Rational and Recognisable Power Series

  • Jacques Sakarovitch
Part of the Monographs in Theoretical Computer Science. An EATCS Series book series (EATCS)


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’.


Rational Series Formal Power Series Division Ring Proper Part Word Base 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.LTCI, ENST/CNRSParis Cedex 13France

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