A model-theoretic characterization of monadic second order logic on infinite words

TL;DR

This paper provides a model-theoretic characterization of monadic second order logic on infinite words, linking it to algebraic structures and linear temporal logic, and leveraging classical automata theory results.

Contribution

It establishes that the first-order theory of a certain structure related to S1S is the model companion of algebras associated with LTL without until, connecting logic, automata, and algebra.

Findings

First-order theory of the structure is the model companion of LTL-related algebras.

Uses completeness of LTL over natural numbers.

Relates S1S formulas to Büchi automata.

Abstract

Monadic second order logic and linear temporal logic are two logical formalisms that can be used to describe classes of infinite words, i.e., first-order models based on the natural numbers with order, successor, and finitely many unary predicate symbols. Monadic second order logic over infinite words (S1S) can alternatively be described as a first-order logic interpreted in $\mathcal{P}(\omega)$, the power set Boolean algebra of the natural numbers, equipped with modal operators for 'initial', 'next' and 'future' states. We prove that the first-order theory of this structure is the model companion of a class of algebras corresponding to the appropriate version of linear temporal logic (LTL) without until. The proof makes crucial use of two classical, non-trivial results from the literature, namely the completeness of LTL with respect to the natural numbers, and the correspondence between S1S-formulas and B\"uchi automata.