Refinement Modal Logic

TL;DR

Refinement modal logic introduces a new operator 'all' to quantify over model refinements, combining modal logic with second-order features, with applications in verification and epistemic logic.

Contribution

The paper develops a new refinement modal logic with an 'all' operator, providing axiomatizations, extensions to mu-calculus, and complexity analysis.

Findings

Sound and complete axiomatization of multi-agent refinement modal logic

Extension to the modal mu-calculus with axiomatization

Complexity results for satisfiability and succinctness

Abstract

In this paper we present {\em refinement modal logic}. A refinement is like a bisimulation, except that from the three relational requirements only `atoms' and `back' need to be satisfied. Our logic contains a new operator 'all' in addition to the standard modalities 'box' for each agent. The operator 'all' acts as a quantifier over the set of all refinements of a given model. As a variation on a bisimulation quantifier, this refinement operator or refinement quantifier 'all' can be seen as quantifying over a variable not occurring in the formula bound by it. The logic combines the simplicity of multi-agent modal logic with some powers of monadic second-order quantification. We present a sound and complete axiomatization of multi-agent refinement modal logic. We also present an extension of the logic to the modal mu-calculus, and an axiomatization for the single-agent version of this logic. Examples and applications are also discussed: to software verification and design (the set of agents can also be seen as a set of actions), and to dynamic epistemic logic. We further give detailed results on the complexity of satisfiability, and on succinctness.