We consider two natural problems about nondeterministic finite automata (NFA). First, given an NFA M of n states, and a length ℓ, does M accept a word of length ℓ? We show that the classic problem of triangle-free graph recognition reduces to this problem, and give an $O(n^{\omega }{(\log n)}^{1+ϵ}\log \ell )$-time algorithm to solve it, where ω is the optimal exponent for matrix multiplication. Second, provided $L(M)$ is finite, we consider the problem of listing the lengths of all words accepted by M. Although this problem seems like it might be significantly harder, we show that in the unary case this problem can be solved in $O(n^{\omega }{(\log n)}^{2+ϵ})$ time. Finally, we give a connection between NFA acceptance and the strong exponential-time hypothesis.

我们考虑了关于非确定性有限自动机（NFA）的两个自然问题。首先，给定一个NFA中号的ñ状态，长度ℓ，不中号接受长度的字ℓ？我们证明了无三角图识别的经典问题可以简化为该问题，并给出$Ø（{ñ}^{\omega }{（\mathrm{日志}ñ）}^{\mathrm{1个}+ϵ}\mathrm{日志}\ell ）$时间算法求解，其中ω是矩阵乘法的最佳指数。二，提供$\mathrm{大号}（\mathrm{中号}）$是有限的，我们考虑列出M接受的所有单词的长度的问题。尽管这个问题似乎要困难得多，但我们表明，在一元情况下，可以通过以下方法解决此问题：$Ø（{ñ}^{\omega }{（\mathrm{日志}ñ）}^{2+ϵ}）$时间。最后，我们将NFA接受度与强大的指数时间假设联系起来。