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

2025-11-28

AI Summary

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).

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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 ($ exttt{NL}$). Next, we apply a recent breakthrough from R. Williams (2025) on the space efficient simulation of deterministic time to show an unconditional $Ω(frac{n^2}{log^3(n) loglog^2(n)})$ 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 $ exttt{PTIME} subseteq exttt{DSPACE}(n^c)$ for some $c in mathbb{N}$ and $ exttt{PSPACE} = exttt{EXPTIME}$.

Problem

• Strengthen conditional time lower bounds for DFA intersection non-emptiness
• Prove unconditional time complexity lower bound using recent simulation breakthrough
• Explore implications of hardness for fixed polynomial time classes

Innovation

• Strengthen conditional time lower bounds using NL algorithms.
• Apply space-time simulation for unconditional lower bound.
• Explore implications of DFA intersection hardness.