Clausal Resolution for Modal Logics of Confluence

TL;DR

This paper introduces a resolution-based method for multimodal logics of confluence, providing a systematic way to handle inference rules, ensuring soundness, completeness, and termination, with practical examples.

Contribution

The paper develops a modular, resolution-based approach for multimodal logics of confluence, linking inference rules to frame conditions and demonstrating soundness and completeness.

Findings

Method is sound and complete for eight families of logics.

Inference rules are systematically derived from axioms.

Examples illustrate practical application of the method.

Abstract

We present a clausal resolution-based method for normal multimodal logics of confluence, whose Kripke semantics are based on frames characterised by appropriate instances of the Church-Rosser property. Here we restrict attention to eight families of such logics. We show how the inference rules related to the normal logics of confluence can be systematically obtained from the parametrised axioms that characterise such systems. We discuss soundness, completeness, and termination of the method. In particular, completeness can be modularly proved by showing that the conclusions of each newly added inference rule ensures that the corresponding conditions on frames hold. Some examples are given in order to illustrate the use of the method.