Towards practical FPRAS for #NFA: Exploiting the Power of Dependence

TL;DR

This paper introduces a new, more practical FPRAS algorithm for counting words accepted by an NFA, significantly reducing the time complexity and making it more feasible for real-world applications.

Contribution

The paper presents a novel FPRAS for #NFA with a substantially improved time complexity, approaching practical implementation.

Findings

New FPRAS with $O(n^2m^3 ext{log}(nm) ext{ε}^{-2} ext{log}( ext{δ}^{-1}))$ complexity

Achieves sub-quadratic complexity relative to membership checks

Marks progress towards practical approximation algorithms for #NFA

Abstract

#NFA refers to the problem of counting the words of length $n$ accepted by a non-deterministic finite automaton. #NFA is #P-hard, and although fully-polynomial-time randomized approximation schemes (FPRAS) exist, they are all impractical. The first FPRAS for #NFA had a running time of $\tilde{O}(n^{17}m^{17}\varepsilon^{-14}\log(\delta^{-1}))$, where $m$ is the number of states in the automaton, $\delta \in (0,1]$ is the confidence parameter, and $\varepsilon > 0$ is the tolerance parameter (typically smaller than $1$). The current best FPRAS achieved a significant improvement in the time complexity relative to the first FPRAS and obtained FPRAS with time complexity $\tilde{O}((n^{10}m^2 + n^6m^3)\varepsilon^{-4}\log^2(\delta^{-1}))$. The complexity of the improved FPRAS is still too intimidating to attempt any practical implementation. In this paper, we pursue the quest for practical FPRAS for #NFA by presenting a new algorithm with a time complexity of $O(n^2m^3\log(nm)\varepsilon^{-2}\log(\delta^{-1}))$. Observe that evaluating whether a word of length $n$ is accepted by an NFA has a time complexity of $O(nm^2)$. Therefore, our proposed FPRAS achieves sub-quadratic complexity with respect to membership checks.