A Saturation Method for the Modal Mu-Calculus with Backwards Modalities over Pushdown Systems

We present an algorithm for computing directly the denotation of a modal μcalculus formula χ with backwards modalities over the configuration graph of a pushdown system. Our method gives the first extension of the saturation technique to the full modal μ-calculus with backwards modalities. Finite word automata are used to represent sets of pushdown configurations. Starting from an initial automaton, we perform a series of automaton manipulations which compute the denotation by recursion over the structure of the formula. We introduce notions of under-approximation (soundness) and over-approximation (completeness) that apply to automaton transitions rather than runs. Our algorithm is relatively simple and direct, and avoids an immediate exponential blow up.