Are Depth-2 Regular Expressions Hard to Intersect?

2025-07-04

AI Summary

This paper investigates the computational complexity of determining nonemptiness of the intersection of depth-2 regular expressions. We develop a unified analytical framework based on nondeterministic finite automaton (NFA) constructions, systematically covering all structural variants—distinguishing whether Kleene star, concatenation (+), or disjunction (OR) operators are present. Under the Strong Exponential Time Hypothesis (SETH), we establish conditional fine-grained complexity bounds. Our key contributions are: (i) the first Ω(n²) time lower bound for intersection nonemptiness of star-free regular expressions using only + and OR, revealing intrinsic quadratic hardness; and (ii) a complete classification of all depth-2 cases—each is either solvable in linear time or admits a tight quadratic lower bound. This work precisely delineates the “hardest regime” for regular expression intersection, providing a foundational benchmark at the intersection of formal language theory and fine-grained complexity.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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 $Θ(mn)$ 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, Grønlund, 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.

Problem

• Study intersection testing for depth-2 regular expressions
• Classify time complexity as linear or quadratic under SETH
• Prove quadratic lower bound for specific expression combinations

Innovation

• Studies intersection of depth-2 regular expressions
• Classifies complexity as linear or quadratic
• Quadratic lower bound for specific expression types