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Hardness of Busy Beaver Value BB(15)

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Reachability Problems (RP 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 15050))

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Abstract

The busy beaver value BB(n) is the maximum number of steps made by any n-state, 2-symbol deterministic halting Turing machine starting on blank tape. The busy beaver function \(n \mapsto \text {BB}(n)\) is uncomputable and, from below, only 4 of its values, \(\text {BB}(1)\, \dots \, \text {BB}(4)\), are known to date. This leads one to ask: from above, what is the smallest BB value that encodes a major mathematical challenge? Knowing BB(4,888) has been shown by Yedidia and Aaronson [32] to be at least as hard as solving Goldbach’s conjecture, with a subsequent improvement, as yet unpublished, to BB(27) [2, 6]. We prove that knowing BB(15) is at least as hard as solving the following Collatz-related conjecture by Erdős, open since 1979 [10]: for all \(n>8\) there is at least one digit 2 in the base 3 representation of \(2^n\). We do so by constructing an explicit 15-state, 2-symbol Turing machine that halts if and only if the conjecture is false. This 2-symbol Turing machine simulates a conceptually simpler 5-state, 4-symbol machine which we construct first. This makes, to date, BB(15) the smallest busy beaver value that is related to a natural open problem in mathematics, bringing to light one of the many challenges underlying the quest of knowing busy beaver values.

T. Stérin and D. Woods—Hamilton Institute and Department of Computer Science, Maynooth University, Ireland. Research supported by European Research Council (ERC, grant agreement No 772766, Active-DNA project) under the European Union’s Horizon 2020 research and innovation programme, European Innovation Council (EIC, DISCO, No 101115422); and Science Foundation Ireland (SFI) under Grant number 18/ERCS/5746. Research sponsored by prgm.dev.

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Notes

  1. 1.

    Radó originally used the notation S(n). The more modern [2] notation \(\textrm{BB}(n)\) can be a source of confusion since some authors, for example [12, 14], use BB to mean Radó’s \(\varSigma \) which counts the number of 1 s on the final tape of a halting Turing machine [24].

  2. 2.

    Trivial values include those where \(n=1\) since for all \(k \ge 1\), \(\text {BB}(1,k) = 1\).

  3. 3.

      Current busy beaver champion [19] for machines with 5 states (A–E) and 2 symbols (0,1), it halts in \(\text {47,176,870}\) steps starting from all-0 input and state \({\textbf {A}}\), which gives the lowerbound \(\text {BB}(5) \ge \text {47,176,870}\). Proving that bound tight is the goal of the bbchallenge project [1]. Is the 5-state, 2 symbol (0,1) busy beaver. Marxen [19] showed it halts in \(\text {47,176,870}\) steps starting from all-0 input and state \({\textbf {A}}\), and the bbchallenge collective [1] proved that no machine halts in more steps, i.e. that \(\text {BB}(5) = \text {47,176,870}\).

  4. 4.

    The situation can be likened to the hunt for small, and fast, universal Turing machines [31], or simple models like Post tag systems, with earlier literature often missing proofs of correctness, but some later papers using induction on machine configurations to do the job, for example refs [23, 30].

  5. 5.

    Simulator and machines available here: https://github.com/tcosmo/bbsim.

  6. 6.

    Such an effort has been done for 5-state Turing machines [1].

  7. 7.

    These machines are also available here: https://github.com/tcosmo/bbsim.

  8. 8.

    Said otherwise: given current knowledge there are no reason to believe that the set of counterexamples to the Collatz conjecture is recursively enumerable.

  9. 9.

    As in [2], we took the liberty of not defining a symbol to write and a direction to move the tape head to when the halting instruction is performed as this does not change the number of transitions that the machine executed.

  10. 10.

    State \(\textbf{mul2}\_\textbf{G}\) was not designed to kick-start the process, but starting with it happens to give us what we need.

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  1. Hamilton Institute and Department of Computer Science, Maynooth University, Maynooth, Ireland

    Tristan Stérin & Damien Woods

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  1. TU Wien, Vienna, Austria

    Laura Kovács

  2. Paris Lodron University of Salzburg, Salzburg, Austria

    Ana Sokolova

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Stérin, T., Woods, D. (2024). Hardness of Busy Beaver Value BB(15). In: Kovács, L., Sokolova, A. (eds) Reachability Problems. RP 2024. Lecture Notes in Computer Science, vol 15050. Springer, Cham. https://doi.org/10.1007/978-3-031-72621-7_9

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