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

TL;DR

This paper characterizes the fine-grained complexity of ambiguity problems in automata, showing optimality of existing algorithms under certain hypotheses and providing efficient solutions for unary alphabets.

Contribution

It provides a nearly complete complexity classification for ambiguity problems in automata and confirms the optimality of existing algorithms under standard hypotheses.

Findings

Quadratic and cubic algorithms are optimal under the Orthogonal Vectors and k-Cycle hypotheses.

Ambiguity problems are decidable in almost linear time for unary alphabets.

Verifying determinisability of unambiguous weighted automata is optimally quadratic, with linear time for unary alphabets.

Abstract

In the field of computational logic, two classes of finite automata are considered fundamental: deterministic and nondeterministic automata (DFAs and NFAs). In a more fine-grained approach three natural intermediate classes were introduced, defined by restricting the number of accepting runs of the input NFA. The classes are called: unambiguous, finitely ambiguous, and polynomially ambiguous finite automata. It was observed that central problems, like equivalence, become tractable when the input NFA is restricted to some of these classes. This naturally brought interest into problems determining whether an input NFA belongs to the intermediate classes. Our first result is a nearly complete characterization of the fine-grained complexity of these problems. We show that the respective quadratic and cubic running times of Allauzen et al. are optimal under the Orthogonal Vectors hypothesis or the k-Cycle hypothesis, for alphabets with at least two symbols. In contrast, for unary alphabets we show that all aforementioned variants of ambiguity can be decided in almost linear time. Finally, we study determinisability of unambiguous weighted automata. We positively resolve a conjecture of Allauzen and Mohri, proving that their quadratic-time algorithm for verifying determinisability of unambiguous weighted automata is optimal, assuming the Orthogonal Vectors hypothesis or the k-Cycle hypothesis. We additionally show that for unary alphabets, this can be decided in linear time.