The logical strength of Büchi's decidability theorem
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by
Leszek Kołodziejczyk, Henryk Michalewski, Pierre Pradic, Michał
Skrzypczak
2019
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:
(1) the induction scheme for Σ^0_2 formulae of arithmetic,
(2) a variant of Ramsey's Theorem for pairs restricted to so-called additive
colourings,
(3) Büchi's complementation theorem for nondeterministic automata on
infinite words,
(4) 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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