The Complexity of Equivalence and Isomorphism of Systems of Equations over Finite Groups

  • Gustav Nordh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3153)


We study the computational complexity of the isomorphism and equivalence problems on systems of equations over a fixed finite group. We show that the equivalence problem is in P if the group is Abelian, and coNP-complete if the group is non-Abelian. We prove that if the group is non-Abelian, then the problem of deciding whether two systems of equations over the group are isomorphic is coNP-hard. If the group is Abelian, then the isomorphism problem is graph isomorphism hard. Moreover, if we impose the restriction that all equations are of bounded length, then we prove that the isomorphism problem for systems of equations over finite Abelian groups is graph isomorphism complete. Finally we prove that the problem of counting the number of isomorphisms of systems of equations is no harder than deciding whether there exist any isomorphisms at all.


Computational Complexity Abelian Group Polynomial Time Regular Semigroup Equivalence Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agrawal, M., Thierauf, T.: The formula isomorphism problem. SIAM Journal on Computing 30(3), 990–1009 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Böhler, E., Hemaspaandra, E., Reith, S., Vollmer, H.: Equivalence and isomorphism for boolean constraint satisfaction. In: Conference for Computer Science Logic, pp. 412–426 (2002)Google Scholar
  3. 3.
    Böhler, E., Hemaspaandra, E., Reith, S., Vollmer, H.: The complexity of boolean constraint isomorphism. In: Proceedings of the 21st Symposium on Theoretical Aspects of Computer Science, pp. 164–175 (2004)Google Scholar
  4. 4.
    Booth, K.S., Colbourn, C.J.: Problems polynomially equivalent to graph isomorphism. Technical report, CS-77-04, Computer Science Dept., University of Waterloo (1979)Google Scholar
  5. 5.
    Goldmann, M., Russel, A.: The complexity of solving equations over finite groups. Information and Computation 178(1), 253–262 (2002)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Köbler, H., Schöning, U., Torán, J.: The Graph Isomorphism Problem: Its Structural Complexity. Birkhäuser, Basel (1993)zbMATHGoogle Scholar
  7. 7.
    Mathon, R.: A note on the graph isomorphism counting problem. Information Processing Letters 8(3), 131–132 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Moore, C., Tesson, P., Thérien, D.: Satisfiability of systems of equations over finite monoids. In: Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science, pp. 537–547 (2001)Google Scholar
  9. 9.
    Nordh, G., Jonsson, P.: The complexity of counting solutions to systems of equations over finite semigroups. In: Proceedings of the 10th International Computing and Combinatorics Conference (2004)Google Scholar
  10. 10.
    Tesson, P.: Computational Complexity Questions Related to Finite Monoids and Semigroups. PhD thesis, School of Computer Science, McGill University, Montreal (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Gustav Nordh
    • 1
  1. 1.Department of Computer and Information ScienceLinköpings UniversitetLinköpingSweden

Personalised recommendations