On the size of disjunctive formulas in the $\mu$-calculus

TL;DR

This paper investigates the normalization of mu-calculus formulas into disjunctive form, showing that an integrated automata-theoretic approach can achieve single-exponential size bounds, improving over previous doubly exponential methods.

Contribution

The authors present an integrated automata-based normalization procedure that reduces the size of disjunctive formulas to single-exponential, improving efficiency over existing methods.

Findings

The integrated approach achieves single-exponential size bounds.

Previous methods were doubly exponential in size.

The new procedure simplifies the normalization process.

Abstract

A key result in the theory of the modal mu-calculus is the disjunctive normal form theorem by Janin & Walukiewicz, stating that every mu-calculus formula is semantically equivalent to a so-called disjunctive formula. These disjunctive formulas have good computational properties and play a pivotal role in the theory of the modal mu-calculus. It is therefore an interesting question what the best normalisation procedure is for rewriting a formula into an equivalent disjunctive formula of minimal size. The best constructions that are known from the literature are automata-theoretic in nature and consist of a guarded transformation, i.e., the constructing of an equivalent guarded alternating automaton from a mu-calculus formula, followed by a Simulation Theorem stating that any such alternating automaton can be transformed into an equivalent non-deterministic one. Both of these transformations are exponential constructions, making the best normalisation procedure doubly exponential. Our key contribution presented here shows that the two parts of the normalisation procedure can be integrated, leading to a procedure that is single-exponential in the closure size of the formula.