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Theory of Computing Systems

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

On the Complexity of Automatic Complexity

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

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 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 .

Keywords

Automatic complexity Context-free languages Computational complexity NP-completeness 

Notes

Acknowledgements

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.

References

  1. 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
  2. 2.
    Angluin, D.: On the complexity of minimum inference of regular sets. Inform. Control 39(3), 337–350 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bar-Hillel, Y., Perles, M., Shamir, E.: On formal properties of simple phrase structure grammars. Z. Phonetik Sprachwiss Kommunikat. 14, 143–172 (1961)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Berstel, J.: Every iterated morphism yields a co-CFL. Inform. Process. Lett. 22(1), 7–9 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cai, J.-Y., Gundermann, T., Hartmanis, J., Hemachandra, L.A., Sewelson, V., Wagner, K., Wechsung, G.: The Boolean hierarchy. I. Structural properties. SIAM J. Comput. 17(6), 1232–1252 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Downey, R.G., Hirschfeldt, D.R.: Algorithmic randomness and complexity. Theory and Applications of Computability. Springer, New York (2010)CrossRefzbMATHGoogle Scholar
  7. 7.
    Fernau, H., Heggernes, P., Villanger, Y.: A multi-parameter analysis of hard problems on deterministic finite automata. J Comput. System Sci. 81(4), 747–765 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Mark Gold, E.: Complexity of automaton identification from given data. Inform. Control 37(3), 302–320 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hyde, K.K., Kjos-Hanssen, B.: Nondeterministic automatic complexity of overlap-free and almost square-free words. Electron. J. Combin. 22(3), Paper 3.22, 18 (2015)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Shallit, J., Wang, M.-W.: Automatic complexity of strings. J. Autom. Lang. Comb. 6(4), 537–554 (2001). 2nd Workshop on Descriptional Complexity of Automata, Grammars and Related Structures (London, ON, 2000)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Sipser, M.: Instructor’s Solutions Manual: Introduction to the Theory of Computation. Cengage Learning 3rd edition (2013)Google Scholar
  12. 12.
    Thue, A.: Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske Vid. Skrifter I Mat.-Nat. Kl., Christiania 1, 1–67 (1912)zbMATHGoogle Scholar
  13. 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

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Hawaii at ManoaHonoluluUSA

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