On the size of disjunctive formulas in the μ-calculus

Clemens Kupke
(University of Strathclyde, Scotland)
Johannes Marti
(ILLC, University of Amsterdam, The Netherlands)
Yde Venema
(ILLC, University of Amsterdam, The Netherlands)

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.

In Pierre Ganty and Davide Bresolin: Proceedings 12th International Symposium on Games, Automata, Logics, and Formal Verification (GandALF 2021), Padua, Italy, 20-22 September 2021, Electronic Proceedings in Theoretical Computer Science 346, pp. 291–307.
Published: 17th September 2021.

ArXived at: http://dx.doi.org/10.4204/EPTCS.346.19 bibtex PDF
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