Games for Succinctness of Regular Expressions

TL;DR

This paper introduces formula size games for regular expressions to analyze their succinctness, establishing hierarchies and succinctness gaps compared to first-order logic and star-free expressions.

Contribution

It develops a new game-based framework to measure regular expression complexity and demonstrates hierarchies and non-elementary succinctness gaps.

Findings

Established a hierarchy based on the number of stars in expressions

Provided a simple proof of the non-elementary succinctness gap between first-order logic and regular expressions

Introduced a measure counting only the number of stars, leading to new insights into expressive power

Abstract

We present a version of so called formula size games for regular expressions. These games characterize the equivalence of languages up to expressions of a given size. We use the regular expression size game to give a simple proof of a known non-elementary succinctness gap between first-order logic and regular expressions. We also use the game to only count the number of stars in an expression instead of the overall size. For regular expressions this measure trivially gives a hierarchy in terms of expressive power. We obtain such a hierarchy also for what we call RE over star-free expressions, where star-free expressions, that is ones with complement but no stars, are combined using the operations of regular expressions.