We observe that the classical Cartesian product construction for the intersection of (languages of) nondeterministic finite automata (NFA) is non-optimal in the worst case, if the automata have many transitions. For a fixed alphabet, the product of two NFA may have $Θ(m^2)$ transitions if these NFA have at most $n$ states and $m$ transitions each. We describe alternative constructions with $O(m n)$ transitions: or $O(m n^{k-1})$ for the intersection of $k$ NFA (for fixed $k \ge 2$ and alphabet $Σ$). This gives a faster algorithm for deciding NFA intersection emptiness. The new algorithm is optimal, unless there exists a breakthrough combinatorial algorithm for detecting $(k+1)$-cliques in undirected graphs. This also leads to a more efficient certification scheme for NFA intersection emptiness.

翻译：我们观察到，对于具有多条转移的非确定性有限自动机（NFA）的语言交集，经典笛卡尔积构造在最坏情况下并非最优。对于固定字母表，当两个NFA各有至多$n$个状态和$m$条转移时，其乘积可能包含$Θ(m^2)$条转移。我们描述了具有$O(m n)$条转移的替代构造；对于k个NFA的交集（固定$k \ge 2$和字母表$Σ$），该构造具有$O(m n^{k-1})$条转移。这为判定NFA交集非空性提供了更快的算法。新算法是最优的，除非存在检测无向图中$(k+1)$-团问题的突破性组合算法。此外，该结果还导致了一种更高效的NFA交集非空性认证方案。