Towards Finite Satisfiability of PDL with Loops

Abstract

Propositional Dynamic Logic (PDL) is a well-established modal logic of programs. Among its many extensions, the loop operator stands out for capturing the cyclic behaviour of programs: a world satisfies loop(pi) precisely when it can return to itself along a path matching the regular expression pi. While the satisfiability problem for the resulting logic LPDL was settled already in the 1980s, the decidability of its finite satisfiability problem has remained open ever since. The core difficulty is inherent to the finite setting: requiring and forbidding cycles at once is easily reconciled over arbitrary models, but over finite ones it defeats the existing finite-model-theoretic techniques. We make the first substantial progress towards closing this gap. We prove that finite satisfiability is ExpTime-complete for LPDL formulae (also in the presence of the converse operator) in negation normal form, under the restriction that every negated non-atomic loop has the form loop(pi^+) for some arbitrarily complex program pi -- a generalised form of self-reachability; positive occurrences of loops remain unrestricted. At the heart of our approach lies a novel way of representing intended finite models as their infinite, cactus-like unravellings: these are recognisable by two-way tree automata and satisfy a tailored omega-regular foldability condition that lets us finitise the accepted structures.