The Validity of Weighted Automata

This paper addresses the problem of the validity of weighted automata in which the presence of ε-circuits results in infinite summations. Earlier works either rule out such automata or characterise the semirings in which these infinite sums are all well-defined. By means of a topological approach, we take here a definition of validity that is strong enough to insure that in any kind of semirings, any closure algorithm will succeed on every valid weighted automaton and turn it into an equivalent proper automaton. This definition is stable with respect to natural transformations of automata. The classical closure algorithms, in particular algorithms based on the computation of the star of the matrix of ε-transitions, can not be used to decide validity. This decision problem remains open for general topological semirings. We present a closure algorithm that yields a decision procedure for the validity of automata in the case where the weights are taken in Q or R, two cases that had never been treated before. These algoritm and procedure are implemented in the Vaucanson platform. ∗Work supported by ANR Project 10-INTB-0203 VAUCANSON 2. †A preliminary version has been presented at the CIAA 2012 conference, under the title The removal of weighted ε-transitions [25]. ‡Published in the International Journal of Algebra and Computation, vol. 23 (4), 2013. DOI: 10.1142/S0218196713400146. §LaBRI, IPB / Université de Bordeaux. ¶LTCI, CNRS / Telecom ParisTech.