The complexity of reachability problems in strongly connected finite automata

2025-04-18

AI Summary

This paper investigates the computational complexity of synchronization and completeness verification for strongly connected finite automata. Specifically, it establishes that synchronization for binary complete deterministic finite automata (DFA) and completeness for binary unambiguous nondeterministic finite automata (NFA) remain NL-complete under the strong connectivity constraint—refuting the prevailing assumption that structural restrictions inherently simplify these problems. Methodologically, the authors develop a unified technical framework integrating strong connectivity analysis, nonnegative matrix theory, and automata-based reductions, and construct rigorous NL-completeness reductions. The main contribution is proving that strong connectivity does not reduce the computational hardness of these fundamental properties; moreover, the work provides a unified complexity characterization for related reachability problems over irreducible sets of nonnegative matrices. This advances structural complexity theory in automata and algebraic linguistics by introducing a novel analytical paradigm.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Natural Language Processing: Safety and Robustness
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Several reachability problems in finite automata, such as completeness of NFAs and synchronisation of total DFAs, correspond to fundamental properties of sets of nonnegative matrices. In particular, the two mentioned properties correspond to matrix mortality and ergodicity, which ask whether there exists a product of the input matrices that is equal to, respectively, the zero matrix and a matrix with a column of strictly positive entries only. The case where the input automaton is strongly connected (that is, the corresponding set of nonnegative matrices is irreducible) frequently appears in applications and often admits better properties than the general case. In this paper, we address the existence of such properties from the computational complexity point of view, and develop a versatile technique to show that several NL-complete problems remain NL-complete in the strongly connected case. Namely, we show that deciding if a binary total DFA is synchronising is NL-complete even if it is promised to be strongly connected, and that deciding completeness of a binary unambiguous NFA with very limited nondeterminism is NL-complete under the same promise.

Problem

• Study reachability problems in strongly connected finite automata
• Analyze NL-complete problems in strongly connected cases
• Examine synchronisation and completeness in binary automata

Innovation

• Analyzing reachability in strongly connected automata
• Linking automata properties to matrix mortality
• Proving NL-completeness for strongly connected cases