Unconditional Time and Space Complexity Lower Bounds for Intersection Non-Emptiness
This paper investigates the computational complexity of the DFA intersection non-emptiness problem. We establish the first unconditional time lower bound of Ω(n²/log³n loglog²n), breaking prior conditional lower bounds that relied on unproven hypotheses, and derive tight space lower bounds. Technically, our approach combines nondeterministic logspace reductions with Williams’ (2025) deterministic time–space-efficient simulation framework, while strengthening the intrinsic connection between time–space trade-offs in simulation. Our main contributions are: (1) the first unconditional quadratic-time lower bound for this problem; and (2) a structural result showing that if DFA intersection non-emptiness is not solvable in fixed-polynomial time, then major complexity class collapses follow—including PTIME ⊆ DSPACE(nᶜ) for some constant c and PSPACE = EXPTIME—thereby deepening the foundational links between automata theory and central complexity classes (P, PSPACE, EXPTIME).