Regular expression length via arithmetic formula complexity

TL;DR

This paper establishes new lower bounds on the length of regular expressions for finite languages by leveraging techniques from arithmetic circuit complexity, connecting regular expression size to arithmetic formula complexity.

Contribution

It introduces a novel reduction from regular expression length to arithmetic formula size and adapts lower bound methods to regular expressions, providing tight bounds for specific languages.

Findings

Lower bounds for regular expressions of binomial and Dyck languages.

Analysis of language operation blow-ups (intersection and shuffle).

Almost tight bounds for languages of binary numbers divisible by p, and permutations.

Abstract

We prove lower bounds on the length of regular expressions for finite languages by methods from arithmetic circuit complexity. First, we show a reduction: the length of a regular expression for a language $L\subseteq \{0,1\}^n$ is bounded from below by the minimum size of a monotone arithmetic formula computing a polynomial that has $L$ as its set of exponent vectors, viewing words as vectors. This result yields lower bounds for the binomial language of all words with exactly $k$ ones and $n-k$ zeros and for the language of all Dyck words of length $2n$. We also determine the blow-up of language operations (intersection and shuffle) of regular expressions for finite languages. Second, we adapt a lower bound method for multilinear arithmetic formulas by so-called log-product polynomials to regular expressions. With this method we show almost tight lower bounds for the language of all binary numbers with $n$ bits that are divisible by a given odd integer $p$, for the language of all words of length $n$ over a $k$ letter alphabet with an even number of occurrences of each letter and for the language of all permutations of $\{1,\dots, n\}$.