Are Depth-2 Regular Expressions Hard to Intersect?

TL;DR

This paper investigates the computational complexity of testing intersections of depth-2 regular expressions, revealing a specific quadratic lower bound for certain concatenation patterns under SETH, advancing understanding of regex intersection hardness.

Contribution

It classifies the complexity of intersection testing for all depth-2 regular expression types, identifying a unique hard case without Kleene star operators.

Findings

Quadratic lower bound established for a specific regex intersection case

Classification of all depth-2 regex intersection complexities under SETH

Identification of a hard case not involving Kleene star operators

Abstract

We study the basic regular expression intersection testing problem, which asks to determine whether the intersection of the languages of two regular expressions is nonempty. A textbook solution to this problem is to construct the nondeterministic finite automaton that accepts the language of both expressions. This procedure results in a $\Theta(mn)$ running time, where $m$ and $n$ are the sizes of the two expressions, respectively. Following the approach of Backurs and Indyk [FOCS'16] and Bringmann, Gr{\o}nlund, and Larsen [FOCS'17] on regular expression matching and membership testing, we study the complexity of intersection testing for homogeneous regular expressions of bounded depth involving concatenation, OR, Kleene star, and Kleene plus. Specifically, we consider all combinations of types of depth-2 regular expressions and classify the time complexity of intersection testing as either linear or quadratic, assuming SETH. The most interesting result is a quadratic conditional lower bound for testing the intersection of a ''concatenation of +s'' expression with a ''concatenation of ORs'' expression: this is the only hard case that does not involve the Kleene star operator and is not implied by existing lower bounds for the simpler membership testing problem.