Intersecting Dense Automata

TL;DR

This paper introduces more efficient algorithms for intersecting nondeterministic finite automata, reducing transition complexity from quadratic to linear in many cases, and improves decision procedures for NFA intersection emptiness.

Contribution

It presents alternative constructions for NFA intersection with lower transition complexity and establishes the optimality of these algorithms under certain complexity assumptions.

Findings

New constructions with O(m n) transitions for NFA intersection

Faster algorithm for deciding NFA intersection emptiness

Optimality of the algorithm unless a major graph problem breakthrough occurs

Abstract

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 $\Theta(m^2)$ 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 $\Sigma$). 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.