Theory of Computing Systems

, Volume 61, Issue 4, pp 1427–1439 | Cite as

On the Complexity of Automatic Complexity

  • Bjørn Kjos-Hanssen
Part of the following topical collections:
  1. Special Issue on Computability, Complexity and Randomness (CCR 2015)


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 .


Automatic 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.


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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Hawaii at ManoaHonoluluUSA

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