Beeping Busy Beaver

A Beeping Busy Beaver (BBB) is a concept defined on a beeping Turing machine, which is a Turing machine that has a special state named "beep state". The goal of a BBB is as follows: when starting from a totally blank tape, we want the final beep to happen as late as possible. The phrasing "final beep" means that the machine must beep finitely many times. It was coined by Scott Aaronson in his 2020 "The Busy Beaver Frontier" survey.

Formally, we define the $n$^th Beeping Busy Beaver number as

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

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

Note that these Turing machines need not ever halt, so the Tree Normal Form algorithm needs to be modified (to allow TMs with no halt transitions) when searching for BBB champions.

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.

Significance

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

Values $>\operatorname {BB} (n)$ may occur when the machine emits a beep before entering non-halting behavior.

Moreover, it turns out that $\operatorname {BBB}$ grows uncomputably faster than $\operatorname {BB}$ (as fast as BB augmented with a halting oracle).

Results

• BBB(1) = 1
• BBB(2) = 6
• BBB(3) = 55
• BBB(4) ≥ 32,779,478
• BBB(5) ≥ 10^14,006 and probably $BBB(5)\geq 10^{10^{10^{5}}}$ due to a "probviously" halting Cryptid.

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