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

#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 Õ(n 17 m 17 ε -14 łog(δ -1 )), where m is the number of states in the automaton, δ ∈ (0,1] is the confidence parameter, and ε > 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 Õ((n 10 m 2 + n 6 m 3 )ε -4 łog 2 (δ -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 2 m 3 łog(nm)ε -2 łog(δ -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.