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The Intersection Problem for Finite Semigroups

  • Lukas Fleischer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11088)

Abstract

We investigate the intersection problem for finite semigroups, which asks for a given set of regular languages, represented by recognizing morphisms to finite semigroups, whether there exists a word contained in their intersection. We introduce compressibility measures as a useful tool to classify the intersection problem for certain classes of finite semigroups into circuit complexity classes and Turing machine complexity classes. Using this framework, we obtain a new and simple proof that for groups and commutative semigroups, the problem is contained in \(\mathsf {NP}\). We uncover certain structural and non-structural properties determining the complexity of the intersection problem for varieties of semigroups containing only trivial submonoids. More specifically, we prove \(\mathsf {NP}\)-hardness for classes of semigroups having a property called unbounded order and for the class of all nilpotent semigroups of bounded order. On the contrary, we show that bounded order and commutativity imply containment in the circuit complexity class \(\mathsf {qAC}^k\) (for some \(k \in \mathbb {N}\)) and decidability in quasi-polynomial time. We also establish connections to the monoid variant of the problem.

Notes

Acknowledgements

I would like to thank the anonymous referees for providing helpful comments that improved the paper.

References

  1. 1.
    Babai, L., Luks, E.M., Seress, Á.: Permutation groups in NC. In: Proceedings of STOC 1987, pp. 409–420 (1987)Google Scholar
  2. 2.
    Babai, L., Szemeredi, E.: On the complexity of matrix group problems I. In: 25th Annual Symposium on Foundations of Computer Science, pp. 229–240, October 1984Google Scholar
  3. 3.
    Barrington, D.A.M.: Quasipolynomial size circuit classes. In: Proceedings of the Seventh Annual Structure in Complexity Theory Conference, pp. 86–93, June 1992Google Scholar
  4. 4.
    Beaudry, M.: Membership testing in transformation monoids. Ph.D. thesis, McGill University, Montreal, Quebec (1988)Google Scholar
  5. 5.
    Beaudry, M., McKenzie, P., Thérien, D.: The membership problem in aperiodic transformation monoids. J. ACM 39(3), 599–616 (1992)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Charikar, M., et al.: The smallest grammar problem. IEEE Trans. Inf. Theory 51(7), 2554–2576 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fleischer, L.: On the complexity of the Cayley semigroup membership problem. In: Proceedings of CCC 2018, pp. 25:1–25:12. Dagstuhl Publishing (2018)Google Scholar
  8. 8.
    Fleischer, L.: The Intersection Problem for Finite Semigroups. CoRR, abs/1806.04996 (2018)Google Scholar
  9. 9.
    Fleischer, L., Kufleitner, M.: The intersection problem for finite monoids. In: Proceedings of STACS 2018, pp. 30:1–30:14. Dagstuhl Publishing (2018)Google Scholar
  10. 10.
    Furst, M., Hopcroft, J., Luks, E.: Polynomial-time algorithms for permutation groups. In: Proceedings of SFCS 1980, pp. 36–41, October 1980Google Scholar
  11. 11.
    Impagliazzo, R., Paturi, R.: Proceedings of CCC (1999)Google Scholar
  12. 12.
    Kozen, D.: Lower bounds for natural proof systems. In: Proceedings of FOCS 1977, pp. 254–266, Providence, Rhode Island. IEEE Computer Society Press (1977)Google Scholar
  13. 13.
    Sims, C.C.: Computational methods in the study of permutation groups. In: Proceedings of the Conference on Computational Problems in Abstract Algebra 1967, Oxford, United Kingdom, pp. 169–183. Pergamon, New York (1968)Google Scholar
  14. 14.
    Straubing, H.: Finite semigroup varieties of the form \(\mathbf{V}\ast \mathbf{D}\). J. Pure Appl. Algebra 36(1), 53–94 (1985)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Straubing, H.: Finite Automata, Formal Logic, and Circuit Complexity. Birkhäuser, Boston (1994)CrossRefGoogle Scholar
  16. 16.
    Tesson, P., Thérien, D.: Diamonds are forever: the variety \({\rm DA}\). In: Proceedings of Semigroups, Algorithms, Automata and Languages, pp. 475–500. World Scientific (2002)Google Scholar
  17. 17.
    Vollmer, H.: Introduction to Circuit Complexity. Springer, Berlin (1999).  https://doi.org/10.1007/978-3-662-03927-4CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.FMIUniversity of StuttgartStuttgartGermany

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