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Rational and Recognisable Power Series

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Handbook of Weighted Automata

Abstract

This chapter presents the theory of weighted automata over graded monoids and with weights taken in arbitrary semirings. The first benefit of broadening the scope beyond free monoids is that it makes clearer the distinction between the rational and the recognisable series. As the topological machinery is set anyway, the star of series is defined in a slightly more general setting than cycle-free series. The main subjects covered in the chapter are then: the notion of covering of automata (also called bisimulation by some authors) and its relationship with the conjugacy of automata; the closure of recognisable series by Hadamard and shuffle products; the derivation of weighted rational expressions over a free monoid; the reduction theory of series over a free monoid and with weights in a (skew) field, that leads to a procedure for the decidability of equivalence (with a cubic complexity); and the basics for a theory of weighted rational relations. As a result, this chapter, among other things, lays the bases for the proof of the decidability of the equivalence of deterministic k-tape transducers which is one of the most striking examples of the application of algebra to ‘machine theory’.

This chapter is adapted from Chaps. III and IV of the book Elements of Automata Theory, Jacques Sakarovitch, 2009, ©Cambridge University Press, where missing proofs, detailed examples and further developments can be found.

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Authors and Affiliations

  1. LTCI, ENST/CNRS, 46 rue Barrault, 75634, Paris Cedex 13, France

    Jacques Sakarovitch

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  1. Jacques Sakarovitch
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Correspondence to Jacques Sakarovitch .

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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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Sakarovitch, J. (2009). Rational and Recognisable 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_4

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  • DOI: https://doi.org/10.1007/978-3-642-01492-5_4

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