Fined-Grained Complexity of Ambiguity Problems on Automata and Directed Graphs

2025-01-24

AI Summary

This work investigates the fine-grained complexity of ambiguity detection for finite automata and directed graphs, addressing unambiguity, finite ambiguity, polynomial ambiguity, and efficient twins-property verification in weighted automata. Methodologically, it leverages fine-grained complexity assumptions and integrates tropical algebra, ring-weighted analysis, and graph traversal techniques; it further uncovers a novel connection between the ℓ¹-norm of GCD matrices and ambiguity classification. Contributions include: (i) the first tight lower bounds—quadratic for unambiguity and polynomial ambiguity, cubic for finite ambiguity—under standard hypotheses; (ii) matching upper bounds, proving quadratic and cubic time-optimal algorithms for the respective problems; and (iii) near-linear (unambiguous) and linear-time optimal algorithms for unary automata across all ambiguity classes and the twins property. The results achieve simultaneous theoretical depth and algorithmic efficiency, advancing both complexity-theoretic understanding and practical verification methods.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• 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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• 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

Two fundamental classes of finite automata are deterministic and nondeterministic ones (DFAs and NFAs). Natural intermediate classes arise from bounds on an NFA's allowed ambiguity, i.e. number of accepting runs per word: unambiguous, finitely ambiguous, and polynomially ambiguous finite automata. It is known that deciding whether a given NFA is unambiguous and whether it is polynomially ambiguous is possible in quadratic time, and deciding finite ambiguity is possible in cubic time. We provide matching lower bounds showing these running times to be optimal, assuming popular fine-grained complexity hypotheses. We improve the upper bounds for unary automata, which are essentially directed graphs with a source and a target. In this view, unambiguity asks whether all walks from the source to the target have different lengths. The running time analysis of our algorithm reduces to bounding the entry-wise 1-norm of a GCD matrix, yielding a near-linear upper bound. For finite and polynomial ambiguity, we provide simple linear-time algorithms in the unary case. Finally, we study the twins property for weighted automata over the tropical semiring, which characterises the determinisability of unambiguous weighted automata. It occurs naturally in our context as deciding the twins property is an intermediate step in determinisability algorithms for weighted automata with bounded ambiguity. We show that Allauzen and Mohri's quadratic-time algorithm checking the twins property is optimal up to the same fine-grained hypotheses as for unambiguity. For unary automata, we show that the problem can be rephrased to whether all cycles in a weighted directed graph have the same average weight and give a linear-time algorithm.

Problem

• Finite Automata
• Confusion Properties
• Weighted Automata Twin Property

Innovation

• Finite Automata
• Confusion Degree Evaluation
• Weighted Automata Twin Property