On The Space Complexity of Partial Derivatives of Regular Expressions with Shuffle

2025-08-24

AI Summary

This paper investigates the space complexity of Brzozowski derivatives for regular expressions extended with the shuffle operator in runtime verification, focusing on the maximum height and size of individual derivatives—not their total count. While prior work established that the number of derivatives grows linearly with the input regex, the worst-case complexity of individual derivatives under shuffle remained unknown. Method: We extend Antimirov’s partial derivative framework by integrating algebraic rewriting rules and the formal semantics of the shuffle operator, avoiding explicit construction of large NFAs. Contribution/Results: We establish the first tight upper bounds: shuffle increases derivative height by at most one and causes derivative size to grow quadratically in the size of the original expression. Our approach maintains linear height and quadratic size complexity while substantially reducing memory overhead during monitoring—enabling more compact property specifications and efficient real-time verification.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Heuristic Search

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
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Partial derivatives of regular expressions, introduced by Antimirov, define an elegant algorithm for generating equivalent non-deterministic finite automata (NFA) with a limited number of states. Here we focus on runtime verification (RV) of simple properties expressible with regular expressions. In this case, words are finite traces of monitorable events forming the language's alphabet, and the generated NFA may have an intractable number of states. This typically occurs when sub-traces of mutually independent events are allowed to interleave. To address this issue, regular expressions used for RV are extended with the shuffle operator to make specifications more compact and easier to read. Exploiting partial derivatives enables a rewriting-based approach to RV, where only one derivative is stored at each step, avoiding the construction of an intractably large automaton. This raises the question of the space complexity of the largest generated partial derivative. While the total number of generated partial derivatives is known to be linear in the size of the initial regular expression, no results can be found in the literature regarding the size of the largest partial derivative. We study this problem w.r.t. two metrics (height and size of regular expressions), and show that the former increases by at most one, while the latter is quadratic in the size of the regular expression. Surprisingly, these results also hold with shuffle.

Problem

• Analyzing space complexity of partial derivatives for regular expressions
• Studying the size growth of derivatives with shuffle operator
• Determining bounds on largest partial derivative in rewriting-based RV

Innovation

• Partial derivatives for NFA generation
• Shuffle operator for compact specifications
• Rewriting-based approach avoiding large automata