Given some integer k, intersection emptiness of k Nondeterministic Finite Automata (NFA k-IE) is a fundamental problem in automata theory, with applications across Computer science from Arithmetic theories and model checking to graph database queries. In this talk, I will discuss some results regarding the fine-grained complexity of NFA k-IE. Informally, what I mean by fine-grained complexity is that we want to know (upto logarithmic factors) the runtime complexity of our algorithms, and argue why they cannot be improved by relating it to the state of the art in solving long-standing hard problems like SAT.

I will demonstrate (what we believe to be) a new algorithm for solving NFA k-IE. If all the NFAs have n states and m transitions, our algorithm runs in time O(n^{k-1}m), compared to the O(m^k) runtime of the classic Cartesian product approach. I will also present a matching lower bound subject to the Combinatorial k-Clique hypothesis, and a barrier to tight SETH-based lower bounds. This is joint work with Dmitry Chistikov, at the University of Warwick, UK.