Lower Bounds on Regular Expression Size

TL;DR

This paper develops a linear programming framework to establish lower bounds on the size of regular expressions for finite languages, providing new insights into their complexity and limitations.

Contribution

It introduces a linear programming approach to bound regular expression size and applies it to binomial and threshold languages for the first time.

Findings

Linear programs encode regular expressions of finite languages.

The optimal value provides a lower bound on regular expression size.

Applied to binomial and threshold languages, the method yields new lower bounds.

Abstract

We introduce linear programs encoding regular expressions of finite languages. We show that, given a language, the optimum value of the associated linear program is a lower bound on the size of any regular expression of the language. Moreover we show that any regular expression can be turned into a dual feasible solution with an objective value that is equal to the size of the regular expression. For binomial languages we can relax the associated linear program using duality theorem. We use this relaxation to prove lower bounds on the size of regular expressions of binomial and threshold languages.