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Valuations of Weighted Automata: Doing It in a Rational Way

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Algebraic Foundations in Computer Science

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7020))

Abstract

We study Kleene’s theorem about the equivalence of automata and expressions in a quantitative setting both for finite and infinite words. The quantities originate from valuation monoids and ω-indexed valuation monoids which cover not only semirings but also cost models like average cost, long-run peaks of resource consumption, or discounting sums of rewards. For finite words we deduce the characterization of weighted automata by regular weighted expressions directly from Kleene’s theorem. For infinite words we define three different behaviors of weighted Büchi automata depending on the way runs are evaluated. Depending on the properties of the underlying ω-indexed valuation monoid, we explore the connections between the different behaviors of weighted Büchi automata and ω-regular weighted expressions. Again, we use classical results on ω-languages to derive results in the quantitative setting.

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Author information

Authors and Affiliations

  1. Institut für Informatik, Universität Leipzig, D-04109, Leipzig, Germany

    Ingmar Meinecke

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  1. Ingmar Meinecke
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Editors and Affiliations

  1. Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Austria

    Werner Kuich

  2. Department of Mathematics, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece

    George Rahonis

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Meinecke, I. (2011). Valuations of Weighted Automata: Doing It in a Rational Way. In: Kuich, W., Rahonis, G. (eds) Algebraic Foundations in Computer Science. Lecture Notes in Computer Science, vol 7020. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24897-9_14

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  • DOI: https://doi.org/10.1007/978-3-642-24897-9_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-24896-2

  • Online ISBN: 978-3-642-24897-9

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