Hardness Results for Intersection Non-Emptiness

Abstract

We carefully reexamine a construction of Karakostas, Lipton, and Viglas (2003) to show that the intersection non-emptiness problem for DFA’s (deterministic finite automata) characterizes the complexity class NL. In particular, if restricted to a binary work tape alphabet, then there exist constants c ₁ and c ₂ such that for every k intersection non-emptiness for k DFA’s is solvable in c ₁ k log(n) space, but is not solvable in c ₂ k log(n) space. We optimize the construction to show that for an arbitrary number of DFA’s intersection non-emptiness is not solvable in $o(\frac{n}{\log(n)\log(\log(n))})$ space. Furthermore, if there exists a function f(k) = o(k) such that for every k intersection non-emptiness for k DFA’s is solvable in n ^f(k) time, then P ≠ NL. If there does not exist a constant c such that for every k intersection non-emptiness for k DFA’s is solvable in n ^c time, then P does not contain any space complexity class larger than NL.