Tableaux for type PDL

Abstract

The system of Type PDL (τPDL) is an extension of Propositional Dynamic Logic (PDL) and its main goal is to provide a formal basis for reasoning about types of actions (modeled by their preconditions and effects) and agent capabilities. The system has two equivalent interpretations, namely the standard relational semantics and the type semantics, where process terms are interpreted as types, i.e. sets of binary relations. Its satisfiability problem is decidable, as a NE T decision procedure was provided based on a filtration argument and it was suggested that the satisfiability problem for τPDL should be solvable in deterministic, single exponential time. In this paper, we address the problem of the complexity of the satisfiability problem of τPDL. We present a deterministic tableau-based satisfiability algorithm and prove that it is sound and complete and that it runs in E T . Additionally, the algorithm detects satisfiability as earlier as possible, by restricting or-branching whenever possible. IntroductionThe satisfiability problem of Propositional Dynamic Logic (PDL) [FL79, HKT00]) is E T -complete. Various decision procedures have been given such as best-case exponential [Pra79], working in multiple stages [Pra80, NS09], on-the-fly [GM00, AGW09] and with analytic cut-rule for the converse [GM00, NS09], most of them based on and-or tableaux [Gor14]. The algorithm of [GW09] for PDL and its extension for the converse (CPDL) [GW10] reveal the crucial role that global caching plays to the achievement of an optimal algorithm.The system of τPDL (Type PDL) [Har13, Har14] is an extension of PDL. The main goal of such a system is to provide a formal basis for reasoning about types of actions instead of merely actions. As described in [Har13], the system is inspired by the OWL-S process model of services [MBH + 04], where concrete actions are hidden from public view and what is publicly known is only the type of actions a service can perform, modeled by their preconditions and effects. Syntactically, the language of τPDL extends that of PDL with abstract process types designated by their preconditions and effects, written in the form ϕ ⇒ ψ, with agent capabilities statements C ı A which declare that agent ı can do A and with a backwards possibility operator, a weak form of converse.The interpretation of the introduced system of τPDL in [Har13] is different from the standard PDL semantics. As the main motivation of τPDL is reasoning about types of processes, type semantics has been proposed where process terms are interpreted as types, i.e. sets of binary relations. On the other hand, in [Har14], it was shown that the standard relational semantics is equivalent to the type semantics.As far as the capabilities operator is concerned, the semantic options made in [Har13, Har14] are slightly different from those made in the KARO Framework [vdHvLM94,vdHvLM99,vdHW03] and it is that difference in semantics, as detailed in [Har13, Har14] that allows for a finitary axiomatization and a decidability proof, provided in [Har13, Har...