Model-Checking the Higher-Dimensional Modal mu-Calculus

TL;DR

This paper introduces the higher-dimensional modal mu-calculus, a logic that extends mu-calculus to tuples of states, enabling polynomial-time model checking and broad applications in verification and decision problems.

Contribution

It formalizes the higher-dimensional modal mu-calculus, demonstrates its polynomial-time model checking, and shows its applicability to various decision problems across multiple fields.

Findings

Model checking is polynomial-time for the logic.

The logic can encode all polynomial-time bisimulation-invariant problems.

Applications include process equivalences, automata, parsing, and games.

Abstract

The higher-dimensional modal mu-calculus is an extension of the mu-calculus in which formulas are interpreted in tuples of states of a labeled transition system. Every property that can be expressed in this logic can be checked in polynomial time, and conversely every polynomial-time decidable problem that has a bisimulation-invariant encoding into labeled transition systems can also be defined in the higher-dimensional modal mu-calculus. We exemplify the latter connection by giving several examples of decision problems which reduce to model checking of the higher-dimensional modal mu-calculus for some fixed formulas. This way generic model checking algorithms for the logic can then be used via partial evaluation in order to obtain algorithms for theses problems which may benefit from improvements that are well-established in the field of program verification, namely on-the-fly and symbolic techniques. The aim of this work is to extend such techniques to other fields as well, here exemplarily done for process equivalences, automata theory, parsing, string problems, and games.