root of the tree. This "machine" halts after 0 steps, writing 0 non-blank symbols to the tape.
* The first instruction starts from A, the implied starting state, reads a 0 on the tape (which is blank at machine start) and must shift right (to avoid consideration of trivial mirror-image machines).
** This allows the following four possibilities: A0→0RA, A0→0RB, A0→1RA, and A0→1RB. The two machines transitioning to state A never halt, while the two machines transitioning to state B halt. The two halting machines are next expecting an instruction for B0, but since no such instruction exists, the machines halt after a single step.
* Following the enumeration of machines for each previous number of instructions, each of the halting machines is extended from the instruction at which it halted. For n=2, this instruction must be B0. (Non-halting machines cannot be extended since they can never reach a new instruction to be added.)
* As per normal TNF procedure, the new instruction can write any symbol to tape, shift left or right, and transition to any state as long as:
** The symbol written can be at most one greater than the highest symbol previously read or written, and
** The state to which the instruction transitions can be at most one state greater than the current maximum state.
* For the BBi problem, any instruction that transitions to a "new" state (one not yet containing any instructions) can be interpreted as a "halting instruction". (e.g. The one-instruction machine A0→1RB is exactly the same as A0→1RH.) Similarly, this new state can be considered a "halt state", as there is functionally no difference between a halt state and an operational state having no instructions. (B = H.)
** Unlike the traditional BB problem, where a halting instruction is normally used in the last remaining cell in an n x m transition table, it is necessary to consider all possible halting transitions for BBi, (not just those of the form →1RH), as this allows the most possible extensions of n-instruction machines to (n+1)-instruction machines.

== Instruction-Limited Busy Beaver Variants ==
The concept of classifying a Turing machine's size by number of instructions, rather than by the number of states and symbols in its transition table, can also be utilized for Busy Beaver variants. The notation for the instruction-limited version of a variant can conveniently be denoted by appending an "i" to the already existing notation for the variant. Some known results for Instruction-Limited BB variants are as follows:

* Instruction-Limited Greedy Busy Beaver. gBBi(n) is the largest number of steps taken by an n-instruction Turing machine, but where its first n-1 instructions (in the order encountered during running the machine) must also be a machine taking gBBi(n-1) steps. In other words, once the greedy n-instruction champion machines are identified, all machines having n+1 instructions must be extended from one of those n-instruction champions.
** gBBi(n) for n=1 through n=13 are: 1, 3, 5, 13, 19, 25, 41, 55, 238, 941, 1341, 10465, 10675. gBBi(14) must be at least 9,874,580 steps for current champion machine {{TM|0RB6RB1LB---3LA1RB7RB2LB_1LA2RB3LA4RB5RB1LB5LA---}}.
* Instruction-Limited Blanking Busy Beaver. BLBi(n) is the largest of steps done by an n-instruction Turing machine before blanking the tape.
** BLBi(8) is at least 1,367,361,263,049.

== References ==
<references />
[[category:Functions]]

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-08-15T17:11:28Z

MrBrain: Applied TM template to another machine.

An '''n-instruction Turing machine''' is a [[Turing machine]] having an arbitrary number of states and symbols, but having only ''n'' defined transitions/instructions in its transition table (all others are undefined).

The '''Instruction-Limited Busy Beaver function (BBi(n))''' is the maximum steps taken by any n-instruction Turing machines before eventually halting (when started on an initially blank tape). Similarly, the '''Instruction-Limited Symbols Busy Beaver function (Σi(n))''' is the maximum [[sigma score]] (number of non-blank symbols left on the tape at halting) for all halting n-instruction Turing machines (when started on an initially blank tape).

* The currently known values of BBi(n) for n>=0 are: 0, 1, 3, 5, 16, 37, 123, 3932963, .... BBi(8) is at least 6.889 x 101565, which is the number of steps taken by the current 8-instruction champion machine, discovered by Nick Drozd.
* The currently known values of Σi(n) for n>=0 are: 0, 1, 2, 4, 5, 9, 14, 2050, .... Σi(8) is at least 1.355 x 10783, which is the number of non-blank symbols written to tape by the same 8-instruction champion machine.

As with the traditional Busy Beaver sequences, BBi(n) and Σi(n) are both uncomputable, with each growing faster than any computable function.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') <ref>Tibor Radó (May 1962). "[https://computation4cognitivescientists.weebly.com/uploads/6/2/8/3/6283774/rado-on_non-computable_functions.pdf On non-computable functions]" (PDF). ''Bell System Technical Journal''. '''41''' (3): 877–884. https://doi.org/10.1002%2Fj.1538-7305.1962.tb00480.x</ref>. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n=1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m=1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? The current BBi(8) champion is a 3-state, 4-symbol machine, but this may still be surpassed by a 3-state, 3-symbol machine yet to be found.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!Step Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|0RH}} {{TM|1RH---}}
|BB(1,2)
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''3'''
|{{TM|0RB---_1LA---}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''5'''
|{{TM|1RB1LB_1LA---}} (and 13 others)
|[[BB(2)|BB(2,2)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''16'''
|{{TM|1RB---_0RC---_1LC0LA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''37'''
|{{TM|1RB2LB---_2LA2RB1LB}}
|[[BB(2,3)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''123'''
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''3,932,963'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|[[BB(2,4)]] - 1, and
A [[BB(3,3)]] high-ranking machine
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 6.889 \times 10^{1565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

{| class="wikitable"
!n
!'''Σ'''i(n)
!Sigma Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|1RH---}}
|Σ(1,2)
|[[oeis:A384766|A384766]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''2'''
|{{TM|1RB---_1LA---}} (and 7 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''4'''
|{{TM|1RB1LB_1LA---}} (and 4 others)
|[[BB(2)|Σ(2,2)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''5'''
|{{TM|1RB0LB---_1LA2RA---}} (and 40 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''9'''
|{{TM|1RB2LB---_2LA2RB1LB}}
|Σ[[BB(2,3)|(2,3)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''14'''
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''2,050'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|Σ[[BB(2,4)|(2,4)]], and
A Σ[[BB(3,3)|(3,3)]] high-ranking machine
|[[oeis:A384766|A384766]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 1.355 \times 10^{783}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

Notes:
* Proven values for BBi(n) and Σi(n) are shown as bold in the above two tables.
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue. With the BBi(8) champion it appears that this trend has been broken (it is believed that <math>BB(3,3) < 10^{1565}</math>)
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5)-1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* The convention for writing TMs in standard text format here is to include as many states and symbols as are referenced. So for example, we write {{TM|1RH---}} instead of {{TM|1RH}} since this TM references 2 symbols (including the implicit blank symbol 0). If one state has no instructions leaving it, the convention is to treat it as the halt state <code>H</code>.

== Tree Normal Form (TNF) for BBi ==
Shown below are the total number of n-instruction Turing machines for each number of instructions n>=0. The numbers of halting and non-halting machines for n=8 are not yet known. The function Tree Normal Form for BBi is denoted BBiTNF(n)
{| class="wikitable"
!n
!All Machines
!All Cumulative
!Halting Machines
!Halting Cumulative
!Non-Halting Machines
!Non-Halting Cumulative
|-
| style="text-align: right;" |0
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |0
| style="text-align: right;" |0
|-
| style="text-align: right;" |1
| style="text-align: right;" |4
| style="text-align: right;" |5
| style="text-align: right;" |2
| style="text-align: right;" |3
| style="text-align: right;" |2
| style="text-align: right;" |2
|-
| style="text-align: right;" |2
| style="text-align: right;" |30
| style="text-align: right;" |35
| style="text-align: right;" |17
| style="text-align: right;" |20
| style="text-align: right;" |13
| style="text-align: right;" |15
|-
| style="text-align: right;" |3
| style="text-align: right;" |378
| style="text-align: right;" |413
| style="text-align: right;" |260
| style="text-align: right;" |280
| style="text-align: right;" |118
| style="text-align: right;" |133
|-
| style="text-align: right;" |4
| style="text-align: right;" |7,640
| style="text-align: right;" |8,053
| style="text-align: right;" |5,581
| style="text-align: right;" |5,861
| style="text-align: right;" |2,059
| style="text-align: right;" |2,192
|-
| style="text-align: right;" |5
| style="text-align: right;" |205,580
| style="text-align: right;" |213,633
| style="text-align: right;" |160,952
| style="text-align: right;" |166,813
| style="text-align: right;" |44,628
| style="text-align: right;" |46.820
|-
| style="text-align: right;" |6
| style="text-align: right;" |7,149,820
| style="text-align: right;" |7,363,453
| style="text-align: right;" |5,843,696
| style="text-align: right;" |6,010,509
| style="text-align: right;" |1,306,124
| style="text-align: right;" |1,352,944
|-
| style="text-align: right;" |7
| style="text-align: right;" |305,333,128
| style="text-align: right;" |312,696,581
| style="text-align: right;" |258,044,501
| style="text-align: right;" |264,055,010
| style="text-align: right;" |47,288,627
| style="text-align: right;" |48,641,571
|-
| style="text-align: right;" |8
| style="text-align: right;" |15,561,793,526
| style="text-align: right;" |15,874,490,107
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
|}
Enumeration of Turing machines in Tree Normal Form ([[TNF]]) for the Limited-Instruction Busy Beaver problem uses the following procedure:

* The "null Turing machine" with 0 instructions is the root of the tree. This "machine" halts after 0 steps, writing 0 non-blank symbols to the tape.
* The first instruction starts from A, the implied starting state, reads a 0 on the tape (which is blank at machine start) and must shift right (to avoid consideration of trivial mirror-image machines).
** This allows the following four possibilities: A0→0RA, A0→0RB, A0→1RA, and A0→1RB. The two machines transitioning to state A never halt, while the two machines transitioning to state B halt. The two halting machines are next expecting an instruction for B0, but since no such instruction exists, the machines halt after a single step.
* Following the enumeration of machines for each previous number of instructions, each of the halting machines is extended from the instruction at which it halted. For n=2, this instruction must be B0. (Non-halting machines cannot be extended since they can never reach a new instruction to be added.)
* As per normal TNF procedure, the new instruction can write any symbol to tape, shift left or right, and transition to any state as long as:
** The symbol written can be at most one greater than the highest symbol previously read or written, and
** The state to which the instruction transitions can be at most one state greater than the current maximum state.
* For the BBi problem, any instruction that transitions to a "new" state (one not yet containing any instructions) can be interpreted as a "halting instruction". (e.g. The one-instruction machine A0→1RB is exactly the same as A0→1RH.) Similarly, this new state can be considered a "halt state", as there is functionally no difference between a halt state and an operational state having no instructions. (B = H.)
** Unlike the traditional BB problem, where a halting instruction is normally used in the last remaining cell in an n x m transition table, it is necessary to consider all possible halting transitions for BBi, (not just those of the form →1RH), as this allows the most possible extensions of n-instruction machines to (n+1)-instruction machines.

== Instruction-Limited Busy Beaver Variants ==
The concept of classifying a Turing machine's size by number of instructions, rather than by the number of states and symbols in its transition table, can also be utilized for Busy Beaver variants. The notation for the instruction-limited version of a variant can conveniently be denoted by appending an "i" to the already existing notation for the variant. Some known results for Instruction-Limited BB variants are as follows:

* Instruction-Limited Greedy Busy Beaver. gBBi(n) is the largest number of steps taken by an n-instruction Turing machine, but where its first n-1 instructions (in the order encountered during running the machine) must also be a machine taking gBBi(n-1) steps. In other words, once the greedy n-instruction champion machines are identified, all machines having n+1 instructions must be extended from one of those n-instruction champions.
** gBBi(n) for n=1 through n=13 are: 1, 3, 5, 13, 19, 25, 41, 55, 238, 941, 1341, 10465, 10675. gBBi(14) must be at least 9,874,580 steps for current champion machines {{TM|0RB6RB1LB---3LA1RB7RB3LA_1LA2RB3LA4RB5RB1LB5LA---}} and {{TM|0RB6RB1LB---3LA1RB7RB3LA_1LA2RB3LA4RB5RB1LB5LA---}}.
* Instruction-Limited Blanking Busy Beaver. BLBi(n) is the largest of steps done by an n-instruction Turing machine before blanking the tape.
** BLBi(8) is at least 1,367,361,263,049.

== References ==
<references />
[[category:Functions]]

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-08-15T17:09:12Z

MrBrain: Using TM template for a machine listed in variants section

An '''n-instruction Turing machine''' is a [[Turing machine]] having an arbitrary number of states and symbols, but having only ''n'' defined transitions/instructions in its transition table (all others are undefined).

The '''Instruction-Limited Busy Beaver function (BBi(n))''' is the maximum steps taken by any n-instruction Turing machines before eventually halting (when started on an initially blank tape). Similarly, the '''Instruction-Limited Symbols Busy Beaver function (Σi(n))''' is the maximum [[sigma score]] (number of non-blank symbols left on the tape at halting) for all halting n-instruction Turing machines (when started on an initially blank tape).

* The currently known values of BBi(n) for n>=0 are: 0, 1, 3, 5, 16, 37, 123, 3932963, .... BBi(8) is at least 6.889 x 101565, which is the number of steps taken by the current 8-instruction champion machine, discovered by Nick Drozd.
* The currently known values of Σi(n) for n>=0 are: 0, 1, 2, 4, 5, 9, 14, 2050, .... Σi(8) is at least 1.355 x 10783, which is the number of non-blank symbols written to tape by the same 8-instruction champion machine.

As with the traditional Busy Beaver sequences, BBi(n) and Σi(n) are both uncomputable, with each growing faster than any computable function.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') <ref>Tibor Radó (May 1962). "[https://computation4cognitivescientists.weebly.com/uploads/6/2/8/3/6283774/rado-on_non-computable_functions.pdf On non-computable functions]" (PDF). ''Bell System Technical Journal''. '''41''' (3): 877–884. https://doi.org/10.1002%2Fj.1538-7305.1962.tb00480.x</ref>. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n=1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m=1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? The current BBi(8) champion is a 3-state, 4-symbol machine, but this may still be surpassed by a 3-state, 3-symbol machine yet to be found.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!Step Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|0RH}} {{TM|1RH---}}
|BB(1,2)
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''3'''
|{{TM|0RB---_1LA---}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''5'''
|{{TM|1RB1LB_1LA---}} (and 13 others)
|[[BB(2)|BB(2,2)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''16'''
|{{TM|1RB---_0RC---_1LC0LA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''37'''
|{{TM|1RB2LB---_2LA2RB1LB}}
|[[BB(2,3)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''123'''
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''3,932,963'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|[[BB(2,4)]] - 1, and
A [[BB(3,3)]] high-ranking machine
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 6.889 \times 10^{1565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

{| class="wikitable"
!n
!'''Σ'''i(n)
!Sigma Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|1RH---}}
|Σ(1,2)
|[[oeis:A384766|A384766]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''2'''
|{{TM|1RB---_1LA---}} (and 7 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''4'''
|{{TM|1RB1LB_1LA---}} (and 4 others)
|[[BB(2)|Σ(2,2)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''5'''
|{{TM|1RB0LB---_1LA2RA---}} (and 40 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''9'''
|{{TM|1RB2LB---_2LA2RB1LB}}
|Σ[[BB(2,3)|(2,3)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''14'''
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''2,050'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|Σ[[BB(2,4)|(2,4)]], and
A Σ[[BB(3,3)|(3,3)]] high-ranking machine
|[[oeis:A384766|A384766]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 1.355 \times 10^{783}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

Notes:
* Proven values for BBi(n) and Σi(n) are shown as bold in the above two tables.
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue. With the BBi(8) champion it appears that this trend has been broken (it is believed that <math>BB(3,3) < 10^{1565}</math>)
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5)-1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* The convention for writing TMs in standard text format here is to include as many states and symbols as are referenced. So for example, we write {{TM|1RH---}} instead of {{TM|1RH}} since this TM references 2 symbols (including the implicit blank symbol 0). If one state has no instructions leaving it, the convention is to treat it as the halt state <code>H</code>.

== Tree Normal Form (TNF) for BBi ==
Shown below are the total number of n-instruction Turing machines for each number of instructions n>=0. The numbers of halting and non-halting machines for n=8 are not yet known. The function Tree Normal Form for BBi is denoted BBiTNF(n)
{| class="wikitable"
!n
!All Machines
!All Cumulative
!Halting Machines
!Halting Cumulative
!Non-Halting Machines
!Non-Halting Cumulative
|-
| style="text-align: right;" |0
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |0
| style="text-align: right;" |0
|-
| style="text-align: right;" |1
| style="text-align: right;" |4
| style="text-align: right;" |5
| style="text-align: right;" |2
| style="text-align: right;" |3
| style="text-align: right;" |2
| style="text-align: right;" |2
|-
| style="text-align: right;" |2
| style="text-align: right;" |30
| style="text-align: right;" |35
| style="text-align: right;" |17
| style="text-align: right;" |20
| style="text-align: right;" |13
| style="text-align: right;" |15
|-
| style="text-align: right;" |3
| style="text-align: right;" |378
| style="text-align: right;" |413
| style="text-align: right;" |260
| style="text-align: right;" |280
| style="text-align: right;" |118
| style="text-align: right;" |133
|-
| style="text-align: right;" |4
| style="text-align: right;" |7,640
| style="text-align: right;" |8,053
| style="text-align: right;" |5,581
| style="text-align: right;" |5,861
| style="text-align: right;" |2,059
| style="text-align: right;" |2,192
|-
| style="text-align: right;" |5
| style="text-align: right;" |205,580
| style="text-align: right;" |213,633
| style="text-align: right;" |160,952
| style="text-align: right;" |166,813
| style="text-align: right;" |44,628
| style="text-align: right;" |46.820
|-
| style="text-align: right;" |6
| style="text-align: right;" |7,149,820
| style="text-align: right;" |7,363,453
| style="text-align: right;" |5,843,696
| style="text-align: right;" |6,010,509
| style="text-align: right;" |1,306,124
| style="text-align: right;" |1,352,944
|-
| style="text-align: right;" |7
| style="text-align: right;" |305,333,128
| style="text-align: right;" |312,696,581
| style="text-align: right;" |258,044,501
| style="text-align: right;" |264,055,010
| style="text-align: right;" |47,288,627
| style="text-align: right;" |48,641,571
|-
| style="text-align: right;" |8
| style="text-align: right;" |15,561,793,526
| style="text-align: right;" |15,874,490,107
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
|}
Enumeration of Turing machines in Tree Normal Form ([[TNF]]) for the Limited-Instruction Busy Beaver problem uses the following procedure:

* The "null Turing machine" with 0 instructions is the root of the tree. This "machine" halts after 0 steps, writing 0 non-blank symbols to the tape.
* The first instruction starts from A, the implied starting state, reads a 0 on the tape (which is blank at machine start) and must shift right (to avoid consideration of trivial mirror-image machines).
** This allows the following four possibilities: A0→0RA, A0→0RB, A0→1RA, and A0→1RB. The two machines transitioning to state A never halt, while the two machines transitioning to state B halt. The two halting machines are next expecting an instruction for B0, but since no such instruction exists, the machines halt after a single step.
* Following the enumeration of machines for each previous number of instructions, each of the halting machines is extended from the instruction at which it halted. For n=2, this instruction must be B0. (Non-halting machines cannot be extended since they can never reach a new instruction to be added.)
* As per normal TNF procedure, the new instruction can write any symbol to tape, shift left or right, and transition to any state as long as:
** The symbol written can be at most one greater than the highest symbol previously read or written, and
** The state to which the instruction transitions can be at most one state greater than the current maximum state.
* For the BBi problem, any instruction that transitions to a "new" state (one not yet containing any instructions) can be interpreted as a "halting instruction". (e.g. The one-instruction machine A0→1RB is exactly the same as A0→1RH.) Similarly, this new state can be considered a "halt state", as there is functionally no difference between a halt state and an operational state having no instructions. (B = H.)
** Unlike the traditional BB problem, where a halting instruction is normally used in the last remaining cell in an n x m transition table, it is necessary to consider all possible halting transitions for BBi, (not just those of the form →1RH), as this allows the most possible extensions of n-instruction machines to (n+1)-instruction machines.

== Instruction-Limited Busy Beaver Variants ==
The concept of classifying a Turing machine's size by number of instructions, rather than by the number of states and symbols in its transition table, can also be utilized for Busy Beaver variants. The notation for the instruction-limited version of a variant can conveniently be denoted by appending an "i" to the already existing notation for the variant. Some known results for Instruction-Limited BB variants are as follows:

* Instruction-Limited Greedy Busy Beaver. gBBi(n) is the largest number of steps taken by an n-instruction Turing machine, but where its first n-1 instructions (in the order encountered during running the machine) must also be a machine taking gBBi(n-1) steps. In other words, once the greedy n-instruction champion machines are identified, all machines having n+1 instructions must be extended from one of those n-instruction champions.
** gBBi(n) for n=1 through n=13 are: 1, 3, 5, 13, 19, 25, 41, 55, 238, 941, 1341, 10465, 10675. gBBi(14) must be at least 9,874,580 steps for current champion machines {{TM|0RB6RB1LB---3LA1RB7RB3LA_1LA2RB3LA4RB5RB1LB5LA---}} and 0RB6RB1LB---3LA1RB7RB3LA_1LA2RB3LA4RB5RB1LB5LA---.
* Instruction-Limited Blanking Busy Beaver. BLBi(n) is the largest of steps done by an n-instruction Turing machine before blanking the tape.
** BLBi(8) is at least 1,367,361,263,049.

== References ==
<references />
[[category:Functions]]

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-08-15T17:07:39Z

MrBrain: Added some results for BBi variants.

An '''n-instruction Turing machine''' is a [[Turing machine]] having an arbitrary number of states and symbols, but having only ''n'' defined transitions/instructions in its transition table (all others are undefined).

The '''Instruction-Limited Busy Beaver function (BBi(n))''' is the maximum steps taken by any n-instruction Turing machines before eventually halting (when started on an initially blank tape). Similarly, the '''Instruction-Limited Symbols Busy Beaver function (Σi(n))''' is the maximum [[sigma score]] (number of non-blank symbols left on the tape at halting) for all halting n-instruction Turing machines (when started on an initially blank tape).

* The currently known values of BBi(n) for n>=0 are: 0, 1, 3, 5, 16, 37, 123, 3932963, .... BBi(8) is at least 6.889 x 101565, which is the number of steps taken by the current 8-instruction champion machine, discovered by Nick Drozd.
* The currently known values of Σi(n) for n>=0 are: 0, 1, 2, 4, 5, 9, 14, 2050, .... Σi(8) is at least 1.355 x 10783, which is the number of non-blank symbols written to tape by the same 8-instruction champion machine.

As with the traditional Busy Beaver sequences, BBi(n) and Σi(n) are both uncomputable, with each growing faster than any computable function.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') <ref>Tibor Radó (May 1962). "[https://computation4cognitivescientists.weebly.com/uploads/6/2/8/3/6283774/rado-on_non-computable_functions.pdf On non-computable functions]" (PDF). ''Bell System Technical Journal''. '''41''' (3): 877–884. https://doi.org/10.1002%2Fj.1538-7305.1962.tb00480.x</ref>. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n=1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m=1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? The current BBi(8) champion is a 3-state, 4-symbol machine, but this may still be surpassed by a 3-state, 3-symbol machine yet to be found.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!Step Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|0RH}} {{TM|1RH---}}
|BB(1,2)
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''3'''
|{{TM|0RB---_1LA---}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''5'''
|{{TM|1RB1LB_1LA---}} (and 13 others)
|[[BB(2)|BB(2,2)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''16'''
|{{TM|1RB---_0RC---_1LC0LA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''37'''
|{{TM|1RB2LB---_2LA2RB1LB}}
|[[BB(2,3)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''123'''
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''3,932,963'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|[[BB(2,4)]] - 1, and
A [[BB(3,3)]] high-ranking machine
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 6.889 \times 10^{1565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

{| class="wikitable"
!n
!'''Σ'''i(n)
!Sigma Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|1RH---}}
|Σ(1,2)
|[[oeis:A384766|A384766]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''2'''
|{{TM|1RB---_1LA---}} (and 7 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''4'''
|{{TM|1RB1LB_1LA---}} (and 4 others)
|[[BB(2)|Σ(2,2)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''5'''
|{{TM|1RB0LB---_1LA2RA---}} (and 40 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''9'''
|{{TM|1RB2LB---_2LA2RB1LB}}
|Σ[[BB(2,3)|(2,3)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''14'''
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''2,050'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|Σ[[BB(2,4)|(2,4)]], and
A Σ[[BB(3,3)|(3,3)]] high-ranking machine
|[[oeis:A384766|A384766]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 1.355 \times 10^{783}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

Notes:
* Proven values for BBi(n) and Σi(n) are shown as bold in the above two tables.
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue. With the BBi(8) champion it appears that this trend has been broken (it is believed that <math>BB(3,3) < 10^{1565}</math>)
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5)-1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* The convention for writing TMs in standard text format here is to include as many states and symbols as are referenced. So for example, we write {{TM|1RH---}} instead of {{TM|1RH}} since this TM references 2 symbols (including the implicit blank symbol 0). If one state has no instructions leaving it, the convention is to treat it as the halt state <code>H</code>.

== Tree Normal Form (TNF) for BBi ==
Shown below are the total number of n-instruction Turing machines for each number of instructions n>=0. The numbers of halting and non-halting machines for n=8 are not yet known. The function Tree Normal Form for BBi is denoted BBiTNF(n)
{| class="wikitable"
!n
!All Machines
!All Cumulative
!Halting Machines
!Halting Cumulative
!Non-Halting Machines
!Non-Halting Cumulative
|-
| style="text-align: right;" |0
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |0
| style="text-align: right;" |0
|-
| style="text-align: right;" |1
| style="text-align: right;" |4
| style="text-align: right;" |5
| style="text-align: right;" |2
| style="text-align: right;" |3
| style="text-align: right;" |2
| style="text-align: right;" |2
|-
| style="text-align: right;" |2
| style="text-align: right;" |30
| style="text-align: right;" |35
| style="text-align: right;" |17
| style="text-align: right;" |20
| style="text-align: right;" |13
| style="text-align: right;" |15
|-
| style="text-align: right;" |3
| style="text-align: right;" |378
| style="text-align: right;" |413
| style="text-align: right;" |260
| style="text-align: right;" |280
| style="text-align: right;" |118
| style="text-align: right;" |133
|-
| style="text-align: right;" |4
| style="text-align: right;" |7,640
| style="text-align: right;" |8,053
| style="text-align: right;" |5,581
| style="text-align: right;" |5,861
| style="text-align: right;" |2,059
| style="text-align: right;" |2,192
|-
| style="text-align: right;" |5
| style="text-align: right;" |205,580
| style="text-align: right;" |213,633
| style="text-align: right;" |160,952
| style="text-align: right;" |166,813
| style="text-align: right;" |44,628
| style="text-align: right;" |46.820
|-
| style="text-align: right;" |6
| style="text-align: right;" |7,149,820
| style="text-align: right;" |7,363,453
| style="text-align: right;" |5,843,696
| style="text-align: right;" |6,010,509
| style="text-align: right;" |1,306,124
| style="text-align: right;" |1,352,944
|-
| style="text-align: right;" |7
| style="text-align: right;" |305,333,128
| style="text-align: right;" |312,696,581
| style="text-align: right;" |258,044,501
| style="text-align: right;" |264,055,010
| style="text-align: right;" |47,288,627
| style="text-align: right;" |48,641,571
|-
| style="text-align: right;" |8
| style="text-align: right;" |15,561,793,526
| style="text-align: right;" |15,874,490,107
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
|}
Enumeration of Turing machines in Tree Normal Form ([[TNF]]) for the Limited-Instruction Busy Beaver problem uses the following procedure:

* The "null Turing machine" with 0 instructions is the root of the tree. This "machine" halts after 0 steps, writing 0 non-blank symbols to the tape.
* The first instruction starts from A, the implied starting state, reads a 0 on the tape (which is blank at machine start) and must shift right (to avoid consideration of trivial mirror-image machines).
** This allows the following four possibilities: A0→0RA, A0→0RB, A0→1RA, and A0→1RB. The two machines transitioning to state A never halt, while the two machines transitioning to state B halt. The two halting machines are next expecting an instruction for B0, but since no such instruction exists, the machines halt after a single step.
* Following the enumeration of machines for each previous number of instructions, each of the halting machines is extended from the instruction at which it halted. For n=2, this instruction must be B0. (Non-halting machines cannot be extended since they can never reach a new instruction to be added.)
* As per normal TNF procedure, the new instruction can write any symbol to tape, shift left or right, and transition to any state as long as:
** The symbol written can be at most one greater than the highest symbol previously read or written, and
** The state to which the instruction transitions can be at most one state greater than the current maximum state.
* For the BBi problem, any instruction that transitions to a "new" state (one not yet containing any instructions) can be interpreted as a "halting instruction". (e.g. The one-instruction machine A0→1RB is exactly the same as A0→1RH.) Similarly, this new state can be considered a "halt state", as there is functionally no difference between a halt state and an operational state having no instructions. (B = H.)
** Unlike the traditional BB problem, where a halting instruction is normally used in the last remaining cell in an n x m transition table, it is necessary to consider all possible halting transitions for BBi, (not just those of the form →1RH), as this allows the most possible extensions of n-instruction machines to (n+1)-instruction machines.

== Instruction-Limited Busy Beaver Variants ==
The concept of classifying a Turing machine's size by number of instructions, rather than by the number of states and symbols in its transition table, can also be utilized for Busy Beaver variants. The notation for the instruction-limited version of a variant can conveniently be denoted by appending an "i" to the already existing notation for the variant. Some known results for Instruction-Limited BB variants are as follows:

* Instruction-Limited Greedy Busy Beaver. gBBi(n) is the largest number of steps taken by an n-instruction Turing machine, but where its first n-1 instructions (in the order encountered during running the machine) must also be a machine taking gBBi(n-1) steps. In other words, once the greedy n-instruction champion machines are identified, all machines having n+1 instructions must be extended from one of those n-instruction champions.
** gBBi(n) for n=1 through n=13 are: 1, 3, 5, 13, 19, 25, 41, 55, 238, 941, 1341, 10465, 10675. gBBi(14) must be at least 9,874,580 steps for current champion machines 0RB6RB1LB---3LA1RB7RB3LA_1LA2RB3LA4RB5RB1LB5LA--- and 0RB6RB1LB---3LA1RB7RB3LA_1LA2RB3LA4RB5RB1LB5LA---.
* Instruction-Limited Blanking Busy Beaver. BLBi(n) is the largest of steps done by an n-instruction Turing machine before blanking the tape.
** BLBi(8) is at least 1,367,361,263,049.

== References ==
<references />
[[category:Functions]]

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-08-15T16:34:45Z

MrBrain: Updated reference in text so it automatically (I hope) shows up in the references section.

Instruction-Limited Busy Beaver

2025-08-15T16:30:04Z

MrBrain: Started two new sections. One on Instruction-Limited BB variants, and the other for references.

Instruction-Limited Busy Beaver

2025-08-07T05:14:09Z

MrBrain: Slight wording change in introduction section.

Instruction-Limited Busy Beaver

2025-08-07T05:11:36Z

MrBrain: Slight wording change in TNF section, describing the maximum state to which a new instruction can transition to.

Instruction-Limited Busy Beaver

2025-08-06T15:41:56Z

MrBrain: Grammar fix in TNF section.

Instruction-Limited Busy Beaver

2025-08-06T14:36:35Z

MrBrain: Fixed description of the four possible 1-instruction machines.

Instruction-Limited Busy Beaver

2025-08-06T14:25:32Z

MrBrain: Added a description of the TNF table prior to the table. Added an outline of the procedure for TNF generation below the table. Explained how a state having no instructions is functionally equivalent to a halt state, and that a transition to such a state is effectively a halting instruction.

Instruction-Limited Busy Beaver

2025-08-06T03:24:31Z

MrBrain: Right justified values in the TNF enumeration table.

Instruction-Limited Busy Beaver

2025-08-06T03:17:16Z

MrBrain: Added a new section/table for TNF counts for BBi.

Instruction-Limited Busy Beaver

2025-08-05T02:48:47Z

MrBrain: Added new table for Symbols Champions. Some of the symbols champions for low n are different than steps champions for the same n. Added a note explaining that proven values for both BBi(n) and Σi(n) are given as bold in the two tables.

Instruction-Limited Busy Beaver

2025-08-04T21:42:12Z

MrBrain: Bolded proven values in Steps Champions table.

Instruction-Limited Busy Beaver

2025-08-04T21:11:01Z

MrBrain: Added a hyphen to "long-running".

Instruction-Limited Busy Beaver

2025-08-04T21:09:44Z

MrBrain: Updated Shawn's note to reflect the new notation for BBi(1).

Instruction-Limited Busy Beaver

2025-08-04T21:03:36Z

MrBrain: There is no page for BB(1), so removed link for that note.

Instruction-Limited Busy Beaver

2025-08-04T21:02:54Z

MrBrain: Updated link for BB(1,2)

Instruction-Limited Busy Beaver

2025-08-04T21:02:04Z

MrBrain: Updated BB(1) to BB(1,2) for clarity.

Instruction-Limited Busy Beaver

2025-08-04T21:01:02Z

MrBrain: Updated Steps Champions table. Changed standard notation for the BBi(1) machines to accurately show that the only instruction is a halting instruction, transitioning to a state having no instructions. These are both 1-state machines, not 2-state machines, which 0RB and 1RB--- might suggest. Also, updated notes to more clearly show how the steps values for 1, 3, 5, and 7 symbols relate to the traditional BB values for 1, 2, 3, and 4 states.

Instruction-Limited Busy Beaver

2025-08-04T20:55:20Z

MrBrain:

Instruction-Limited Busy Beaver

2025-08-04T20:48:25Z

MrBrain: Fixed notation for symbols sequence.

Instruction-Limited Busy Beaver

2025-08-04T20:46:25Z

MrBrain: Updated the introduction section to more clearly define various terms such as Busy Beaver, Competition, Sequence, in the context of the Instruction-Limited Busy Beaver concept. Also, included both the steps and symbols BBi sequences through the known values of BBi(7) and Σi(7), as well as the currently highest known values for those sequences for BBi(8) and Σi(8).

Instruction-Limited Busy Beaver

2025-07-27T03:42:48Z

MrBrain: Added a note for BBi(8) to clarify that the machine given is merely the current champion.

An '''n-instruction Turing machine''' is a [[Turing machine]] with an arbitrary number of states and symbols, but only ''n'' defined transitions/instructions in its transition table (all others are undefined). The '''Instruction-Limited Busy Beaver''' (BBi(n)) problem is the Busy Beaver problem limited to n-instruction TMs. BBi(n) is the longest runtime for all halting n-instruction TMs when started on a blank tape and Σi(n) is the maximum number of non-blank symbols written by an n-instruction TM when halting and when started on a blank tape.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') [1]. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n=1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m=1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? The current BBi(8) champion is a 3-state, 4-symbol machine, but this may still be surpassed by a 3-state, 3-symbol machine yet to be found.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!TNF Size
!Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | 1
| style="text-align: right;" | 5
|{{TM|0RB_---}} {{TM|1RB---_------}}
|
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | 3
| style="text-align: right;" | 35
|{{TM|0RB---_1LA---}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | 5
| style="text-align: right;" | 413
|{{TM|1RB1LB_1LA---}} (and 13 others)
|[[BB(2)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | 16
| style="text-align: right;" | 8,053
|{{TM|1RB---_0RC---_1LC0LA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | 37
| style="text-align: right;" | 213,633
|{{TM|1RB2LB---_2LA2RB1LB}}
|[[BB(2,3)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | 123
| style="text-align: right;" | 7,363,453
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | 3,932,963
| style="text-align: right;" | 312,696,581
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|[[BB(2,4)]] champion and
its [[BB(3,3)]] doppelganger
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of longrunning BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 6.889 \times 10^{1565}</math>
| style="text-align: right;" | 15,874,490,107
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

Notes:
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue.
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5)-1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* TNF Size is the total number of TMs enumerated in [[TNF]] with n (or fewer) instructions defined.
* BBi(n) ≥ n. A halting (n+1) x 1 TM that chains all its states together (A0 = 0RB, B0 = 0RC, C0 = 0RD, ...) through the penultimate TM state will take n steps to pass through all states exactly once (a repeat would guarantee nonhalting). The penultimate state, the nth state, puts the TM into n+1st state, which is undefined and does not count as a step.

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-07-27T03:10:59Z

MrBrain: Added the current situation for the BBi(8) contest to the end of the relevant paragraph in the Motivation section.

An '''n-instruction Turing machine''' is a [[Turing machine]] with an arbitrary number of states and symbols, but only ''n'' defined transitions/instructions in its transition table (all others are undefined). The '''Instruction-Limited Busy Beaver''' (BBi(n)) problem is the Busy Beaver problem limited to n-instruction TMs. BBi(n) is the longest runtime for all halting n-instruction TMs when started on a blank tape and Σi(n) is the maximum number of non-blank symbols written by an n-instruction TM when halting and when started on a blank tape.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') [1]. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n=1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m=1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? The current BBi(8) champion is a 3-state, 4-symbol machine, but this may still be surpassed by a 3-state, 3-symbol machine yet to be found.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!TNF Size
!Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | 1
| style="text-align: right;" | 5
|{{TM|0RB_---}} {{TM|1RB---_------}}
|
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | 3
| style="text-align: right;" | 35
|{{TM|0RB---_1LA---}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | 5
| style="text-align: right;" | 413
|{{TM|1RB1LB_1LA---}} (and 13 others)
|[[BB(2)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | 16
| style="text-align: right;" | 8,053
|{{TM|1RB---_0RC---_1LC0LA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | 37
| style="text-align: right;" | 213,633
|{{TM|1RB2LB---_2LA2RB1LB}}
|[[BB(2,3)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | 123
| style="text-align: right;" | 7,363,453
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | 3,932,963
| style="text-align: right;" | 312,696,581
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|[[BB(2,4)]] champion and
its [[BB(3,3)]] doppelganger
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of longrunning BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 6.889*10^{1565}</math>
| style="text-align: right;" | 15,874,490,107
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC}}
|
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

Notes:
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue.
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5)-1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* TNF Size is the total number of TMs enumerated in [[TNF]] with n (or fewer) instructions defined.
* BBi(n) ≥ n. A halting (n+1) x 1 TM that chains all its states together (A0 = 0RB, B0 = 0RC, C0 = 0RD, ...) through the penultimate TM state will take n steps to pass through all states exactly once (a repeat would guarantee nonhalting). The penultimate state, the nth state, puts the TM into n+1st state, which is undefined and does not count as a step.

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-07-27T03:00:12Z

MrBrain: Adding more precision based on Shawn's steps calculation.

An '''n-instruction Turing machine''' is a [[Turing machine]] with an arbitrary number of states and symbols, but only ''n'' defined transitions/instructions in its transition table (all others are undefined). The '''Instruction-Limited Busy Beaver''' (BBi(n)) problem is the Busy Beaver problem limited to n-instruction TMs. BBi(n) is the longest runtime for all halting n-instruction TMs when started on a blank tape and Σi(n) is the maximum number of non-blank symbols written by an n-instruction TM when halting and when started on a blank tape.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') [1]. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n=1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m=1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions?

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!TNF Size
!Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | 1
| style="text-align: right;" | 5
|{{TM|0RB_---}} {{TM|1RB---_------}}
|
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | 3
| style="text-align: right;" | 35
|{{TM|0RB---_1LA---}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | 5
| style="text-align: right;" | 413
|{{TM|1RB1LB_1LA---}} (and 13 others)
|[[BB(2)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | 16
| style="text-align: right;" | 8,053
|{{TM|1RB---_0RC---_1LC0LA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | 37
| style="text-align: right;" | 213,633
|{{TM|1RB2LB---_2LA2RB1LB}}
|[[BB(2,3)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | 123
| style="text-align: right;" | 7,363,453
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | 3,932,963
| style="text-align: right;" | 312,696,581
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|[[BB(2,4)]] champion and
its [[BB(3,3)]] doppelganger
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of longrunning BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 6.889*10^{1565}</math>
| style="text-align: right;" | 15,874,490,107
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC}}
|
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

Notes:
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue.
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5)-1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* TNF Size is the total number of TMs enumerated in [[TNF]] with n (or fewer) instructions defined.
* BBi(n) ≥ n. A halting (n+1) x 1 TM that chains all its states together (A0 = 0RB, B0 = 0RC, C0 = 0RD, ...) through the penultimate TM state will take n steps to pass through all states exactly once (a repeat would guarantee nonhalting). The penultimate state, the nth state, puts the TM into n+1st state, which is undefined and does not count as a step.

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-07-27T02:22:05Z

MrBrain: Removed redundant conclusion to Motivation section.

An '''n-instruction Turing machine''' is a [[Turing machine]] with an arbitrary number of states and symbols, but only ''n'' defined transitions/instructions in its transition table (all others are undefined). The '''Instruction-Limited Busy Beaver''' (BBi(n)) problem is the Busy Beaver problem limited to n-instruction TMs. BBi(n) is the longest runtime for all halting n-instruction TMs when started on a blank tape and Σi(n) is the maximum number of non-blank symbols written by an n-instruction TM when halting and when started on a blank tape.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') [1]. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n=1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m=1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions?

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!TNF Size
!Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | 1
| style="text-align: right;" | 5
|{{TM|0RB_---}} {{TM|1RB---_------}}
|
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | 3
| style="text-align: right;" | 35
|{{TM|0RB---_1LA---}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | 5
| style="text-align: right;" | 413
|{{TM|1RB1LB_1LA---}} (and 13 others)
|[[BB(2)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | 16
| style="text-align: right;" | 8,053
|{{TM|1RB---_0RC---_1LC0LA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | 37
| style="text-align: right;" | 213,633
|{{TM|1RB2LB---_2LA2RB1LB}}
|[[BB(2,3)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | 123
| style="text-align: right;" | 7,363,453
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | 3,932,963
| style="text-align: right;" | 312,696,581
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|[[BB(2,4)]] champion and
its [[BB(3,3)]] doppelganger
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of longrunning BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | >1.355 x 10783
| style="text-align: right;" | 15,874,490,107
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC}}
|A 3x4 Turing machine
Given bound is Σi(8) < BBi(8)
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd]
|}

Notes:
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue.
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5)-1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* TNF Size is the total number of TMs enumerated in [[TNF]] with n (or fewer) instructions defined.
* BBi(n) ≥ n. A halting (n+1) x 1 TM that chains all its states together (A0 = 0RB, B0 = 0RC, C0 = 0RD, ...) through the penultimate TM state will take n steps to pass through all states exactly once (a repeat would guarantee nonhalting). The penultimate state, the nth state, puts the TM into n+1st state, which is undefined and does not count as a step.
* Estimate of number of steps for the current BBi(8) champion is not currently known, but must be greater than the number of symbols written to tape Σi(8), which is approximately 1.355 x 10783. The true steps value for the current BBi(8) champion is likely around the square of the symbols value (e.g. ~101566).

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-07-27T00:35:59Z

MrBrain: Clarified note about BBi(8) steps value.

An '''n-instruction Turing machine''' is a [[Turing machine]] with an arbitrary number of states and symbols, but only ''n'' defined transitions/instructions in its transition table (all others are undefined). The '''Instruction-Limited Busy Beaver''' (BBi(n)) problem is the Busy Beaver problem limited to n-instruction TMs. BBi(n) is the longest runtime for all halting n-instruction TMs when started on a blank tape and Σi(n) is the maximum number of non-blank symbols written by an n-instruction TM when halting and when started on a blank tape.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') [1]. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n=1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m=1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? This is just one of the many questions that motivated the instruction-limited Busy Beaver contest.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!TNF Size
!Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | 1
| style="text-align: right;" | 5
|{{TM|0RB_---}} {{TM|1RB---_------}}
|
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | 3
| style="text-align: right;" | 35
|{{TM|0RB---_1LA---}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | 5
| style="text-align: right;" | 413
|{{TM|1RB1LB_1LA---}} (and 13 others)
|[[BB(2)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | 16
| style="text-align: right;" | 8,053
|{{TM|1RB---_0RC---_1LC0LA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | 37
| style="text-align: right;" | 213,633
|{{TM|1RB2LB---_2LA2RB1LB}}
|[[BB(2,3)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | 123
| style="text-align: right;" | 7,363,453
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | 3,932,963
| style="text-align: right;" | 312,696,581
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|[[BB(2,4)]] champion and
its [[BB(3,3)]] doppelganger
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of longrunning BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | >1.355 x 10783
| style="text-align: right;" | 15,874,490,107
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC}}
|A 3x4 Turing machine
Given bound is Σi(8) < BBi(8)
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd]
|}

Notes:
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue.
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5)-1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* TNF Size is the total number of TMs enumerated in [[TNF]] with n (or fewer) instructions defined.
* BBi(n) ≥ n. A halting (n+1) x 1 TM that chains all its states together (A0 = 0RB, B0 = 0RC, C0 = 0RD, ...) through the penultimate TM state will take n steps to pass through all states exactly once (a repeat would guarantee nonhalting). The penultimate state, the nth state, puts the TM into n+1st state, which is undefined and does not count as a step.
* Estimate of number of steps for the current BBi(8) champion is not currently known, but must be greater than the number of symbols written to tape Σi(8), which is approximately 1.355 x 10783. The true steps value for the current BBi(8) champion is likely around the square of the symbols value (e.g. ~101566).

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-07-26T23:19:53Z

MrBrain: Added an example lower bound to illustrate the likely true size of BBi(8) in the explanatory note below the BBi table.

An '''n-instruction Turing machine''' is a [[Turing machine]] with an arbitrary number of states and symbols, but only ''n'' defined transitions/instructions in its transition table (all others are undefined). The '''Instruction-Limited Busy Beaver''' (BBi(n)) problem is the Busy Beaver problem limited to n-instruction TMs. BBi(n) is the longest runtime for all halting n-instruction TMs when started on a blank tape and Σi(n) is the maximum number of non-blank symbols written by an n-instruction TM when halting and when started on a blank tape.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') [1]. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n=1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m=1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? This is just one of the many questions that motivated the instruction-limited Busy Beaver contest.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!TNF Size
!Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | 1
| style="text-align: right;" | 5
|{{TM|0RB_---}} {{TM|1RB---_------}}
|
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | 3
| style="text-align: right;" | 35
|{{TM|0RB---_1LA---}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | 5
| style="text-align: right;" | 413
|{{TM|1RB1LB_1LA---}} (and 13 others)
|[[BB(2)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | 16
| style="text-align: right;" | 8,053
|{{TM|1RB---_0RC---_1LC0LA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | 37
| style="text-align: right;" | 213,633
|{{TM|1RB2LB---_2LA2RB1LB}}
|[[BB(2,3)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | 123
| style="text-align: right;" | 7,363,453
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | 3,932,963
| style="text-align: right;" | 312,696,581
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|[[BB(2,4)]] champion and
its [[BB(3,3)]] doppelganger
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of longrunning BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | >1.355 x 10783
| style="text-align: right;" | 15,874,490,107
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC}}
|A 3x4 Turing machine
Given bound is Σi(8) < BBi(8)
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd]
|}

Notes:
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue.
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5)-1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* TNF Size is the total number of TMs enumerated in [[TNF]] with n (or fewer) instructions defined.
* BBi(n) ≥ n. A halting (n+1) x 1 TM that chains all its states together (A0 = 0RB, B0 = 0RC, C0 = 0RD, ...) through the penultimate TM state will take n steps to pass through all states exactly once (a repeat would guarantee nonhalting). The penultimate state, the nth state, puts the TM into n+1st state, which is undefined and does not count as a step.
* Estimate of number of steps for the current BBi(8) champion is not currently known, but must be greater than the number of symbols written to tape Σi(8), which is approximately 1.355 x 10783. The true value for the current BBi(8) champion is likely around the square of the symbols value (e.g. ~101566).

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-07-26T23:03:02Z

MrBrain: Clarified that the current value for BBi(n) is a lower bound based on the currently known estimate for Σi(8), the number of symbols written to tape.

An '''n-instruction Turing machine''' is a [[Turing machine]] with an arbitrary number of states and symbols, but only ''n'' defined transitions/instructions in its transition table (all others are undefined). The '''Instruction-Limited Busy Beaver''' (BBi(n)) problem is the Busy Beaver problem limited to n-instruction TMs. BBi(n) is the longest runtime for all halting n-instruction TMs when started on a blank tape and Σi(n) is the maximum number of non-blank symbols written by an n-instruction TM when halting and when started on a blank tape.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') [1]. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n=1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m=1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? This is just one of the many questions that motivated the instruction-limited Busy Beaver contest.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!TNF Size
!Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | 1
| style="text-align: right;" | 5
|{{TM|0RB_---}} {{TM|1RB---_------}}
|
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | 3
| style="text-align: right;" | 35
|{{TM|0RB---_1LA---}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | 5
| style="text-align: right;" | 413
|{{TM|1RB1LB_1LA---}} (and 13 others)
|[[BB(2)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | 16
| style="text-align: right;" | 8,053
|{{TM|1RB---_0RC---_1LC0LA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | 37
| style="text-align: right;" | 213,633
|{{TM|1RB2LB---_2LA2RB1LB}}
|[[BB(2,3)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | 123
| style="text-align: right;" | 7,363,453
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | 3,932,963
| style="text-align: right;" | 312,696,581
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|[[BB(2,4)]] champion and
its [[BB(3,3)]] doppelganger
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of longrunning BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | >1.355 x 10783
| style="text-align: right;" | 15,874,490,107
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC}}
|A 3x4 Turing machine
Given bound is Σi(8) < BBi(8)
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd]
|}

Notes:
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue.
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5)-1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* TNF Size is the total number of TMs enumerated in [[TNF]] with n (or fewer) instructions defined.
* BBi(n) ≥ n. A halting (n+1) x 1 TM that chains all its states together (A0 = 0RB, B0 = 0RC, C0 = 0RD, ...) through the penultimate TM state will take n steps to pass through all states exactly once (a repeat would guarantee nonhalting). The penultimate state, the nth state, puts the TM into n+1st state, which is undefined and does not count as a step.
* Estimate of number of steps for the current BBi(8) champion is not currently known, but must be greater than the number of symbols written to tape Σi(8), which is approximately 1.355 x 10783. The true value for the current BBi(8) champion is likely around the square of the symbols value.

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-07-26T21:43:29Z

MrBrain: Clarifying description of the current BBi(8) champion, making clear it is not a current BB(3,4) champion.

An '''n-instruction Turing machine''' is a [[Turing machine]] with an arbitrary number of states and symbols, but only ''n'' defined transitions/instructions in its transition table (all others are undefined). The '''Instruction-Limited Busy Beaver''' (BBi(n)) problem is the Busy Beaver problem limited to n-instruction TMs. BBi(n) is the longest runtime for all halting n-instruction TMs when started on a blank tape and Σi(n) is the maximum number of non-blank symbols written by an n-instruction TM when halting and when started on a blank tape.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') [1]. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n=1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m=1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? This is just one of the many questions that motivated the instruction-limited Busy Beaver contest.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!TNF Size
!Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | 1
| style="text-align: right;" | 5
|{{TM|0RB_---}} {{TM|1RB---_------}}
|
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | 3
| style="text-align: right;" | 35
|{{TM|0RB---_1LA---}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | 5
| style="text-align: right;" | 413
|{{TM|1RB1LB_1LA---}} (and 13 others)
|[[BB(2)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | 16
| style="text-align: right;" | 8,053
|{{TM|1RB---_0RC---_1LC0LA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | 37
| style="text-align: right;" | 213,633
|{{TM|1RB2LB---_2LA2RB1LB}}
|[[BB(2,3)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | 123
| style="text-align: right;" | 7,363,453
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | 3,932,963
| style="text-align: right;" | 312,696,581
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|[[BB(2,4)]] champion and
its [[BB(3,3)]] doppelganger
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of longrunning BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | >1.355 x 10783
| style="text-align: right;" | 15,874,490,107
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC}}
|A 3x4 Turing machine
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd]
|}

Notes:
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue.
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5)-1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* TNF Size is the total number of TMs enumerated in [[TNF]] with n (or fewer) instructions defined.
* BBi(n) ≥ n. A halting (n+1) x 1 TM that chains all its states together (A0 = 0RB, B0 = 0RC, C0 = 0RD, ...) through the penultimate TM state will take n steps to pass through all states exactly once (a repeat would guarantee nonhalting). The penultimate state, the nth state, puts the TM into n+1st state, which is undefined and does not count as a step.

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-07-26T21:18:17Z

MrBrain: Link to bbchallenge discord of machine announcement.

An '''n-instruction Turing machine''' is a [[Turing machine]] with an arbitrary number of states and symbols, but only ''n'' defined transitions/instructions in its transition table (all others are undefined). The '''Instruction-Limited Busy Beaver''' (BBi(n)) problem is the Busy Beaver problem limited to n-instruction TMs. BBi(n) is the longest runtime for all halting n-instruction TMs when started on a blank tape and Σi(n) is the maximum number of non-blank symbols written by an n-instruction TM when halting and when started on a blank tape.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') [1]. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n=1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m=1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? This is just one of the many questions that motivated the instruction-limited Busy Beaver contest.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!TNF Size
!Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | 1
| style="text-align: right;" | 5
|{{TM|0RB_---}} {{TM|1RB---_------}}
|
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | 3
| style="text-align: right;" | 35
|{{TM|0RB---_1LA---}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | 5
| style="text-align: right;" | 413
|{{TM|1RB1LB_1LA---}} (and 13 others)
|[[BB(2)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | 16
| style="text-align: right;" | 8,053
|{{TM|1RB---_0RC---_1LC0LA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | 37
| style="text-align: right;" | 213,633
|{{TM|1RB2LB---_2LA2RB1LB}}
|[[BB(2,3)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | 123
| style="text-align: right;" | 7,363,453
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | 3,932,963
| style="text-align: right;" | 312,696,581
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|[[BB(2,4)]] champion and
its [[BB(3,3)]] doppelganger
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of longrunning BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | >1.355 x 10783
| style="text-align: right;" | 15,874,490,107
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC}}
|Current 3x4 champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd]
|}

Notes:
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue.
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5)-1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* TNF Size is the total number of TMs enumerated in [[TNF]] with n (or fewer) instructions defined.
* BBi(n) ≥ n. A halting (n+1) x 1 TM that chains all its states together (A0 = 0RB, B0 = 0RC, C0 = 0RD, ...) through the penultimate TM state will take n steps to pass through all states exactly once (a repeat would guarantee nonhalting). The penultimate state, the nth state, puts the TM into n+1st state, which is undefined and does not count as a step.

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-07-26T21:16:02Z

MrBrain: Updated description of new current BBi(8) champion.

An '''n-instruction Turing machine''' is a [[Turing machine]] with an arbitrary number of states and symbols, but only ''n'' defined transitions/instructions in its transition table (all others are undefined). The '''Instruction-Limited Busy Beaver''' (BBi(n)) problem is the Busy Beaver problem limited to n-instruction TMs. BBi(n) is the longest runtime for all halting n-instruction TMs when started on a blank tape and Σi(n) is the maximum number of non-blank symbols written by an n-instruction TM when halting and when started on a blank tape.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') [1]. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n=1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m=1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? This is just one of the many questions that motivated the instruction-limited Busy Beaver contest.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!TNF Size
!Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | 1
| style="text-align: right;" | 5
|{{TM|0RB_---}} {{TM|1RB---_------}}
|
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | 3
| style="text-align: right;" | 35
|{{TM|0RB---_1LA---}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | 5
| style="text-align: right;" | 413
|{{TM|1RB1LB_1LA---}} (and 13 others)
|[[BB(2)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | 16
| style="text-align: right;" | 8,053
|{{TM|1RB---_0RC---_1LC0LA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | 37
| style="text-align: right;" | 213,633
|{{TM|1RB2LB---_2LA2RB1LB}}
|[[BB(2,3)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | 123
| style="text-align: right;" | 7,363,453
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | 3,932,963
| style="text-align: right;" | 312,696,581
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|[[BB(2,4)]] champion and
its [[BB(3,3)]] doppelganger
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of longrunning BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | >1.355 x 10783
| style="text-align: right;" | 15,874,490,107
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC}}
|Current 3x4 champion
|A384629
|}

Notes:
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue.
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5)-1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* TNF Size is the total number of TMs enumerated in [[TNF]] with n (or fewer) instructions defined.
* BBi(n) ≥ n. A halting (n+1) x 1 TM that chains all its states together (A0 = 0RB, B0 = 0RC, C0 = 0RD, ...) through the penultimate TM state will take n steps to pass through all states exactly once (a repeat would guarantee nonhalting). The penultimate state, the nth state, puts the TM into n+1st state, which is undefined and does not count as a step.

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-07-26T21:14:15Z

MrBrain: Edited the code for the current champion BBi(8) TM.

An '''n-instruction Turing machine''' is a [[Turing machine]] with an arbitrary number of states and symbols, but only ''n'' defined transitions/instructions in its transition table (all others are undefined). The '''Instruction-Limited Busy Beaver''' (BBi(n)) problem is the Busy Beaver problem limited to n-instruction TMs. BBi(n) is the longest runtime for all halting n-instruction TMs when started on a blank tape and Σi(n) is the maximum number of non-blank symbols written by an n-instruction TM when halting and when started on a blank tape.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') [1]. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n=1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m=1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? This is just one of the many questions that motivated the instruction-limited Busy Beaver contest.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!TNF Size
!Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | 1
| style="text-align: right;" | 5
|{{TM|0RB_---}} {{TM|1RB---_------}}
|
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | 3
| style="text-align: right;" | 35
|{{TM|0RB---_1LA---}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | 5
| style="text-align: right;" | 413
|{{TM|1RB1LB_1LA---}} (and 13 others)
|[[BB(2)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | 16
| style="text-align: right;" | 8,053
|{{TM|1RB---_0RC---_1LC0LA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | 37
| style="text-align: right;" | 213,633
|{{TM|1RB2LB---_2LA2RB1LB}}
|[[BB(2,3)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | 123
| style="text-align: right;" | 7,363,453
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | 3,932,963
| style="text-align: right;" | 312,696,581
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|[[BB(2,4)]] champion and
its [[BB(3,3)]] doppelganger
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of longrunning BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | >1.355 x 10783
| style="text-align: right;" | 15,874,490,107
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC}}
|Current [[BB(3,3)]] champion
|A384629
|}

Notes:
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue.
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5)-1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* TNF Size is the total number of TMs enumerated in [[TNF]] with n (or fewer) instructions defined.
* BBi(n) ≥ n. A halting (n+1) x 1 TM that chains all its states together (A0 = 0RB, B0 = 0RC, C0 = 0RD, ...) through the penultimate TM state will take n steps to pass through all states exactly once (a repeat would guarantee nonhalting). The penultimate state, the nth state, puts the TM into n+1st state, which is undefined and does not count as a step.

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-07-26T21:12:29Z

MrBrain: Number of steps found by Nick for new current BBi(8) champion.

An '''n-instruction Turing machine''' is a [[Turing machine]] with an arbitrary number of states and symbols, but only ''n'' defined transitions/instructions in its transition table (all others are undefined). The '''Instruction-Limited Busy Beaver''' (BBi(n)) problem is the Busy Beaver problem limited to n-instruction TMs. BBi(n) is the longest runtime for all halting n-instruction TMs when started on a blank tape and Σi(n) is the maximum number of non-blank symbols written by an n-instruction TM when halting and when started on a blank tape.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') [1]. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n=1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m=1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? This is just one of the many questions that motivated the instruction-limited Busy Beaver contest.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!TNF Size
!Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | 1
| style="text-align: right;" | 5
|{{TM|0RB_---}} {{TM|1RB---_------}}
|
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | 3
| style="text-align: right;" | 35
|{{TM|0RB---_1LA---}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | 5
| style="text-align: right;" | 413
|{{TM|1RB1LB_1LA---}} (and 13 others)
|[[BB(2)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | 16
| style="text-align: right;" | 8,053
|{{TM|1RB---_0RC---_1LC0LA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | 37
| style="text-align: right;" | 213,633
|{{TM|1RB2LB---_2LA2RB1LB}}
|[[BB(2,3)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | 123
| style="text-align: right;" | 7,363,453
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | 3,932,963
| style="text-align: right;" | 312,696,581
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|[[BB(2,4)]] champion and
its [[BB(3,3)]] doppelganger
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of longrunning BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | >1.355 x 10783
| style="text-align: right;" | 15,874,490,107
|{{TM|0RB2LA1RA_1LA2RB1RC_---1LB1LC}}
|Current [[BB(3,3)]] champion
|A384629
|}

Notes:
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue.
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5)-1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* TNF Size is the total number of TMs enumerated in [[TNF]] with n (or fewer) instructions defined.
* BBi(n) ≥ n. A halting (n+1) x 1 TM that chains all its states together (A0 = 0RB, B0 = 0RC, C0 = 0RD, ...) through the penultimate TM state will take n steps to pass through all states exactly once (a repeat would guarantee nonhalting). The penultimate state, the nth state, puts the TM into n+1st state, which is undefined and does not count as a step.

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Instruction-Limited Busy Beaver

2025-07-26T17:57:13Z

MrBrain: Corrected total number of TNF machines for n<=8.

An '''n-instruction Turing machine''' is a [[Turing machine]] with an arbitrary number of states and symbols, but only ''n'' defined transitions/instructions in its transition table (all others are undefined). The '''Instruction-Limited Busy Beaver''' (BBi(n)) problem is the Busy Beaver problem limited to n-instruction TMs. BBi(n) is the longest runtime for all halting n-instruction TMs when started on a blank tape and Σi(n) is the maximum number of non-blank symbols written by an n-instruction TM when halting and when started on a blank tape.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by Tibor Radó, was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''') [1]. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n=1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m=1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? This is just one of the many questions that motivated the instruction-limited Busy Beaver contest.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!TNF Size
!Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | 1
| style="text-align: right;" | 5
|{{TM|0RB_---}} {{TM|1RB---_------}}
|
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | 3
| style="text-align: right;" | 35
|{{TM|0RB---_1LA---}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | 5
| style="text-align: right;" | 413
|{{TM|1RB1LB_1LA---}} (and 13 others)
|[[BB(2)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | 16
| style="text-align: right;" | 8,053
|{{TM|1RB---_0RC---_1LC0LA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | 37
| style="text-align: right;" | 213,633
|{{TM|1RB2LB---_2LA2RB1LB}}
|[[BB(2,3)]] champion
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | 123
| style="text-align: right;" | 7,363,453
|{{TM|1RB3LA1RA0LA_2LA------3RA}}
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | 3,932,963
| style="text-align: right;" | 312,696,581
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---}}
|[[BB(2,4)]] champion and
its [[BB(3,3)]] doppelganger
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of longrunning BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | >=119,112,334,170,342,540
| style="text-align: right;" | 15,874,490,107
|{{TM|0RB2LA1RA_1LA2RB1RC_---1LB1LC}}
|Current [[BB(3,3)]] champion
|A384629
|}

Notes:
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue.
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5)-1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* TNF Size is the total number of TMs enumerated in [[TNF]] with n (or fewer) instructions defined.
* BBi(n) ≥ n. A halting (n+1) x 1 TM that chains all its states together (A0 = 0RB, B0 = 0RC, C0 = 0RD, ...) through the penultimate TM state will take n steps to pass through all states exactly once (a repeat would guarantee nonhalting). The penultimate state, the nth state, puts the TM into n+1st state, which is undefined and does not count as a step.

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290