Enriched µ-Calculus Pushdown Module Checking

The model checking problem for open systems (called module checking) has been intensively studied in the literature, both for finite–state and infinite–state systems. In this paper, we focus on pushdown module checking with respect to decidable fragments of the fully enriched μ–calculus. We recall that finite–state module checking with respect to fully enriched μ–calculus is undecidable and hence the extension of this problem to pushdown systems remains undecidable as well. On the contrary, for the fragments of the fully enriched μ–calculus we consider here, we show that pushdown module checking is decidable and solvable in double–exponential time in the size of the formula and in exponential time in the size of the system. This result is obtained by exploiting a classical automata–theoretic approach via pushdown nondeterministic parity tree automata. In particular, we reduce in exponential time our problem to the emptiness problem for these automata, which is known to be decidable in Exptime. As a key step of our algorithm, we show an exponential improvement of the construction of a nondeterministic parity tree automaton accepting all models of a formula of the considered logic. This result, does not only allow our algorithm to match the known lower bound, but also to investigate decision problems related to the fragments of the enriched μ-calculus in a greatly simplified manner.