Minimal Proof Search for Modal Logic K Model Checking

TL;DR

This paper introduces Minimal Proof Search (MPS), an optimal algorithm for model checking in Multi-agent Modal Logic K, ensuring minimal proof costs and broad applicability to multi-player game verification.

Contribution

The paper presents MPS, a novel effort number based algorithm that guarantees minimal cost proofs in MMLK model checking, with proven optimality and broad relevance.

Findings

MPS finds minimal cost proofs and disproofs in MMLK model checking.

MPS is proven to be optimal among comparable algorithms.

The algorithm relates to heuristic search and game proof strategies.

Abstract

Most modal logics such as S5, LTL, or ATL are extensions of Modal Logic K. While the model checking problems for LTL and to a lesser extent ATL have been very active research areas for the past decades, the model checking problem for the more basic Multi-agent Modal Logic K (MMLK) has important applications as a formal framework for perfect information multi-player games on its own. We present Minimal Proof Search (MPS), an effort number based algorithm solving the model checking problem for MMLK. We prove two important properties for MPS beyond its correctness. The (dis)proof exhibited by MPS is of minimal cost for a general definition of cost, and MPS is an optimal algorithm for finding (dis)proofs of minimal cost. Optimality means that any comparable algorithm either needs to explore a bigger or equal state space than MPS, or is not guaranteed to find a (dis)proof of minimal cost on every input. As such, our work relates to A* and AO* in heuristic search, to Proof Number Search and DFPN+ in two-player games, and to counterexample minimization in software model checking.