Turing machines deciders, part I

The bbchallenge Collaboration; Justin Blanchard; Konrad Deka; Nathan Fenner; Tony Guilfoyle; Iijil; Maja Kądziołka; Pavel Kropitz; Shawn Ligocki; Pascal Michel; Mateusz Naściszewski; Tristan Stérin

arXiv · 2504.20563

Turing machines deciders, part I

The bbchallenge Collaboration·Justin Blanchard·Konrad Deka·Nathan Fenner·Tony Guilfoyle·Iijil·Maja Kądziołka·Pavel Kropitz·Shawn Ligocki·Pascal Michel·Mateusz Naściszewski·Tristan Stérin

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Abstract

The Busy Beaver Challenge (or bbchallenge) aims at collaboratively solving the following conjecture: "$S(5) = 47{,}176{,}870$" [Rad\'o, 1962], [Marxen and Buntrock, 1990], [Aaronson, 2020]. This conjecture says that if a 5-state Turing machine runs for more than 47,176,870 steps without halting, then it will never halt -- starting from the all-0 tape. Proving this conjecture amounts to deciding whether 181,385,789 Turing machines with 5 states halt or not -- starting from the all-0 tape [bbchallenge, 2025]. To do so, we write $\textit{deciders}$: programs that take as input a Turing machine and output either HALT, NONHALT, or UNKNOWN. Each decider is specialised in recognising a particular type of non-halting behavior. After two years of work, the Busy Beaver Challenge achieved its goal in July 2024 by delivering a proof of "$S(5) = 47{,}176{,}870$" formalised in Coq [bbchallenge, 2025]. In this document, we present deciders that were developed before the Coq proof and which were mainly not used in the proof; nonetheless, they are relevant techniques for analysing Turing machines. Part II of this work is the decider section of our paper showing "$S(5) = 47{,}176{,}870$" [bbchallenge, 2025], presenting the deciders that were used in the Coq proof.

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The bbchallenge Collaboration, Justin Blanchard, Konrad Deka, Nathan Fenner, Tony Guilfoyle, Iijil, Maja Kądziołka, Pavel Kropitz, Shawn Ligocki, Pascal Michel, Mateusz Naściszewski, Tristan Stérin. 2025-04-29. Turing machines deciders, part I. https://arxiv.org/abs/2504.20563

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