Existential and Positive Theories of Equations in Graph Products

  • Volker Diekert
  • Markus Lohrey
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2285)

Abstract

We prove that the existential theory of equations with normalized rational constraints in a fixed graph product of finite monoids, free monoids, and free groups is PSPACE-complete. Under certain restrictions this result also holds if the graph product is part of the input. As the second main result we prove that the positive theory of equations with recognizable constraints in graph products of finite and free groups is decidable.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    IJ. J. Aalbersberg and H. J. Hoogeboom. Characterizations of the decidability of some problems for regular trace languages. Mathematical Systems Theory, 22:1–19, 1989.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    A. V. Anisimov and D. E. Knuth. Inhomogeneous sorting. International Journal of Computer and Information Sciences, 8:255–260, 1979.CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    M. Benois. Parties rationnelles du groupe libre. C. R. Acad. Sci. Paris, Sér. A, 269:1188–1190, 1969.MATHMathSciNetGoogle Scholar
  4. 4.
    J. Berstel. Transductions and context-free languages. Teubner Studienbücher, Stuttgart, 1979.Google Scholar
  5. 5.
    A. V. da Costa. Graph products of monoids. Semigroup Forum, 63(2):247–277, 2001.MATHMathSciNetGoogle Scholar
  6. 6.
    V. Diekert, C. Gutiérrez, and C. Hagenah. The existential theory of equations with rational constraints in free groups is PSPACE-complete. In Proceedings of the 18th Annual Symposium on Theoretical Aspects of Computer Science (STACS 01), number 2010 in Lecture Notes in Computer Science, pages 170–182. Springer, 2001.Google Scholar
  7. 7.
    V. Diekert and M. Lohrey. A note on the existential theory of equations in plain groups. International Journal of Algebra and Computation, 2001. to apear.Google Scholar
  8. 8.
    V. Diekert and M. Lohrey. Existential and positive theories of equations in graph products. Technical Report 2001/10, University of Stuttgart, Germany, 2001.Google Scholar
  9. 9.
    V. Diekert, Y. Matiyasevich, and A. Muscholl. Solving word equations modulo partial commutations. Theoretical Computer Science, 224(1–2):215–235, 1999.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    V. Diekert and A. Muscholl. Solvability of equations in free partially commutative groups is decidable. In Proceedings of the 28th International Colloquium on Automata, Languages and Programming (ICALP 01), number 2076 in Lecture Notes in Computer Science, pages 543–554. Springer, 2001.Google Scholar
  11. 11.
    V. Diekert and G. Rozenberg, editors. The Book of Traces. World Scientific, 1995.Google Scholar
  12. 12.
    C. Droms. Graph groups, coherence and three-manifolds. Journal of Algebra, 106(2):484–489, 1985.CrossRefMathSciNetGoogle Scholar
  13. 13.
    V. G. Durnev. Undecidability of the positive ℬ∃3-theory of a free semi-group. Sibirsky Matematicheskie Jurnal, 36(5):1067–1080, 1995.MathSciNetGoogle Scholar
  14. 14.
    E. R. Green. Graph Products of Groups. PhD thesis, The University of Leeds, 1990.Google Scholar
  15. 15.
    C. Gutiérrez. Satisfiability of equations in free groups is in PSPACE. In 32nd Annual ACM Symposium on Theory of Computing (STOC’2000), pages 21–27. ACM Press, 2000.Google Scholar
  16. 16.
    R. H. Haring-Smith. Groups and simple languages. Transactions of the American Mathematical Society, 279:337–356, 1983.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    D. Kozen. Lower bounds for natural proof systems. In Proceedings of the 18th Annual Symposium on Foundations of Computer Science, (FOCS 77), pages 254–266. IEEE Computer Society Press, 1977.Google Scholar
  18. 18.
    M. Lohrey. Confluence problems for trace rewriting systems. Information and Computation, 170:1–25, 2001.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    G. S. Makanin. The problem of solvability of equations in a free semigroup. Math. Sbornik, 103:147–236, 1977. (Russian); English translation in Math. USSR Sbornik 32 (1977).MathSciNetGoogle Scholar
  20. 20.
    G. S. Makanin. Equations in a free group. Izv. Akad. Nauk SSR, Ser. Math. 46:1199–1273, 1983. (Russian); English translation in Math. USSR Izv. 21 (1983).MathSciNetGoogle Scholar
  21. 21.
    G. S. Makanin. Decidability of the universal and positive theories of a free group. Izv. Akad. Nauk SSSR, Ser. Mat. 48:735–749, 1984. (Russian); English translation in: Math. USSR Izvestija, 25, 75–88, 1985.MathSciNetGoogle Scholar
  22. 22.
    S. S. Marchenkov. Unsolvability of positive ℬ∃-theory of a free semi-group. Sibirsky Matematicheskie Jurnal, 23(1):196–198, 1982.MATHMathSciNetGoogle Scholar
  23. 23.
    Y. I. Merzlyakov. Positive formulas on free groups. Algebra i Logika Sem., 5(4):25–42, 1966. (Russian).MATHMathSciNetGoogle Scholar
  24. 24.
    E. Ochmański. Regular behaviour of concurrent systems. Bulletin of the European Association for Theoretical Computer Science (EATCS), 27:56–67, 1985.Google Scholar
  25. 25.
    W. Plandowski. Satisfiability of word equations with constants is in PSPACE. In Proceedings of the 40th Annual Symposium on Foundations of Computer Science (FOCS 99), pages 495–500. IEEE Computer Society Press, 1999.Google Scholar
  26. 26.
    M. Presburger. Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. In Comptes Rendus du Premier Congrès des Mathématicienes des Pays Slaves, pages 92–101, 395, Warsaw, 1927.Google Scholar
  27. 27.
    K. U. Schulz. Makanin’s algorithm for word equations — Two improvements and a generalization. In Word Equations and Related Topics, number 572 in Lecture Notes in Computer Science, pages 85–150. Springer, 1991.Google Scholar

Copyright information

© Springer-VerlagBerlin Heidelberg 2002

Authors and Affiliations

  • Volker Diekert
    • 1
  • Markus Lohrey
    • 1
  1. 1.Institut für InformatikUniversität StuttgartStuttgartGermany

Personalised recommendations