On the Complexity of Hmelevskii’s Theorem and Satisfiability of Three Unknown Equations

  • Aleksi Saarela
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5583)


We analyze Hmelevskii’s theorem, which states that the general solutions of constant-free equations on three unknowns are expressible by a finite collection of formulas of word and numerical parameters. We prove that the size of the finite representation is bounded by an exponential function on the size of the equation. We also prove that the shortest nontrivial solution of the equation, if it exists, is exponential, and that its existence can be solved in nondeterministic polynomial time.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Albert, M.H., Lawrence, J.: A proof of Ehrenfeucht’s conjecture. Theoret. Comput. Sci. 41, 121–123 (1985)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Choffrut, C., Karhumäki, J.: Combinatorics of words. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages. Springer, Heidelberg (1997)Google Scholar
  3. 3.
    Czeizler, E., Karhumäki, J.: On non-periodic solutions of independent systems of word equations over three unknowns. Internat. J. Found. Comput. Sci. 18, 873–897 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Eilenberg, S., Schützenberger, M.P.: Rational sets in commutative monoids. J. Algebra 13, 173–191 (1969)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    von zur Gathen, J., Sieveking, M.: A bound on solutions of linear integer equalities and inequalities. Proc. Amer. Math. Soc. 72, 155–158 (1978)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Guba, V.S.: Equivalence of infinite systems of equations in free groups and semigroups to finite subsystems. Mat. Zametki 40, 321–324 (1986)MathSciNetMATHGoogle Scholar
  7. 7.
    Harju, T., Karhumäki, J., Plandowski, W.: Independent systems of equations. In: Lothaire, M. (ed.) Algebraic Combinatorics on Words. Cambridge University Press, Cambridge (2002)Google Scholar
  8. 8.
    Hmelevskii, Y.I.: Equations in free semigroups. Proc. Steklov Inst. of Math. 107 (1971); Amer. Math. Soc. Translations (1976)Google Scholar
  9. 9.
    Karhumäki, J., Saarela, A.: An analysis and a reproof of Hmelevskii’s theorem. In: Proc. of 12th International Conference on Developments in Language Theory, pp. 467–478 (2008)Google Scholar
  10. 10.
    Lothaire, M.: Combinatorics on words. Addison-Wesley, Reading (1983)MATHGoogle Scholar
  11. 11.
    Makanin, G.S.: The problem of solvability of equations in a free semigroup. Mat. Sb. 103, 147–236; English transl. in Math. USSR Sb. 32, 129–198 (1977)Google Scholar
  12. 12.
    Plandowski, W.: Satisfiability of word equations with constants is in PSPACE. J. ACM 51, 483–496 (2004)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Plandowski, W., Rytter, W.: Application of Lempel-Ziv encodings to the solution of word equations. In: Proc. of 25th International Colloquium on Automata, Languages, and Programming, pp. 731–742 (1998)Google Scholar
  14. 14.
    Saarela, A.: A new proof of Hmelevskii’s theorem. Licentiate thesis, Univ. Turku (2009), http://users.utu.fi/amsaar/en/licthesis.pdf
  15. 15.
    Spehner, J.-C.: Quelques problemes d’extension, de conjugaison et de presentation des sous-monoides d’un monoide libre. Ph.D. Thesis, Univ. Paris (1976)Google Scholar
  16. 16.
    Spehner, J.-C.: Les presentations des sous-monoides de rang 3 d’un monoide libre. Semigroups. In: Proc. Conf. Math. Res. Inst., pp. 116–155 (1978)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Aleksi Saarela
    • 1
  1. 1.Department of Mathematics and Turku Centre for Computer Science TUCSUniversity of TurkuTurkuFinland

Personalised recommendations