Correctness of an Incremental and Worst-Case Optimal Decision Procedure for Modal Logic with Eventualities

TL;DR

This paper introduces a simple, incremental decision procedure for modal logic with eventualities, which is both correct and worst-case optimal, and is inspired by and related to existing PDL provers.

Contribution

It provides a new, straightforward theoretical framework for an incremental, optimal decision procedure that can be practically implemented for modal logic with eventualities.

Findings

The procedure is both incremental and worst-case optimal.

It effectively determines satisfiability using small tableaux.

The approach is inspired by and extends PDL prover techniques.

Abstract

We present a simple theory explaining the construction and the correctness of an incremental and worst-case optimal decision procedure for modal logic with eventualities. The procedure gives an abstract account of important aspects of Gor\'e and Widmann's PDL prover. Starting from an input formula, the procedure grows a Pratt-style graph tableau until the tableau proves or disproves the satisfiability of the formula. The procedure provides a basis for practical provers since satisfiability and unsatisfiability of formulas can often be determined with small tableaux.