Tableaux with Partial Caching for Hybrid PDL with Satisfaction Statements

Abstract

We give a novel deterministic tableau-based satisfiability algorithm for Hybrid Propositional Dynamic Logic (i.e. PDL with nominals) with satisfaction statements ($\texttt {HPDL}_@$). It builds and-or graphs in which it detects unfulfilled eventualities and unifies nodes (due to nominals) on-the-fly. There are two kinds of nodes: sentential nodes that represent partial descriptions of worlds of a model and unification nodes that deal with nominals. The main technical achievement of this work is the determination of the necessary information that a sentential node should have so that caching is feasible. Each saturated sentential node is available for reuse until it becomes out of date, due to loop dependencies. Thus, the algorithm runs in double exponential time. However, for iteration-free formulas, loops do not occur and thus, it works in exponential time. Nevertheless, despite the iteration operator, thanks to partial caching, the algorithm has the potential to achieve acceptable performance.