# Existential length universality

## TL;DR

This paper investigates the computational complexity of the existential length universality problem across different automata models, revealing complexity classifications and bounds for DFAs, NFAs, and PDAs, and how input encoding affects difficulty.

## Contribution

It provides the first complexity classifications for the existential length universality problem for DFA, NFA, and PDA, including tight bounds and the impact of binary input encoding.

## Key findings

- NEXPTIME-complete for NFA, with doubly exponential smallest length
- Recursively unsolvable for PDA, with unbounded smallest length
- NP-complete for DFA, with tight exponential bounds

## Abstract

We study the following natural variation on the classical universality problem: given a language $L(M)$ represented by $M$ (e.g., a DFA/RE/NFA/PDA), does there exist an integer $\ell \geq 0$ such that $\Sigma^\ell \subseteq L(M)$? In the case of an NFA, we show that this problem is NEXPTIME-complete, and the smallest such $\ell$ can be doubly exponential in the number of states. This particular case was formulated as an open problem in 2009, and our solution uses a novel and involved construction. In the case of a PDA, we show that it is recursively unsolvable, while the smallest such $\ell$ is not bounded by any computable function of the number of states. In the case of a DFA, we show that the problem is NP-complete, and $e^{\sqrt{n \log n} (1+o(1))}$ is an asymptotically tight upper bound for the smallest such $\ell$, where $n$ is the number of states. Finally, we prove that in all these cases, the problem becomes computationally easier when the length $\ell$ is also given in binary in the input: it is polynomially solvable for a DFA, PSPACE-complete for an NFA, and co-NEXPTIME-complete for a PDA.