On the Complexity of Automatic Complexity
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 ε > 0, L ε ∉ coCFL, where L ε is the set of all words whose deterministic automatic complexity A(x) satisfies A(x) ≥ |x|1/2−ε .
KeywordsAutomatic complexity Context-free languages Computational complexity NP-completeness
This work was partially supported by a grant from the Simons Foundation (#315188 to Bjørn Kjos-Hanssen). This material is based upon work supported by the National Science Foundation under Grant No. 1545707.
- 1.Allender, E.: The complexity of complexity. In: Computability and Complexity Symposium in honor of Rodney G. Downey’s 60th Birthday, volume 10010 of Lecture Notes in Computer Science, pp. 79–94. Springer (2017)Google Scholar
- 11.Sipser, M.: Instructor’s Solutions Manual: Introduction to the Theory of Computation. Cengage Learning 3rd edition (2013)Google Scholar
- 13.Wechsung, G.: On the Boolean closure of NP. In: Fundamentals of computation theory (Cottbus, 1985), volume 199 of Lecture Notes in Computer Science, pp. 485–493. Springer, Berlin (1985)Google Scholar