The logical strength of Büchi's decidability theorem release_3s6rnfri4nbbpk7sv2uxy7dayq

by Leszek Kołodziejczyk, Henryk Michalewski, Pierre Pradic, Michał Skrzypczak

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2018  

Abstract

We study the strength of axioms needed to prove various results related to automata on infinite words and Büchi's theorem on the decidability of the MSO theory of (N, <). We prove that the following are equivalent over the weak second-order arithmetic theory RCA_0: - the induction scheme for Σ^0_2 formulae of arithmetic, - a variant of Ramsey's Theorem for pairs restricted to so-called additive colourings, - Büchi's complementation theorem for nondeterministic automata on infinite words, - the decidability of the depth-n fragment of the MSO theory of (N, <), for each n > 5. Moreover, each of (1)-(4) implies McNaughton's determinisation theorem for automata on infinite words, as well as the "bounded-width" version of König's Lemma, often used in proofs of McNaughton's theorem.
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Date   2018-12-13
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arXiv  1608.07514v3
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