ExpTime Tableaux for Type PDL

TL;DR

This paper introduces a deterministic tableau-based algorithm for the Type PDL logic, proving its ExpTime complexity and enhancing satisfiability detection efficiency.

Contribution

It provides the first ExpTime tableau algorithm for Type PDL, ensuring soundness, completeness, and early satisfiability detection.

Findings

The algorithm runs in ExpTime.

It is sound and complete.

Early detection of satisfiability is achieved.

Abstract

The system of Type PDL ($\tau$PDL) 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 NExpTime decision procedure was provided based on a filtration argument and it was suggested that the satisfiability problem for $\tau$PDL should be solvable in deterministic, single exponential time. In this paper, we address the problem of the complexity of the satisfiability problem of $\tau$PDL. We present a deterministic tableau-based satisfiability algorithm and prove that it is sound and complete and that it runs in ExpTime. Additionally, the algorithm detects satisfiability as earlier as possible, by restricting or-branching whenever possible.