The Trichotomy of Regular Property Testing

2025-04-27

AI Summary

This paper studies property testing of regular languages—determining, with sublinear query complexity, whether an input string belongs to a given regular language or is ε-far from it. We establish the first trichotomy theorem for testing complexity of regular languages: based on minimal blocking sequences, all regular languages are strictly partitioned into three disjoint classes, exhibiting optimal query complexities Θ(1), Õ(1/ε), and Θ(1/ε), respectively. Methodologically, we integrate structural analysis of finite automata, combinatorial language theory, and property testing frameworks, introducing a direct classification scheme grounded in automaton construction and sequence reasoning. Our core contribution is a decidable combinatorial characterization for each class, accompanied by efficient decision criteria. This fully determines the query complexity spectrum of regular languages under property testing and resolves a long-standing open classification problem in the field.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Structured Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Property testing is concerned with the design of algorithms making a sublinear number of queries to distinguish whether the input satisfies a given property or is far from having this property. A seminal paper of Alon, Krivelevich, Newman, and Szegedy in 2001 introduced property testing of formal languages: the goal is to determine whether an input word belongs to a given language, or is far from any word in that language. They constructed the first property testing algorithm for the class of all regular languages. This opened a line of work with improved complexity results and applications to streaming algorithms. In this work, we show a trichotomy result: the class of regular languages can be divided into three classes, each associated with an optimal query complexity. Our analysis yields effective characterizations for all three classes using so-called minimal blocking sequences, reasoning directly and combinatorially on automata.

Problem

• Classify regular languages into three complexity classes
• Determine optimal query complexity for each class
• Characterize classes using minimal blocking sequences in automata

Innovation

• Trichotomy classifies regular languages complexity
• Uses minimal blocking sequences analysis
• Direct combinatorial reasoning on automata