The strength of the tree theorem for pairs in reverse mathematics

International audienceNo natural principle is currently known to be strictly between the arithmetic comprehension axiom (ACA0) and Ramsey's theorem for pairs (RT 2 2) in reverse mathematics. The tree theorem for pairs (TT 2 2) is however a good candidate. The tree theorem states that for every finite coloring over tuples of comparable nodes in the full binary tree, there is a monochromatic subtree isomorphic to the full tree. The principle TT 2 2 is known to lie between ACA0 and RT 2 2 over RCA0, but its exact strength remains open. In this paper, we prove that RT 2 2 together with weak König's lemma (WKL0) does not imply TT 2 2 , thereby answering a question of Montálban. This separation is a case in point of the method of Lerman, Solomon and Towsner for designing a computability-theoretic property which discriminates between two statements in reverse mathematics. We therefore put the emphasis on the different steps leading to this separation in order to serve as a tutorial for separating principles in reverse mathematics

The logical strength of Büchi's decidability theorem

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 RCA0: (1) the induction scheme for Σ02 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

Π 0 1 ENCODABILITY AND OMNISCIENT REDUCTIONS

International audienceA set of integers A is computably encodable if every infinite set of integers has an infinite subset computing A. By a result of Solovay, the computably encodable sets are exactly the hyperarithmetic ones. In this paper, we extend this notion of computable encodability to subsets of the Baire space and we characterize the Π01 encodable compact sets as those who admit a non-empty Σ11 subset. Thanks to this equivalence, we prove that weak weak König's lemma is not strongly computably reducible to Ramsey's theorem. This answers a question of Hirschfeldt and Jockusch

The logical strength of Büchi's decidability theorem

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 RCA0: (1) the induction scheme for Σ02 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