Synchronization of Parikh Automata

Synchronizability of automata has been investigated and extended to various models. Here, we investigate notions of synchronizability for Parikh automata, i.e., automata with counters that are checked against a semilinear set at the end of the computation. We consider various notions of synchronizability (or directability) for Parikh automata and show that they give decidable and PSPACE-complete problems. We then show that for deterministic and complete Parikh automata on letters the synchronization problems are NP-complete over a binary alphabet and solvable in polynomial time for a unary alphabet when the dimension is fixed and the semilinear set is encoded in unary. For a binary encoding the problem remains NP- hard even for unary two-state deterministic and complete Parikh automata of dimension one.