Decidable first-order transition logics for PA-processes

We show the decidability of model checking PA-processes against several first-order logics based upon the reachability predicate. The main tool for this result is the recognizability by tree automata of the reachability relation. The tree automata approach and the transition logics we use allow a smooth and general treatment of parameterized model checking for PA. This approach is extended to handle a quite general notion of costs of PA-steps. In particular, when costs are Parikh images of traces, we show decidability of a transition logic extended by some form of first-order reasoning over costs.