Infinite and Bi-infinite Words with Decidable Monadic Theories

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.