Finite Convergence of the Modal Mu-Calculus on Almost-Periodic Words

2026-07-09

AI Summary

This study addresses the finite convergence problem of the modal μ-calculus over infinite words. By integrating automata theory, the analysis of almost periodic sequences from combinatorics on words, and the semantics of the modal μ-calculus, it establishes for the first time that an infinite word satisfies finite convergence if and only if it is almost periodic. This characterization precisely delineates the semantic boundary under which the modal μ-calculus exhibits finite convergence over infinite structures. Building on this result, the paper provides a novel proof of Semenov’s (1984) decidability theorem for this logic, thereby highlighting the central role of almost periodicity in the decidability analysis of fixed-point logics.

Abstract

A formula of the modal mu-calculus enjoys finite convergence on a structure if there is some finite unfolding of the formula that defines the same set. A structure enjoys finite convergence if all formulas of the mu-calculus enjoy finite convergence on said structure. It is known that there are words that are not ultimately periodic, but have finite convergence. An almost-periodic word w is one in which each finite word v either appears only finitely often, or within each factor of some length that only depends only on w and v. It is immediate that words that have finite convergence must be almost periodic. In this paper we show the converse, namely that all almost-periodic words have finite convergence. This characterizes finite convergence on infinite words, and also re-proves a decidability result due to Semenov ('84).

Problem

• modal mu-calculus
• finite convergence
• almost-periodic words
• infinite words
• decidability

Innovation

• modal mu-calculus
• finite convergence
• almost-periodic words
• decidability
• infinite words