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

Fabian Lehr; Florian Bruse

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

Logic in Computer Science 2026-07-09 v1 Formal Languages and Automata Theory

Authors: Fabian Lehr , Florian Bruse

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).

Keywords

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

Cite

@article{arxiv.2607.08181,
  title  = {Finite Convergence of the Modal Mu-Calculus on Almost-Periodic Words},
  author = {Fabian Lehr and Florian Bruse},
  journal= {arXiv preprint arXiv:2607.08181},
  year   = {2026}
}