Beeping Busy Beaver

The Beeping Busy Beaver (BBB) function is a variant of the Busy Beaver function proposed by Scott Aaronson in his 2020 Busy Beaver Frontier survey. It is notable because it is uncomputable even if you have access to a halting oracle for Turing Machines. These can simulate problems in Π₂ and Σ₂ of the arithmetic hierarchy, such as the Collatz conjecture and twin prime conjecture.

Definition

Consider a beeping Turing machine, which is a Turing machine that has a special state named "beep state". Every time the TM enters the "beep state" it beeps. There are two possibilities, either this TM beeps a finite number of times (and thus there is a final beep) or it never stops beeping. Nick Drozd coined the term quasihalting to describe the event when a TM last beeps. A TM quasihalts if it beeps only a finite number of times. The Beeping Busy Beaver problem is analogous to the Busy Beaver problem, replacing halting with quasihalting. In other words, let $b(M)$ be the number of steps the machine M takes until it quasihalts (beeps for the last time) if it quasihalts (we will say $b(M)=\mathrm{\infty }$ if the TM never stops beeping). Then

$$\mathrm{BBB}(n):=\underset{M\in T(n):b(M)<\mathrm{\infty }}{\max }b(M)$$

where $T(n)$ is the set of Turing machines with $n$ states and two symbols.

Note that quasihalting does not require halting, so we can consider Turing machines which have no halt transitions. Thus the Tree Normal Form algorithm needs to be modified (to allow TMs with no halt transitions) when searching for BBB champions.

Another way to define BBB is to have Turing machines have a special beep transition instead of an entire beep state.

Growth rate

It is easy to see that $\mathrm{BBB}(n)\ge \mathrm{BB}(n)$, by letting the beep state be the state that is reached immediately before the halt state.

In fact, BBB grows much faster than BB. BBB eventually dominates any computable function augmented with an oracle for computing BB. So, for example, there exists some N such that for all n > N:

• $BBB(n)>BB(BB(n))$
• $BBB(n)>B{B}^n(n)$, where $B{B}^k$ represents k iterations of BB
• $BBB(n)>B{B}^{BB(n)}(n)$

etc.

Champions

This table considers beep states.

Domain	Max Steps	Champions	Notes
BBB(1)	1	1RZ--- (bbch)	Case analysis by Scott Aaronson.
BBB(2)	6	1RB1LB_1LB1LA (bbch)	Case analysis by Scott Aaronson.
BBB(3)	55	1RB0LB_1LA0RC_1LC1LA (bbch)	Claimed as a lower bound by Scott Aaronson, verified with exhaustive search by Nick Drozd.
BBB(4)	32,779,478	1RB1LD_1RC1RB_1LC1LA_0RC0RD (bbch)	Proven in Coq.
BBB(5)	> 1014,006	1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA (bbch)	Best known, per Nick Drozd. Probably >$10^{10^{10^5}}$ due to four probviously quasihalting Cryptids.
BBB(2,3)	59	1RB2LB1LA_2LB2RA0RA (bbch)	Per Nick Drozd.
BBB(3,3)	> 10 ↑↑ 6.74	1RB0LB2LA_1LA0RC0LB_2RC2RB0LC (bbch)	Best known, per Nick Drozd.
BBB(2,4)	> $2\times 10^{23}$	1RB2LA1RA1LB_0LB2RB3RB1LA (bbch)	Best known, per Nick Drozd.

All known champions quasihalt by becoming Translated Cyclers, a property which is known to be weaker than the general quasihalting condition.

Beeping Booping Busy Beavers

A further extension devised by Bram Cohen is as follows: a Turing machine has two special transitions, a beep transition and a boop transition, both of which may repeat infinitely often. The machine "outputs" an integer sequence, indicated by the number of beeps between successive boops. If any such integer is repeated an infinite number of times, the machine is said to pseudoquasihalt. Now if we define $g(M)$ as the number of transitions for machine M to finish outputting that infinitely-repeated value for the first time (or ∞ if there is no such value), the Beeping Booping Busy Beaver number is:

$$\mathrm{BBBB}(n):=\underset{M\in T(n):g(M)<\mathrm{\infty }}{\max }g(M)$$

These machines are equivalent to Turing machines with second-order oracles, and so their pseudoquasihalting behavior can describe sets in Π₃ and Σ₃ of the arithmetic hierarchy.

There are multiple models to choose from, depending on whether one allows multiple beep transitions and multiple boop transitions, or only one of each, or only one boop transition while making every transition a beep transition. All of these models have the same computational strength and corresponding growth rate, but each has its own advantages. The first is the most general, including all machines from the other two, and in that sense it is more natural, but that makes the space of candidate machines to search through significantly larger. The second is the original definition and it is relatively simple. The third is equivalent to simply considering the amount of time between successive boops, instead of the number of beeps between successive boops, and has the smallest space of candidate machines. It is not yet decided which of these models (or perhaps a model different from all three) should be chosen for "the" BBBB function.

BBBB champions

Domain	Value	Champions	Notes
BBBB(1)	2	0RA--- (bbch) (beep: any, boop: A0)	Confirmed.
BBBB(2)	18	0RB1LB_1LA1RB (bbch) (beep: A0, boop: A1)	Confirmed by exhaustive search.
BBBB(3)	≥ 127	1RB1LB_0RC0LA_1LC0LA (bbch) (beep: A0, boop: B1)	Holdouts: 935.
BBBB(2, 3)	≥ 187	2RB2LA1RA_1LB1LA0RB (bbch) (beep: B0, boop: B2)	Holdouts: >2000.

Higher-order Beavers

For higher order functions, using https://katelyndoucette.com/articles/building-the-busy-beaver-ladder definition, we can get BBBBB(1) = 2 and BBBBB(2) ≥ 18.