Synchronizing weighted automata

We introduce two generalizations of synchronizability to automata with transitions weighted in an arbitrary semiring K = (K,+, ·,0,1). For states p,q and word u, let (pu)q ∈ K denote the sum of the weights of all u-labeled paths from p to q, with the weigth of a path being the product of the weights of its edges, as usual. We call the automaton A location-synchronizable if ∃q,u: ∀p,r (pu)r 6= 0 iff r = q and synchronizable if ∃q,u,k 6= 0: ∀p,r (pu)q = k and (pu)r = 0 for each r 6= q. Note that these notions coincide for stochastic automata and also in the Boolean semiring. Both problems are PSPACE-hard for any nontrivial semiring, and in any semiring, the length of the shortest (location) synchronizing word can be exponential. 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).