Synchronizing weighted automata

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 Knxn is called (location-) synchronizing if M generates a matrix subsemigroup containing a (location-) synchronizingmatrix. The K-(location) synchronizability problem is the following: given a finite setM 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).