Locality Testing for NFAs is PSPACE-complete

2025-11-10

AI Summary

This paper investigates the locality decision problem for nondeterministic finite automata (NFAs). We establish that NFA locality is PSPACE-complete: first, we prove PSPACE-hardness via a polynomial-time reduction from regular expression equivalence; second, we present a PSPACE upper-bound algorithm, thereby completing the complexity characterization. In contrast, the analogous problem for deterministic finite automata (DFAs) is solvable in polynomial time, highlighting a fundamental complexity jump induced by nondeterminism. This work fills a longstanding gap in the complexity landscape of structural decision problems for formal languages—specifically, the computational complexity of NFA locality—and provides a critical theoretical benchmark for automata-based language classification and verification.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Local Search
• Natural Language Processing: Safety and Robustness

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed 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
• Security and Privacy: Large-scale security measurements

Abstract

The class of local languages is a well-known subclass of the regular languages that admits many equivalent characterizations. In this short note we establish the PSPACE-completeness of the problem of determining, given as input a nondeterministic finite automaton (NFA) A, whether the language recognized by A is local or not. This contrasts with the case of deterministic finite automata (DFA), for which the problem is known to be in PTIME.

Problem

• Testing NFA locality is PSPACE-complete
• Determining if NFA recognizes local language is hard
• DFA locality testing is easier than NFA testing

Innovation

• Proves NFA locality testing is PSPACE-complete
• Establishes computational complexity for automata analysis
• Contrasts with PTIME complexity for DFAs