Infinite and Bi-infinite Words with Decidable Monadic Theories

TL;DR

This paper investigates the logical complexity of infinite and bi-infinite words with decidable monadic theories, revealing their classification, transferability of properties, existence across Turing degrees, and structural indistinguishability.

Contribution

It extends known characterizations of decidable monadic theories from infinite to bi-infinite words and analyzes their structural and computational properties.

Findings

Set of recursive ω-words with decidable monadic theories is Σ₃-complete.

Characterizations of ω-words with decidable theories transfer to bi-infinite words.

Such predicates exist in every Turing degree.

Abstract

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 $\Sigma_3$-complete. (b) Known characterisations of the $\omega$-words with decidable monadic second order theories are transfered to the corresponding question for bi-infinite words. (c) 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.