Formula size games for modal logic and $\mu$-calculus

TL;DR

This paper introduces a new formula size game for modal logic and $$-calculus that simplifies previous approaches, enabling proofs of non-elementary succinctness gaps between first-order logic and modal logics.

Contribution

It presents a novel, genuine two-player formula size game for modal logic and $$-calculus, with simpler winning conditions and applications to succinctness gap proofs.

Findings

Established a non-elementary succinctness gap between FO and ML.

Extended the game to the modal $$-calculus showing similar results.

Provided a simpler, genuine two-player game compared to Adler-Immerman.

Abstract

We propose a new version of formula size game for modal logic. The game characterizes the equivalence of pointed Kripke-models up to formulas of given numbers of modal operators and binary connectives. Our game is similar to the well-known Adler-Immerman game. However, due to a crucial difference in the definition of positions of the game, its winning condition is simpler, and the second player does not have a trivial optimal strategy. Thus, unlike the Adler-Immerman game, our game is a genuine two-person game. We illustrate the use of the game by proving a non-elementary succinctness gap between bisimulation invariant first-order logic $\mathrm{FO}$ and (basic) modal logic $\mathrm{ML}$. We also present a version of the game for the modal $\mu$-calculus $\mathrm{L}_\mu$ and show that $\mathrm{FO}$ is also non-elementarily more succinct than $\mathrm{L}_\mu$.