Advertisement

Acta Mathematica Hungarica

, Volume 159, Issue 1, pp 246–256 | Cite as

Notes on extended equation solvability and identity checking for groups

  • M. KompatscherEmail author
Article
  • 29 Downloads

Abstract

Every finite non-nilpotent group can be extended by a term operation such that solving equations in the resulting algebra is NP-complete and checking identities is co-NP-complete. This result was firstly proven by Horváth and Szabó; the term constructed in their proof depends on the underlying group. In this paper we provide a uniform term extension that induces hard problems. In doing so we also characterize a big class of solvable, non-nilpotent groups for which extending by the commutator operation suffices.

Key words and phrases

equation solvability identity checking solvable group 

Mathematics Subject Classification

20F10 20F12 20D10 08A70 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgement

The author would like to thank Attila Földvári and Gábor Horváth for their valuable feedback on earlier versions of this paper.

References

  1. 1.
    Aichinger, E., Mudrinski, N., Opršal, J.: Complexity of term representations of finitary functions. Int. J. Algebra Comput. 28, 1101–1118 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baer, R.: Engelsche Elemente Noetherscher Gruppen. Math. Ann. 133, 256–270 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    D. M. Barrington, P. McKenzie, C. Moore, P. Tesson, and D. Thérien, Equation satisfiability and program satisfiability for finite monoids, in: International Symposium on Mathematical Foundations of Computer Science (Bratislava, 2000), Lecture Notes in Comput. Sci., 1893, Springer (Berlin, 2000), pp. 172–181Google Scholar
  4. 4.
    Burris, S., Lawrence, J.: The equivalence problem for finite rings. J. Symbolic Comput. 15, 67–71 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Burris, S., Lawrence, J.: Results on the equivalence problem for finite groups. Algebra Universalis 52, 495–500 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Földvári, A.: The complexity of the equation solvability problem over semipattern groups. Int. J. Algebra Comput. 27, 259–272 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Földvári, A.: The complexity of the equation solvability problem over nilpotent groups. J. Algebra 495, 289–303 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Goldmann, M., Russell, A.: The complexity of solving equations over finite groups. Inform. and Comput. 178, 253–262 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gorazd, T., Krzaczkowski, J.: The complexity of problems connected with two-element algebras. Rep. Math. Logic 46, 91–108 (2011)MathSciNetzbMATHGoogle Scholar
  10. 10.
    G. Horváth, Functions and Polynomials over Finite Groups from the Computational Perspective, PhD thesis, University of Hertfordshire (2008)Google Scholar
  11. 11.
    Horváth, G.: The complexity of the equivalence and equation solvability problems over nilpotent rings and groups. Algebra Universalis 66, 391–403 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    G. Horváth, J. Lawrence, and R. Willard, The complexity of the equation solvability problem over finite rings, preprint, http://real.mtak.hu/28210/ (2015)
  13. 13.
    Horváth, G., Mérai, L., Szabó, C., Lawrence, J.: The complexity of the equivalence problem for nonsolvable groups. Bull. London Math. Soc. 39, 433–438 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Horváth, G., Szabó, C.: The extended equivalence and equation solvability problems for groups. Discrete Math. Theor. Comput. Sci. 13, 23–32 (2011)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Horváth, G., Szabó, C.: Equivalence and equation solvability problems for the alternating group \(A_4\). J. Pure Appl. Algebra 216, 2170–2176 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hunt III, H.B., Stearns, R.E.: The complexity of equivalence for commutative rings. J. Symbolic Comput. 10, 411–436 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    P. M. Idziak and J. Krzaczkowski, Satisfiability in multi-valued circuits, in: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, ACM (2018), pp. 550–558Google Scholar
  18. 18.
    Kompatscher, M.: The equation solvability problem over supernilpotent algebras with Mal'cev term. Int. J, Algebra Comput (2018)zbMATHGoogle Scholar
  19. 19.
    D.-J. Robinson, A Course in the Theory of Groups, Springer (Berlin–Heidelberg, 1996)Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Department of AlgebraMFF UKPraha 8Czech Republic

Personalised recommendations