Unconditional Time and Space Complexity Lower Bounds for Intersection Non-Emptiness

TL;DR

This paper strengthens known lower bounds for the Intersection Non-Emptiness problem for DFA's, providing both conditional and unconditional complexity bounds, and explores implications for complexity class relationships.

Contribution

It improves existing lower bounds for the problem and applies recent breakthroughs to establish unconditional time complexity limits.

Findings

Unconditional lower bound of rac{n^2}{\u221a{ ext{log}^3(n)} ext{loglog}^2(n)} time.

Strengthened conditional lower bounds based on rac{ ext{NL}}{ ext{NL}} assumptions.

Exploration of complexity class implications if the problem is hard for fixed polynomial classes.

Abstract

We reinvestigate known lower bounds for the Intersection Non-Emptiness Problem for Deterministic Finite Automata (DFA's). We first strengthen conditional time complexity lower bounds from T. Kasai and S. Iwata (1985) which showed that Intersection Non-Emptiness is not solvable more efficiently unless there exist more efficient algorithms for non-deterministic logarithmic space ($\texttt{NL}$). Next, we apply a recent breakthrough from R. Williams (2025) on the space efficient simulation of deterministic time to show an unconditional $\Omega(\frac{n^2}{\log^3(n) \log\log^2(n)})$ time complexity lower bound for Intersection Non-Emptiness. Finally, we consider implications that would follow if Intersection Non-Emptiness for a fixed number of DFA's is computationally hard for a fixed polynomial time complexity class. These implications include $\texttt{PTIME} \subseteq \texttt{DSPACE}(n^c)$ for some $c \in \mathbb{N}$ and $\texttt{PSPACE} = \texttt{EXPTIME}$.