Existential length universality

We study the following natural variation on the classical universality problem: given an automaton M of some type (DFA/NFA/PDA), does there exist an integer l ≥ 0 such that Σl ⊆ L(M)? The case of an NFA was an open problem since 2009. Here, using a novel and deep construction, we prove that the problem is NEXPTIME-complete, and the smallest such l can be doubly exponential in the number of states. In the case of a DFA the problem is NP-complete, and there exist examples for which the smallest such l is of the form e √ n logn(1+o(1)), which is best possible, where n is the number of states. In the case of a PDA this problem is recursively unsolvable, while the smallest such l cannot be bounded in the number of states by any computable function. Finally, we show that in all the cases the problem becomes computationally easier when the length l is also given in the input.