Non-halting Turing machine: Difference between revisions

From BusyBeaverWiki
Jump to navigation Jump to search
VisualWikitext
(Added links)
(Added some more pictures)
Line 92: Line 92:
!Count
!Count
!Notes and Notable examples
!Notes and Notable examples
!Example space-time diagram
|-
|-
|[[Translated cycler]]
|[[Translated cycler]]
Line 99: Line 100:
* {{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}, the current preperiod record holder, with a preperiod of 119,120,230,102. Before phasing into a translated cycle, this machine appears to be a [[spaghetti]].
* {{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}, the current preperiod record holder, with a preperiod of 119,120,230,102. Before phasing into a translated cycle, this machine appears to be a [[spaghetti]].
* {{TM|1RB1RC_1LC0RA_0LB0LD_1LA1LD}} starts out as an irregular bell, but phase transitions into a translated cycler with period 4222 at step 29754825.
* {{TM|1RB1RC_1LC0RA_0LB0LD_1LA1LD}} starts out as an irregular bell, but phase transitions into a translated cycler with period 4222 at step 29754825.
|
|-
|-
|[[Cycler]]
|[[Cycler]]
Line 105: Line 107:
* {{TM|1RB0RB_1LC0RD_1LA1LB_0LC1RD}}, likely period record holder, with a period of 120.
* {{TM|1RB0RB_1LC0RD_1LA1LB_0LC1RD}}, likely period record holder, with a period of 120.
* {{TM|1RB1LB_1LC1RD_1LA0RD_0LA0RB}}, likely preperiod record holder, with a preperiod of 146.
* {{TM|1RB1LB_1LC1RD_1LA0RD_0LA0RB}}, likely preperiod record holder, with a preperiod of 146.
|
|-
|-
|[[Bouncer]]
|[[Bouncer]]
Line 112: Line 115:
* {{TM|1RB0RC_0RC0RB_1LC0LD_1LA0RA}}, starts out as irregular-side bells before phasing into a bouncer at step 3350.
* {{TM|1RB0RC_0RC0RB_1LC0LD_1LA0RA}}, starts out as irregular-side bells before phasing into a bouncer at step 3350.
* {{TM|1RB1LC_0RD0LC_1LB0LA_1LD1RA}}, a bouncer with very complex runs. Start step 145,729.
* {{TM|1RB1LC_0RD0LC_1LB0LA_1LD1RA}}, a bouncer with very complex runs. Start step 145,729.
|
|-
|-
|[[Counter]]
|[[Counter]]
Line 129: Line 133:


* {{TM|1RB1LC_0RD0RB_1LA0LA_1LD0LA}}, an example of a superexponential counter.
* {{TM|1RB1LC_0RD0RB_1LA0LA_1LD0LA}}, an example of a superexponential counter.
|
|-
|-
|[[Bell]]
|[[Bell]]
Line 137: Line 142:
* {{TM|1RB0LC_1RC1RB_1LA1LD_0RA0RB}}, a grow-and-shrink bell.
* {{TM|1RB0LC_1RC1RB_1LA1LD_0RA0RB}}, a grow-and-shrink bell.
* {{TM|1RB0RC_1LC0RA_1RA1LD_0LC0LA}}, starts out as an irregular bell before phasing into a bell.
* {{TM|1RB0RC_1LC0RA_1RA1LD_0LC0LA}}, starts out as an irregular bell before phasing into a bell.
|
|-
|-
|[[Cubic bell]]
|[[Cubic bell]]
Line 143: Line 149:
* {{TM|1RB1RC_1LC0RC_0RA1LD_0LC0LB}}, a cubic inverted bell.
* {{TM|1RB1RC_1LC0RC_0RA1LD_0LC0LB}}, a cubic inverted bell.
* {{TM|1RB0RC_0LD0RA_0LA1RC_1LA1LD}}, a cubic grow-and-shrink bell.
* {{TM|1RB0RC_0LD0RA_0LA1RC_1LA1LD}}, a cubic grow-and-shrink bell.
|
|-
|-
|Bouncer + X
|Bouncer + X
Line 153: Line 160:
* {{TM|1RB1LB_0LC0RD_0RA1LC_1RA1RD}}, a bouncer + cubic bell, leading to quartic tape growth on the left.
* {{TM|1RB1LB_0LC0RD_0RA1LC_1RA1RD}}, a bouncer + cubic bell, leading to quartic tape growth on the left.
* {{TM|1RB0LC_1LA1RD_1RA1LD_0LA0RB}}, a bouncer + unclassified. (If you can classify it, let me know in the talk page!)
* {{TM|1RB0LC_1LA1RD_1RA1LD_0LA0RB}}, a bouncer + unclassified. (If you can classify it, let me know in the talk page!)
|<div class="text-align: center">[[File:1RB0LB 1LC1RA 1RD0LC 0LA0RB.png|frameless|300x300px]]</div>
|-
|-
|Bouncing counter
|Bounce-counter
|≈330
|≈330
|
|
* {{TM|1RB1LC_1LC0RB_1RA0LD_1RA1LA}}, a typical bouncing binary counter.
* {{TM|1RB1LC_1LC0RB_1RA0LD_1RA1LA}}, a typical binary bounce-counter.
* {{TM|1RB1LB_1RC1RD_0LA0RC_1LD0LB}}, a typical bouncing quaternary counter.
* {{TM|1RB1LB_1RC1RD_0LA0RC_1LD0LB}}, a typical quaternary bounce-counter.
* {{TM|1RB1RA_0RC0LC_1LA0LD_0RA1LC}}, a bouncing ternary counter, which is more rare.
* {{TM|1RB1RA_0RC0LC_1LA0LD_0RA1LC}}, a ternary bounce-counter, which is more rare.
* {{TM|1RB0LA_0RC1LA_0RD1RB_1LD1LA}}, a hybrid quaternary-octal bouncing counter.
* {{TM|1RB0LA_0RC1LA_0RD1RB_1LD1LA}}, a hybrid quaternary-octal bounce-counter.
* {{TM|1RB0LC_1RD0RB_1LA1RC_1LC1RB}}, a bouncing 3/2-counter.
* {{TM|1RB0LC_1RD0RB_1LA1RC_1LC1RB}}, a 3/2-bounce-counter.
* {{TM|1RB0LC_1LC0RD_1RA1LA_0RA0LA}}, a bouncing binary counter with stationary counter digits.
* {{TM|1RB0LC_1LC0RD_1RA1LA_0RA0LA}}, a binary bounce-counter with stationary counter digits.
|<div class="text-align: center">[[File:1RB1LC 1LC0RB 1RA0LD 1RA1LA.png|frameless|300x300px]]</div>
|-
|-
|Fractal
|Fractal
Line 168: Line 177:
|
|
* {{TM|1RB1RC_0RC0RB_0LD1LA_1LD0LA}}, a typical example.
* {{TM|1RB1RC_0RC0RB_0LD1LA_1LD0LA}}, a typical example.
|<div class="text-align: center">[[File:1RB1RC 0RC0RB 0LD1LA 1LD0LA.png|frameless|300x300px]]</div>
|-
|-
|[[Shift overflow bouncer counter|Tetration counter]]
|[[Shift overflow bouncer counter|Tetration counter]]
Line 173: Line 183:
|
|
* {{TM|1RB1LC_0RD0RD_0RC0LA_1LD1RA}}, a typical example.
* {{TM|1RB1LC_0RD0RD_0RC0LA_1LD1RA}}, a typical example.
|<div class="text-align: center">[[File:1RB1LC 0RD0RD 0RC0LA 1LD1RA.png|frameless|300x300px]]</div>
|-
|-
|Cubic bouncing counter
|Cubic bounce-counter
|13
|13
|
|
* {{TM|1RB1RA_0LC0RB_0RA0LD_1LC1RD}}, a typical example. Note that these share many of the same properties as [[Dekaheptoid|dekaheptoids]].
* {{TM|1RB1RA_0LC0RB_0RA0LD_1LC1RD}}, a typical example. Note that these share many of the same properties as [[Dekaheptoid|dekaheptoids]].
|<div class="text-align: center">[[File:1RB1RA 0LC0RB 0RA0LD 1LC1RD.png|frameless|300x300px]]</div>
|}
|}


==== Chaotic ====
==== Chaotic ====
{| class="wikitable"
{{Table alignment}}
{| class="wikitable col4center"
!Classification
!Classification
!Count
!Count
Line 192: Line 205:
* {{TM|1RB0RC_1LC1RA_1RA1LD_0LC0LA}}, a typical irregular bell.
* {{TM|1RB0RC_1LC1RA_1RA1LD_0LC0LA}}, a typical irregular bell.
* {{TM|1RB1LA_0RC0RD_1LD1RC_1LD0LA}}, a typical irregular inverted bell.
* {{TM|1RB1LA_0RC0RD_1LD1RC_1LD0LA}}, a typical irregular inverted bell.
|[[File:1RB0RC 1LC1RA 1RA1LD 0LC0LA.png|frameless]]
|<div class="text-align: center">[[File:1RB0RC 1LC1RA 1RA1LD 0LC0LA.png|frameless|300x300px]]</div>
|-
|-
|[[Spaghetti]]
|[[Spaghetti]]
Line 199: Line 212:


* {{TM|1RB0RB_1LC1RC_0RA1LD_1RC0LD}}, a typical spaghetti.
* {{TM|1RB0RB_1LC1RC_0RA1LD_1RC0LD}}, a typical spaghetti.
* {{TM|1RB0RA_1RC0RD_1LD1LC_1RA0LC}}, a spaghetti that seems to converge to a bouncing counter.
* {{TM|1RB0RA_1RC0RD_1LD1LC_1RA0LC}}, a spaghetti that seems to converge to a bounce-counter.
* {{TM|1RB1LC_0LA0RD_1LA0LB_1LA1RD}}, a spaghetti whose envelope seems to converge to that of a bouncer.
* {{TM|1RB1LC_0LA0RD_1LA0LB_1LA1RD}}, a spaghetti whose envelope seems to converge to that of a bouncer.
* {{TM|1RB0RC_1LC1LD_1RD1LB_1RA0LB}}, a cellular-automaton-like bouncer. The spaghetti nature of this machine is local.
* {{TM|1RB0RC_1LC1LD_1RD1LB_1RA0LB}}, a cellular-automaton-like bouncer. The spaghetti nature of this machine is local.
* {{TM|1RB1LC_1LA1RD_1RA0LC_1LB0RD}}, a "spaghetti sandwich" -- a spaghetti sandwiched on the left and right by a growing predictable repeating pattern.
* {{TM|1RB1LC_1LA1RD_1RA0LC_1LB0RD}}, a "spaghetti sandwich" -- a spaghetti sandwiched on the left and right by a growing predictable repeating pattern.
|[[File:1RB0RB 1LC1RC 0RA1LD 1RC0LD.png|frameless]]
|<div class="text-align: center">[[File:1RB0RB 1LC1RC 0RA1LD 1RC0LD.png|frameless|300x300px]]</div>
|-
|-
|Chaotic counter
|Chaotic counter
Line 219: Line 232:
* {{TM|1RB1LC_0LA0RB_1RD0LC_1LA0RD}}
* {{TM|1RB1LC_0LA0RB_1RD0LC_1LA0RD}}
* {{TM|1RB1LC_1RD0RB_1LA0LC_0LA0RD}}
* {{TM|1RB1LC_1RD0RB_1LA0LC_0LA0RD}}
|[[File:1RB1LC 1RD0RB 1LA0LC 0LA0RD.png|frameless]]
|<div class="text-align: center">[[File:1RB1LC 1RD0RB 1LA0LC 0LA0RD.png|frameless|300x300px]]</div>
|}
|}


Revision as of 19:18, 9 November 2024

A non-halting Turing machine is a Turing machine that does not halt. These include halt-free Turing machines, meaning those without an undefined or halt transition, as well as non-halt-free Turing machines that never enter an undefined or halt transition.

The crux of the Busy Beaver problem, of finding BB(n, k) for a given n and k, is to prove that all non-halting Turing machines with n states and k symbols are, in fact, non-halting.

The zoology of non-halting Turing machines is extremely rich. See Translated cycler, Bouncer, Bell, Counter, Shift overflow counter, Shift overflow bouncer counter for a sample. In this page, we provide a detailed zoology for some low numbers of states and symbols.

Zoology

Machines are enumerated in TNF-1RB, and we exclude halting machines. In particular, a transition is defined if and only if it is reachable; unreachable transitions are undefined. This avoids duplicates.

2 × 2

There are 106 TNF-1RB machines with 2 states and 2 symbols, with the following breakdown:

Classification Count Notable examples
Translated cycler 88
  • 1RB0RB_1LB1RA (bbch), unique TM with a record period of 9.
  • 1RB0LB_1LA0RB (bbch), unique TM with a record preperiod of 9.
Cycler 14
  • 1RB0RB_1LB1LA (bbch), one of the 7 TMs with a record period of 4.
  • 1RB1LB_1LA1RA (bbch), unique TM with a record preperiod of 5.
Bouncer 3
  • 1RB1LA_0LA1RB (bbch)
  • 1RB1LA_1LA1RB (bbch)
  • 1RB0LB_1LA0RA (bbch)
Counter 1
  • 1RB1LA_0LA0RB (bbch), a binary counter.

3 × 2

There are 15,064 TNF-1RB machines with 3 states and 2 symbols, with the following breakdown:

Classification Count Notable examples
Translated cycler 12,427
  • 1RB0LA_0RC1LA_1LC0RB (bbch), unique TM with a record period of 92.
  • 1RB1LB_0RC0LA_1LC0LA (bbch), unique TM with a record preperiod of 101.
Cycler 1,969
  • 1RB0LB_1LB1LC_0RC1RA (bbch), unique TM with a record period of 18.
  • 1RB1RC_1LC0LB_1RA1LA (bbch), unique TM with a record preperiod of 22.
Bouncer 558
  • 1RB0LC_1RC0RB_1LA1LC (bbch), a bouncer with two distinct shift rules.
Counter 95
  • 1RB1LA_0LA0RC_0LA1RB (bbch), a Fibonacci counter.
  • 1RB1LA_0LA1RC_0LA0RB (bbch), a two-phase binary counter.
  • 1RB1LA_0LA1RC_1LB0RB (bbch), a translating binary counter.
Cubic bell 10
  • 1RB1LA_0LA0RC_0LA1RC (bbch), a typical cubic bell.
Bell 5
  • 1RB0LC_1RC1RA_1LA0RB (bbch), a typical bell.
  • 1RB1LA_0RC0RC_1LC0LA (bbch), a typical inverted bell.

4 × 2

There are 2,744,516 TNF-1RB machines with 4 states and 2 symbols, with the following breakdown. This breakdown is not complete due the chaotic Turing machines. These are the irregular bells, spaghetti, and chaotic counters. These are informal names for chaotic behaviors which escape analysis, and such may well become a translated cycler, or more rarely, a bouncer, after a large number of steps.

Regular (non-chaotic)

Classification Count Notes and Notable examples Example space-time diagram
Translated cycler ≥2,253,849
  • 1RB0LA_0RC1RD_1LD0RB_1LA1RB (bbch), the current period record holder, with a period of 212,081,736. Before phasing into a translated cycle, this machine appears to be a spaghetti.
  • 1RB1LC_0LA1RD_0RB0LC_1LA0RD (bbch), the current preperiod record holder, with a preperiod of 119,120,230,102. Before phasing into a translated cycle, this machine appears to be a spaghetti.
  • 1RB1RC_1LC0RA_0LB0LD_1LA1LD (bbch) starts out as an irregular bell, but phase transitions into a translated cycler with period 4222 at step 29754825.
Cycler ≥341,617
  • 1RB0RB_1LC0RD_1LA1LB_0LC1RD (bbch), likely period record holder, with a period of 120.
  • 1RB1LB_1LC1RD_1LA0RD_0LA0RB (bbch), likely preperiod record holder, with a preperiod of 146.
Bouncer ≈132,000
  • 1RB1LA_1LC0RC_0RA1LD_1RC0LD (bbch), the current record holder for longest time to settle into a bouncer, with a start step of 83,158,409.
  • 1RB0RC_0RC0RB_1LC0LD_1LA0RA (bbch), starts out as irregular-side bells before phasing into a bouncer at step 3350.
  • 1RB1LC_0RD0LC_1LB0LA_1LD1RA (bbch), a bouncer with very complex runs. Start step 145,729.
Counter ≈14,700
  • 1RB1LC_0LC1RD_1LA1LB_0LC0RD (bbch), a ternary counter.
  • 1RB1LC_0LA1RB_1LD0RB_1LA1RA (bbch), a quaternary counter.
  • 1RB0LB_1RC0LD_1LB1RA_0RB1LD (bbch), a quinary counter.
  • 1RB1LC_0LD1RB_1LD0RD_1LA0RB (bbch), a senary counter.
  • 1RB1LC_0LD0RB_1RD1LA_1RA0LC (bbch), a 3/2-counter.
  • 1RB0RA_1LC1RA_1LD0LC_1LA1LD (bbch), a binary bi-counter.
  • 1RB1LC_1RC1RB_1RD0LC_1LA0RD (bbch), a binary-ternary bi-counter.
  • 1RB1LA_0LA0RC_0LA0RD_0LA1RC (bbch), a counter encoding a recurrence with characteristic polynomial .
  • 1RB1LA_0LA0RC_0LA1RD_0LA0RB (bbch), a counter encoding a recurrence with characteristic polynomial .
  • 1RB1LC_1LD1RA_0RA0LC_0RB0LD (bbch), a counter encoding a recurrence that grows like .
  • 1RB0RC_0LC0RA_1LA0LD_1RA1LD (bbch), a tri-phasic binary counter.
  • 1RB1LC_0RD0RB_1LA0LA_1LD0LA (bbch), an example of a superexponential counter.
Bell ≈2,350
  • 1RB1LA_0RC0LD_1LC0LA_1RC0RD (bbch), a typical inverted bell.
  • 1RB1LB_1LC0RA_1RD0LB_1LA1RC (bbch), alternates between bell and half-bell.
  • 1RB0LC_1RC1RB_1LA1LD_0RA0RB (bbch), a grow-and-shrink bell.
  • 1RB0RC_1LC0RA_1RA1LD_0LC0LA (bbch), starts out as an irregular bell before phasing into a bell.
Cubic bell ≈1,376
  • 1RB1RC_1LC0RC_0RA1LD_0LC0LB (bbch), a cubic inverted bell.
  • 1RB0RC_0LD0RA_0LA1RC_1LA1LD (bbch), a cubic grow-and-shrink bell.
Bouncer + X ≈365
  • 1RB1LA_1RC0RB_0LC1LD_0LD1RA (bbch), a bouncer + binary counter.
  • 1RB0LA_0RC1LA_1RD0RA_0LB1RB (bbch), a bouncer + bell.
  • 1RB1LC_0RC1RD_1LA0LA_1RC0RB (bbch), a bouncer + cellular automaton. This could be universal.
  • 1RB1LC_0RC1RD_1LA0LA_0LA0RB (bbch), a bouncer + cellular automaton with a fractal nature.
  • 1RB1LB_0LC0RD_0RA1LC_1RA1RD (bbch), a bouncer + cubic bell, leading to quartic tape growth on the left.
  • 1RB0LC_1LA1RD_1RA1LD_0LA0RB (bbch), a bouncer + unclassified. (If you can classify it, let me know in the talk page!)
Bounce-counter ≈330
  • 1RB1LC_1LC0RB_1RA0LD_1RA1LA (bbch), a typical binary bounce-counter.
  • 1RB1LB_1RC1RD_0LA0RC_1LD0LB (bbch), a typical quaternary bounce-counter.
  • 1RB1RA_0RC0LC_1LA0LD_0RA1LC (bbch), a ternary bounce-counter, which is more rare.
  • 1RB0LA_0RC1LA_0RD1RB_1LD1LA (bbch), a hybrid quaternary-octal bounce-counter.
  • 1RB0LC_1RD0RB_1LA1RC_1LC1RB (bbch), a 3/2-bounce-counter.
  • 1RB0LC_1LC0RD_1RA1LA_0RA0LA (bbch), a binary bounce-counter with stationary counter digits.
Fractal 20
  • 1RB1RC_0RC0RB_0LD1LA_1LD0LA (bbch), a typical example.
Tetration counter 19
  • 1RB1LC_0RD0RD_0RC0LA_1LD1RA (bbch), a typical example.
Cubic bounce-counter 13
  • 1RB1RA_0LC0RB_0RA0LD_1LC1RD (bbch), a typical example. Note that these share many of the same properties as dekaheptoids.

Chaotic

Template:Table alignment

Classification Count Notes and Notable examples Example space-time diagram
Irregular bell 39
  • 1RB0RC_1LC1RA_1RA1LD_0LC0LA (bbch), a typical irregular bell.
  • 1RB1LA_0RC0RD_1LD1RC_1LD0LA (bbch), a typical irregular inverted bell.
Spaghetti 26 This is an informal description for spaghetti-code Turing machines that seem to have no predictable behavior, instead winding back and forth like a spaghetti. Any of these machines could potentially end up proven as one of the regular classifications. Indeed, many translated cyclers start their life out as spaghetti.
  • 1RB0RB_1LC1RC_0RA1LD_1RC0LD (bbch), a typical spaghetti.
  • 1RB0RA_1RC0RD_1LD1LC_1RA0LC (bbch), a spaghetti that seems to converge to a bounce-counter.
  • 1RB1LC_0LA0RD_1LA0LB_1LA1RD (bbch), a spaghetti whose envelope seems to converge to that of a bouncer.
  • 1RB0RC_1LC1LD_1RD1LB_1RA0LB (bbch), a cellular-automaton-like bouncer. The spaghetti nature of this machine is local.
  • 1RB1LC_1LA1RD_1RA0LC_1LB0RD (bbch), a "spaghetti sandwich" -- a spaghetti sandwiched on the left and right by a growing predictable repeating pattern.
Chaotic counter 10

Chaotic counters have slow-growing tapes like counters, but the behavior seems to be chaotic and is as of yet unknown:

  • 1RB0RC_0LD1RC_1LD0RB_0LA1LA (bbch)
  • 1RB1RA_0LC0LD_1LD0RB_0RA1LC (bbch)
  • 1RB0RC_1LA1RC_0LD0RB_0LA1LD (bbch)
  • 1RB0RB_1LC1RA_1LD0LC_0RA0LD (bbch)
  • 1RB1LA_0RC1RC_0LD0RB_0LA1LD (bbch)
  • 1RB1LC_0LC0RD_0LA1LA_0RB1RD (bbch)
  • 1RB0RB_0LC1RA_1LD1LC_0RA0LD (bbch)
  • 1RB1LC_1LD0RB_1RA0LC_0RA0LD (bbch)
  • 1RB1LC_0LA0RB_1RD0LC_1LA0RD (bbch)
  • 1RB1LC_1RD0RB_1LA0LC_0LA0RD (bbch)

Records

Translated cycler preperiod

  • 3x2: 101 (proven): 1RB1LB_0RC0LA_1LC0LA (bbch) (period = 24)
  • 4x2: 119,120,230,102 (current champion): 1RB1LC_0LA1RD_0RB0LC_1LA0RD (bbch) (period = 966,716)
  • 2x4: 293,225,660,896 (current champion): 1RB2LA0RA3LA_1LA1LB3RB1RA (bbch) (period = 483,328)

Translated cycler period

  • 3x2: 92 (proven): 1RB0LA_0RC1LA_1LC0RB (bbch) (preperiod = 0)
  • 4x2: 212,081,736 (current champion): 1RB0LA_0RC1RD_1LD0RB_1LA1RB (bbch) (preperiod = 5,248,647,886)
  • 2x4: 33,209,131 (current champion): 1RB0RA3LB1RB_2LA0LB1RA2RB (bbch) (preperiod = 63,141,841)
Retrieved from "https://wiki.bbchallenge.org/w/index.php?title=Non-halting_Turing_machine&oldid=1169"