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Derivatives of rational expressions with multiplicity

Abstract : This paper addresses the problem of turning a rational (i.e. regular) expression into a finite automaton. We formalize and generalize the idea of "partial derivatives" introduced in 1995 by V. Antimirov, in order to obtain a construction of an automaton with multiplicity from a rational expression describing a formal power series with coefficients in a semiring. We first define precisely what is such a rational expression with multiplicity and which hypothesis should be put on the semiring of coefficients in order to keep the usual identities. We then define the derivative of such a rational expression as a linear combination of expressions called derived terms and we show that all derivatives of a given expression are generated by a finite set of derived terms, that yields a finite automaton with multiplicity whose behaviour is the series denoted by the expression. We also prove that this automaton is a quotient of the standard (or Glushkov) automaton of the expression. Finally, we propose and discuss some possible modifications to our definition of derivation.
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Contributor : Sylvain Lombardy <>
Submitted on : Tuesday, January 17, 2006 - 4:22:41 PM
Last modification on : Friday, January 8, 2021 - 11:22:05 AM

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Sylvain Lombardy, Jacques Sakarovitch. Derivatives of rational expressions with multiplicity. Theoretical Computer Science, Elsevier, 2004, 332, pp.141-177. ⟨10.1016/j.tcs.2004.10.016⟩. ⟨hal-00017202⟩



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