Infinite and Bi-infinite Words with Decidable Monadic Theories

We study word structures of the form $(D,<,P)$ where $D$ is either $\mathbb{N}$ or $\mathbb{Z}$, $<$ is the natural linear ordering on $D$ and $P\subseteq D$ is a predicate on $D$. In particular we show: (a) The set of recursive $\omega$-words with decidable monadic second order theories is ${\mathrm{\Sigma }}_3$-complete. (b) Known characterisations of the $\omega$-words with decidable monadic second order theories are transfered to the corresponding question for bi-infinite words. © We show that such “tame” predicates $P$ exist in every Turing degree. (d) We determine, for $P\subseteq \mathbb{Z}$, the number of predicates $Q\subseteq \mathbb{Z}$ such that $(\mathbb{Z},\le ,P)$ and $(\mathbb{Z},\le ,Q)$ are indistinguishable. Through these results we demonstrate similarities and differences between logical properties of infinite and bi-infinite words.