Infinite and Bi-infinite Words with Decidable Monadic Theories

12/11/2017

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by   Dietrich Kuske, et al.

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We study word structures of the form (D,<,P) where D is either N or Z, < is the natural linear ordering on D and P⊆ D is a predicate on D. In particular we show: (a) The set of recursive ω-words with decidable monadic second order theories is Σ_3-complete. (b) Known characterisations of the ω-words with decidable monadic second order theories are transfered to the corresponding question for bi-infinite words. (c) We show that such "tame" predicates P exist in every Turing degree. (d) We determine, for P⊆Z, the number of predicates Q⊆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.