On the size of disjunctive formulas in the μ-calculus

Kupke, Clemens and Marti, Johannes and Venema, Yde (2021) On the size of disjunctive formulas in the μ-calculus. Preprint / Working Paper. arXiv.org, Ithaca, NY. (https://doi.org/10.4204/EPTCS.346.19)

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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.

  • Item type:Monograph(Preprint / Working Paper)
    ID code:81848
    Dates:
    Date
    Event
    17 September 2021
    Published
    17 September 2021
    Submitted
    Notes:Paper included in the conference proceedings, as available via arXiv.org: Proceedings 12th International Symposium on Games, Automata, Logics, and Formal Verification (GandALF 2021) - https://arxiv.org/abs/2109.07798
    Subjects:Science > Mathematics
    Department:Faculty of Science > Computer and Information Sciences
    Depositing user: Pure Administrator
    Date deposited:11 Aug 2022 16:02
    Last modified:11 Nov 2024 16:07
    Related URLs:
    URI:https://strathprints.strath.ac.uk/id/eprint/81848
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