Solvability of Context Equations with Two Context Variables Is Decidable

  • Manfred Schmidt-Schauß
  • Klaus U. Schulz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1632)

Abstract

Context unification is a natural variant of second order unification that represents a generalization of word unification at the same time. While second order unification is wellknown to be undecidable and word unification is decidable it is currently open if solvability of context equations is decidable. We show that solvability of systems of context equations with two context variables is decidable. The context variables may have an arbitrary number of occurrences, and the equations may contain an arbitrary number of individual variables as well. The result holds under the assumption that the first-order background signature is finite

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    H. Comon. Completion of rewrite systems with membership constraints, part I: Deduction rules and part II: Constraint solving. Technical Report, CNRS and LRI, Université de Paris Sud, 1993. to appear in JSC.Google Scholar
  2. 2.
    W. Farmer. Simple second-order languages for which unification is undecidable. Theoretical Computer Science, 87:173–214, 1991.CrossRefMathSciNetGoogle Scholar
  3. 3.
    H. Ganzinger, F. Jacquemard, and M. Veanes. Rigid reachability. In J. Hsiang and A. Ohori, editors, Advances in Computing Science-ASIAN’98, Springer LNCS 1538, pages 4–21, 1998.CrossRefGoogle Scholar
  4. 4.
    W. Goldfarb. The undecidability of the second-order unification problem. Theoretical Computer Science, 13:225–230, 1981.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    A. Kościelski and L. Pacholski. Complexity of Makanin’s algorithms. Journal of the Association for Computing Machinery, 43:670–684, 1996.MATHGoogle Scholar
  6. 6.
    J. Levy. Linear second order unification. In Proc. of the 7th Int. Conf. on Rewriting Techniques and Applications, Springer LNCS 1103, pages 332–346, 1996.Google Scholar
  7. 7.
    J. Levy and M. Veanes. On the undecidability of second-order unification. Submitted to Information and Computation, 1999.Google Scholar
  8. 8.
    G. Makanin. The problem of solvability of equations in a free semigroup. Math. USSR Sbornik, 32(2):129–198, 1977.MATHCrossRefGoogle Scholar
  9. 9.
    J. Marcinkowski. Undecidability of the first order theory of one-step right ground rewriting. In H. Comon, editor, International Conference on Rewriting Techniques and Applications, Springer LNCS 1232, pages 241–253, 1997.Google Scholar
  10. 10.
    J. Niehren, M. Pinkal, and P. Ruhrberg. On equality up-to constraints over finite trees, context unification, and one-step rewriting. In Proc. of the Int. Conf. on Automated Deduction, Springer LNCS 1249, pages 34–48, 1997.Google Scholar
  11. 11.
    J. Niehren, M. Pinkal, and P. Ruhrberg. A uniform approach to underspecification and parallelism. Technical Report, 1997.Google Scholar
  12. 12.
    J. Niehren, S. Tison, and R. Treinen. On stratified context unification and rewriting constraints. Talk at CCL’98 Workshop, 1998.Google Scholar
  13. 13.
    M. Schmidt-Schauß. Unification of stratified second-order terms. Internal Report 12/94, Fachb. Informatik, J.W. Goethe-Universität Frankfurt, Germany, 1994.Google Scholar
  14. 14.
    M. Schmidt-Schauß. An algorithm for distributive unification. Theoretical Computer Science, 208:111–148, 1998.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    M. Schmidt-Schauß. Decidability of bounded second order unification. Draft, Fachbereich Informatik, J.W. Goethe-Universität Frankfurt, Germany, 1998.Google Scholar
  16. 16.
    M. Schmidt-Schauß and K. U. Schulz. On the exponent of periodicity of minimal solutions of context equations. In Rewriting Techniques and Applications, Proc. RTA’98, volume 1379 of LNCS, pages 61–75. Springer-Verlag, 1998.CrossRefGoogle Scholar
  17. 17.
    M. Schmidt-Schauß and K. U. Schulz. Solvability of context equations with two context variables is decidable. CIS-Report 98-114, CIS, University of Munich, Germany, 1999. available under ftp://ftp.cis.uni-muenchen.de/pub/cis-berichte/CIS-Bericht-98-114.ps.Google Scholar
  18. 18.
    K. U. Schulz. Makanin’s algorithm-two improvements and a generalization. In Proc. of IWWERT 1990, Springer LNCS 572, pages 85–150, 1990.Google Scholar
  19. 19.
    K. U. Schulz. Word unification and transformation of generalized equations. J. Automated Reasoning, pages 149–184, 1993.Google Scholar
  20. 20.
    R. Treinen. The first-order theory of one-step rewriting is undecidable. In H. Ganzinger, editor, 7th International Conference on Rewriting Techniques and Applications, Springer LNCS 1103, pages 276–286,Rutgers University, NJ, USA, 1996.Google Scholar
  21. 21.
    S. Vorobyov. The first-order theory of one step rewriting in linear noetherian systems is undecidable. In H. Comon, editor, International Conference on Rewriting Techniques and Applications, Springer LNCS 1232, pages 254–268, 1997.Google Scholar
  22. 22.
    S. Vorobyov. The 898-equational theory of context unification is co-recursively enumerable hard. Talk at CCL’98 Workshop, 1998.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Manfred Schmidt-Schauß
    • 1
  • Klaus U. Schulz
    • 2
  1. 1.Fachbereich InformatikJ.-W.-Goethe-UniversitätFrankfurtGermany
  2. 2.CIS, University of MunichMünchenGermany

Personalised recommendations