Synchronizing weighted automata

Szabolcs Iván
(University of Szeged, Hungary)

We introduce two generalizations of synchronizability to automata with transitions weighted in an arbitrary semiring K=(K,+,*,0,1). (or equivalently, to finite sets of matrices in K^nxn.) Let us call a matrix A location-synchronizing if there exists a column in A consisting of nonzero entries such that all the other columns of A are filled by zeros. If additionally all the entries of this designated column are the same, we call A synchronizing. Note that these notions coincide for stochastic matrices and also in the Boolean semiring. A set M of matrices in K^nxn is called (location-)synchronizing if M generates a matrix subsemigroup containing a (location-)synchronizing matrix. The K-(location-)synchronizability problem is the following: given a finite set M of nxn matrices with entries in K, is it (location-)synchronizing?

Both problems are PSPACE-hard for any nontrivial semiring. We give sufficient conditions for the semiring K when the problems are PSPACE-complete and show several undecidability results as well, e.g. synchronizability is undecidable if 1 has infinite order in (K,+,0) or when the free semigroup on two generators can be embedded into (K,*,1).

In Zoltán Ésik and Zoltán Fülöp: Proceedings 14th International Conference on Automata and Formal Languages (AFL 2014), Szeged, Hungary, May 27-29, 2014, Electronic Proceedings in Theoretical Computer Science 151, pp. 301–313.
Published: 21st May 2014.

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