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Games, Probability, and the Quantitative μ-Calculus qMμ

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Logic for Programming, Artificial Intelligence, and Reasoning (LPAR 2002)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 2514))

Abstract

The μ-calculus is a powerful tool for specifying and verifying transition systems, including those with demonic (universal) and angelic (existential) choice; its quantitative generalisation qMμ extends that to probabilistic choice. We show for a finite-state system that the straightforward denotational interpretation of the quantitative μ-calculus is equivalent to an operational interpretation given as a turn-based gambling game between two players.

Kozen defined the standard Boolean-typed calculus denotationally; later Stirling gave it an operational interpretation as a turn-based game between two players, and showed the two interpretations equivalent. By doing the same for the quantitative real-typed calculus, we set it on a par with the standard calculus, in that it too can benefit from a solid interface linking the logical and operational frameworks.

Stirling’s game analogy, as an aid to intuition, continues in the more general context to provide a surprisingly practical specification tool, meeting for example Vardi’s challenge to “figure out the meaning of AF AXp” as a branching-time formula.

We also show that memoriless strategies suffice for achieving the minimax value of a quantitative game, when the state space is finite.

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Author information

Authors and Affiliations

  1. Dept. Computer Science, Macquarie University, NSW, 2109, Australia

    A. K. McIver

  2. Dept. Comp. Sci. & Eng., University of New South Wales, NSW, 2052, Australia

    C. C. Morgan

Authors
  1. A. K. McIver
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  2. C. C. Morgan
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Editor information

Editors and Affiliations

  1. Abteilung für Theoretische Informatik, Institut für Algebra und Diskrete Mathematik, Wiedner Hauptstr. 8-10, 1040, Wien, Austria

    Matthias Baaz

  2. Department of Computer Science, University of Manchester, Kilburn Building, Oxford Road, Manchester, M13, 9PL, UK

    Andrei Voronkov

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© 2002 Springer-Verlag Berlin Heidelberg

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McIver, A.K., Morgan, C.C. (2002). Games, Probability, and the Quantitative μ-Calculus qMμ . In: Baaz, M., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2002. Lecture Notes in Computer Science(), vol 2514. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36078-6_20

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  • DOI: https://doi.org/10.1007/3-540-36078-6_20

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-00010-5

  • Online ISBN: 978-3-540-36078-0

  • eBook Packages: Springer Book Archive

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