Model checking synchronized products of infinite transition systems

Formal verification using the model-checking paradigm has to deal with two aspects. The systems models are structured, often as products of components, and the specification logic has to be expressive enough to allow the formalization of reachability properties. The present paper is a study on what can be achieved for infinite transition systems under these premises. As models, we consider products of infinite transition systems with different synchronization constraints. We introduce finitely synchronized transition systems, i.e. product systems which contain only finitely many synchronized transitions, and show that the decidability of FO(R), first-order logic extended by reachability predicates, of the product system can be reduced to the decidability of FO(R) of the components in a Feferman-Vaught like style. This result is optimal in the following sense. (1) If we allow semifinite synchronization, i.e. just in one component infinitely many transitions are synchronized, the FO(R)-theory of the product system is in general undecidable. (2) We cannot extend the expressive power of the logic under consideration. Already a weak extension of first-order logic with transitive closure, where we restrict the transitive closure operators to arity one and nesting depth two, is undecidable for an asynchronous (and hence finitely synchronized) product, namely for the infinite grid.