Semirings and Formal Power Series (original) (raw)

Abstract

This chapter presents basic foundations for the theory of weighted automata: semirings and formal power series. A fundamental question is how to extend the star operation (Kleene iteration) from languages to series. For this, we investigate ordered, complete and continuous semirings and the related concepts of star semirings and Conway semirings. We derive natural properties for the Kleene star of cycle-free series and also of matrices often used to analyze the behavior of weighted automata. Finally, we investigate cycle-free linear equations which provide a useful tool for proving identities for formal power series.

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Authors and Affiliations

  1. Institut für Informatik, Universität Leipzig, 04009, Leipzig, Germany
    Manfred Droste
  2. Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, 1040, Wien, Austria
    Werner Kuich

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  1. Manfred Droste
  2. Werner Kuich

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Correspondence toManfred Droste .

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Editors and Affiliations

  1. Inst. Informatik, Universität Leipzig, Augustusplatz 10-11, Leipzig, 04109, Germany
    Manfred Droste
  2. Institut für Diskrete, TU Wien, Wiedner Hauptstr. 8-10, Wien, 1040, Austria
    Werner Kuich
  3. Fak. Informatik, TU Dresden, Nöthnitzer Str. 46, Dresden, 01187, Germany
    Heiko Vogler

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Droste, M., Kuich, W. (2009). Semirings and Formal Power Series. In: Droste, M., Kuich, W., Vogler, H. (eds) Handbook of Weighted Automata. Monographs in Theoretical Computer Science. An EATCS Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01492-5\_1

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