Scalar and Vectorial mu-calculus with Atoms

TL;DR

This paper extends the modal μ-calculus to structures with atoms, establishing decidability of model checking on orbit-finite structures and exploring expressive limitations influenced by atom structures and syntax choices.

Contribution

It introduces an atom-enriched μ-calculus, analyzes its properties, and clarifies how atom structures and syntax affect its expressive power and decidability.

Findings

Model checking is decidable on orbit-finite structures.

Satisfiability is undecidable in the extended calculus.

Expressive power depends on atom structure and syntax choice.

Abstract

We study an extension of modal $\mu$-calculus to sets with atoms and we study its basic properties. Model checking is decidable on orbit-finite structures, and a correspondence to parity games holds. On the other hand, satisfiability becomes undecidable. We also show expressive limitations of atom-enriched $\mu$-calculi, and explain how their expressive power depends on the structure of atoms used, and on the choice between basic or vectorial syntax.