Unconditional Time and Space Complexity Lower Bounds for Intersection Non-Emptiness
Michael Wehar
TL;DR
The paper investigates the Intersection Non-Emptiness problem for DFAs (DFA-INT), revisiting conditional lower bounds and proving unconditional space/time lower bounds by leveraging recent space-time simulations. It strengthens Kasai–Iwata’s conditional bounds, establishes an unconditional subquadratic-time barrier via space-to-time simulations, and proposes a hardness hypothesis whose consequences would collapse key complexity classes (e.g., PTIME ⊆ DSPACE(n^β) and PSPACE = EXPTIME). The work links automata-theoretic intersection problems to broader structural complexity results, offering a framework to pursue unconditional lower bounds for other PSPACE-hard problems. It also outlines how advances in space-efficient simulations could translate into sharper time lower bounds if the hardness hypothesis holds for fixed-k DFA-INT. Overall, the results illuminate deep connections between time, space, and hardness in fundamental decision problems.
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 $Ω(\frac{n^2}{\log^3(n) \log\log^2(n)})$ 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 $\texttt{PSPACE} = \texttt{EXPTIME}$.