The Complexity of Enriched Mu-Calculi

TL;DR

This paper investigates the decidability and complexity of various fragments of the enriched mu-calculus, identifying maximal decidable subsets and introducing new automata models to analyze satisfiability.

Contribution

It introduces two new automata models, 2GAPTs and FEAs, and proves their ExpTime-completeness, enabling decidability results for fragments of the enriched mu-calculus.

Findings

Decidability and ExpTime-completeness of all fragments obtained by dropping at least one construct.

Introduction of two automata models: 2GAPTs and FEAs.

Reduction of satisfiability problems to automata emptiness problems.

Abstract

The fully enriched μ-calculus is the extension of the propositional μ-calculus with inverse programs, graded modalities, and nominals. While satisfiability in several expressive fragments of the fully enriched μ-calculus is known to be decidable and ExpTime-complete, it has recently been proved that the full calculus is undecidable. In this paper, we study the fragments of the fully enriched μ-calculus that are obtained by dropping at least one of the additional constructs. We show that, in all fragments obtained in this way, satisfiability is decidable and ExpTime-complete. Thus, we identify a family of decidable logics that are maximal (and incomparable) in expressive power. Our results are obtained by introducing two new automata models, showing that their emptiness problems are ExpTime-complete, and then reducing satisfiability in the relevant logics to these problems. The automata models we introduce are two-way graded alternating parity automata over infinite trees (2GAPTs) and fully enriched automata (FEAs) over infinite forests. The former are a common generalization of two incomparable automata models from the literature. The latter extend alternating automata in a similar way as the fully enriched μ-calculus extends the standard μ-calculus.