Abstract
Generalizing the notion of automatic complexity of individual words due to Shallit and Wang, we define the automatic complexity A(E) of an equivalence relation E on a finite set S of words. We prove that the problem of determining whether A(E) equals the number |E| of equivalence classes of E is NP-complete. The problem of determining whether A(E) = |E| + k for a fixed k ≥ 1 is complete for the second level of the Boolean hierarchy for NP, i.e., BH 2-complete. Let L be the language consisting of all words of maximal nondeterministic automatic complexity. We characterize the complexity of infinite subsets of L by showing that they can be co-context-free but not context-free, i.e., L is CFL-immune, but not coCFL-immune. We show that for each e > 0, L e ∉ coCFL, where L e is the set of all words whose deterministic automatic complexity A(x) satisfies A(x) ≥ |x|1/2−e .
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