The Relative Succinctness and Expressiveness of Modal Logics Can Be Arbitrarily Complex

TL;DR

This paper investigates the complexity of relationships between modal logics, demonstrating they can be arbitrarily complex in terms of succinctness and expressiveness using formal tools like formula size games and bisimulations.

Contribution

It introduces a framework showing the potential complexity of modal logic relationships and applies formal methods to analyze these properties.

Findings

Relationships can be as complex as any countable partial order

Uses two formal systems to analyze modal operators

Employs formula size games and bisimulations in proofs

Abstract

We study the relative succinctness and expressiveness of modal logics, and prove that these relationships can be as complex as any countable partial order. For this, we use two uniform formalisms to define modal operators, and obtain results on succinctness and expressiveness in these two settings. Our proofs are based on formula size games introduced by Adler and Immerman and bisimulations.