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2015•Common Library Network (Der Gemeinsame Bibliotheksverbund)Open access

Infinite and Bi-infinite Words with Decidable Monadic Theories

Dietrich Kuske, Jiamou Liu, Anastasia Moskvina

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Abstract

We study word structures of the form (D,<=,P) where D is either N or Z, <= is a linear ordering on D and P in D is a predicate on D. In particular we show: (a) The set of recursive omega-words with decidable monadic second order theories is Sigma_3-complete. (b) We characterise those sets P subset of Z that yield bi-infinite words (Z,<=,P) with decidable monadic second order theories. (c) We show that such "tame" predicates P exist in every Turing degree. (d) We determine, for P subset of Z, the number of predicates Q subset of Z such that (Z,<=,P) and (Z,<=,Q) are indistinguishable. Through these results we demonstrate similarities and differences between logical properties of infinite and bi-infinite words.

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We study word structures of the form (D,<=,P) where D is either N or Z, <= is a linear ordering on D and P in D is a predicate on D. In particular we show: (a) The set of recursive omega-words with decidable monadic second order theories is Sigma_3-complete. (b) We characterise those sets P subset of Z that yield bi-infinite words (Z,<=,P) with decidable monadic second order theories. (c) We show that such "tame" predicates P exist in every Turing degree. (d) We determine, for P subset of Z, the number of predicates Q subset of Z such that (Z,<=,P) and (Z,<=,Q) are indistinguishable. Through these results we demonstrate similarities and differences between logical properties of infinite and bi-infinite words.

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Available abstract

We study word structures of the form (D,<=,P) where D is either N or Z, <= is a linear ordering on D and P in D is a predicate on D. In particular we show: (a) The set of recursive omega-words with decidable monadic second order theories is Sigma_3-complete. (b) We characterise those sets P subset of Z that yield bi-infinite words (Z,<=,P) with decidable monadic second order theories. (c) We show that such "tame" predicates P exist in every Turing degree. (d) We determine, for P subset of Z, the number of predicates Q subset of Z such that (Z,<=,P) and (Z,<=,Q) are indistinguishable. Through these results we demonstrate similarities and differences between logical properties of infinite and bi-infinite words.

Key concepts: Decidability, Monadic predicate calculus, Mathematics, Predicate (mathematical logic), Combinatorics, Discrete mathematics, Order (exchange), Sigma

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