Backwards-reachability for cooperating multi-pushdown systems

A cooperating multi-pushdown system consists of a tuple of pushdown systems that can delegate the execution of recursive procedures to sub-tuples; control returns to the calling tuple once all sub-tuples finished their task. This allows the concurrent execution since disjoint sub-tuples can perform their task independently. Because of the concrete form of recursive descent into sub-tuples, the content of the multi-pushdown does not form an arbitrary tuple of words, but can be understood as a Mazurkiewicz trace. For such systems, we prove that the backwards reachability relation efficiently preserves recognizability, generalizing a result and proof technique by Bouajjani et al. for single-pushdown systems. It follows that the reachability relation is decidable for cooperating multi-pushdown systems in polynomial time and the same holds, e.g., for safety and liveness properties given by recognizable sets of configurations.