Stronger bounds on the degree of ambiguity of finite automata

2026-10-06

AI Summary

This study addresses a longstanding theoretical gap in estimating the upper bound on the maximum number of accepting paths in nondeterministic finite automata (NFAs). Specifically, the exponential upper bound of $2^{O(n \log n)}$ established by Weber and Seidl for finitely ambiguous NFAs is not optimal. Drawing upon formal language theory and combinatorial analysis of automata, this work provides a refined characterization of the path growth mechanism in finitely ambiguous NFAs and systematically reconstructs the derivation framework for ambiguity upper bounds. The proposed approach reduces the ambiguity upper bound from $2^{O(n \log n)}$ to an asymptotically tight $2^{O(n)}$, while proving that ambiguity grows linearly with the number of states. By establishing the optimal exponential upper bound for finite automaton ambiguity, this research provides a complete theoretical foundation for complexity analysis in this domain.

Abstract

Ambiguity measures the number of accepting runs in nondeterministic finite automata (NFA). We consider finitely ambiguous NFA, where there exists a constant $N$ such that over every word $w$ there are at most $N$ accepting runs. In such a case we also say that the NFA is $N$-ambiguous. Importantly $N$ depends only on the NFA, it does not depend on the length of the word. Weber and Seidl showed that every NFA is $N$-ambiguous for $N = 2^{O(n \log n)}$, where $n$ is the number of states. We improve this to $N = 2^{O(n)}$, which is asymptotically tight.

Problem

• finite automata
• ambiguity degree
• nondeterministic finite automata
• finitely ambiguous NFA
• upper bound

Innovation

• finite automata
• ambiguity
• nondeterministic finite automata
• asymptotically tight bound
• accepting runs