Stratified Context Unification Is NP-Complete

  • Jordi Levy
  • Manfred Schmidt-Schauß
  • Mateu Villaret
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4130)


Context Unification is the problem to decide for a given set of second-order equations E where all second-order variables are unary, whether there exists a unifier, such that for every second-order variable X, the abstraction λx. r instantiated for X has exactly one occurrence of the bound variable x in r. Stratified Context Unification is a specialization where the nesting of second-order variables in E is restricted.

It is already known that Stratified Context Unification is decidable, NP-hard, and in PSPACE, whereas the decidability and the complexity of Context Unification is unknown. We prove that Stratified Context Unification is in NP by proving that a size-minimal solution can be represented in a singleton tree grammar of polynomial size, and then applying a generalization of Plandowski’s polynomial algorithm that compares compacted terms in polynomial time. This also demonstrates the high potential of singleton tree grammars for optimizing programs maintaining large terms.

A corollary of our result is that solvability of rewrite constraints is NP-complete.


Context Variable Function Symbol Decision Algorithm Surface Equation Main Path 
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. [And86]
    Andrews, P.: An introduction to mathematical logic and type theory: to truth through proof. Academic Press, London (1986)MATHGoogle Scholar
  2. [BLM05]
    Busatto, G., Lohrey, M., Maneth, S.: Efficient memory representation of XML documents. In: Bierman, G., Koch, C. (eds.) DBPL 2005. LNCS, vol. 3774, pp. 199–216. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  3. [CDG+97]
    Comon, H., Dauchet, M., Gilleron, R., Jacquemard, F., Lugiez, D., Tison, S., Tommasi, M.: Tree automata techniques and applications (1997) (release, October 1, 2002), available on
  4. [Dow01]
    Dowek, G.: Higher-order unification and matching. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, ch. 16, vol. II, pp. 1009–1062. Elsevier Science, Amsterdam (2001)CrossRefGoogle Scholar
  5. [EN00]
    Erk, K., Niehren, J.: Parallelism constraints. In: Bachmair, L. (ed.) RTA 2000. LNCS, vol. 1833, pp. 110–126. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  6. [Far91]
    Farmer, W.M.: Simple second-order languages for wich unification is undecidable. Theoretical Computer Science 87, 173–214 (1991)CrossRefMathSciNetGoogle Scholar
  7. [Gol81]
    Goldfarb, W.D.: The undecidability of the second-order unification problem. Theoretical Computer Science 13, 225–230 (1981)MATHCrossRefMathSciNetGoogle Scholar
  8. [Hue75]
    Huet, G.: A unification algorithm for typed λ-calculus. Theoretical Computer Science 1, 27–57 (1975)CrossRefMathSciNetGoogle Scholar
  9. [KNT98]
    Koller, A., Niehren, J., Treinen, R.: Dominance constraints: Algorithms and complexity. In: Moortgat, M. (ed.) LACL 1998. LNCS (LNAI), vol. 2014, pp. 106–125. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  10. [KP96]
    Kościelski, A., Pacholski, L.: Complexity of Makanin’s algorithm. Journal of the ACM 43(4), 670–684 (1996)MATHCrossRefMathSciNetGoogle Scholar
  11. [Lev96]
    Levy, J.: Linear second order unification. In: Ganzinger, H. (ed.) RTA 1996. LNCS, vol. 1103, pp. 332–346. Springer, Heidelberg (1996)Google Scholar
  12. [LNV05]
    Levy, J., Niehren, J., Villaret, M.: Well-nested context unification. In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS (LNAI), vol. 3632, pp. 149–163. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  13. [LSSV04]
    Levy, J., Schmidt-Schauß, M., Villaret, M.: Monadic second-order unification is NP-complete. In: van Oostrom, V. (ed.) RTA 2004. LNCS, vol. 3091, pp. 55–69. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  14. [LSSV06]
    Levy, J., Schmidt-Schauß, M., Villaret, M.: Bounded second-order unification is NP-complete. In: Pfenning, F. (ed.) RTA 2006. LNCS, vol. 4098, Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. [LV00]
    Levy, J., Veanes, M.: On the undecidability of second-order unification. Information and Computation 159, 125–150 (2000)MATHCrossRefMathSciNetGoogle Scholar
  16. [LV02]
    Levy, J., Villaret, M.: Currying second-order unification problems. In: Tison, S. (ed.) RTA 2002. LNCS, vol. 2378, pp. 326–339. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  17. [Mak77]
    Makanin, G.S.: The problem of solvability of equations in a free semigroup. Math. USSR Sbornik 32(2), 129–198 (1977)MATHCrossRefGoogle Scholar
  18. [NK01]
    Niehren, J., Koller, A.: Dominance constraints in context unification. In: Moortgat, M. (ed.) LACL 1998. LNCS (LNAI), vol. 2014, pp. 199–218. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  19. [NPR97a]
    Niehren, J., Pinkal, M., Ruhrberg, P.: On equality up-to constraints over finite trees, context unification, and one-step rewriting. In: McCune, W. (ed.) CADE 1997. LNCS, vol. 1249, pp. 34–48. Springer, Heidelberg (1997)Google Scholar
  20. [NPR97b]
    Niehren, J., Pinkal, M., Ruhrberg, P.: A uniform approach to underspecification and parallelism. In: 35th ACL 1997, Madrid, pp. 410–417 (1997)Google Scholar
  21. [NTT00]
    Niehren, J., Tison, S., Treinen, R.: On rewrite constraints and context unification. Information Processing Letters 74, 35–40 (2000)MATHCrossRefMathSciNetGoogle Scholar
  22. [Pau94]
    Paulson, L.C.: Isabelle. LNCS, vol. 828. Springer, Heidelberg (1994)MATHCrossRefGoogle Scholar
  23. [Pla94]
    Plandowski, W.: Testing equivalence of morphisms in context-free languages. In: van Leeuwen, J. (ed.) ESA 1994. LNCS, vol. 855, pp. 460–470. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  24. [Pla95]
    Plandowski, W.: The Complexity of the Morphism Equivalence Problem for Context-Free Languages. PhD thesis, Dept. of Mathematics, Informatics and Mechanics, Warsaw University (1995)Google Scholar
  25. [Pla04]
    Plandowski, W.: Satisfiability of word equations with constants is in PSPACE. Journal of the ACM 51(3), 483–496 (2004)MATHCrossRefMathSciNetGoogle Scholar
  26. [PS99]
    Pfenning, F., Schürmann, C.: System description: Twelf - a meta-logical framework for deductive systems. In: Ganzinger, H. (ed.) CADE 1999. LNCS (LNAI), vol. 1632, pp. 202–206. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  27. [SAT06] (2006)Google Scholar
  28. [SS98]
    Schmidt-Schauß, M.: A decision algorithm for distributive unification. TCS 208, 111–148 (1998)MATHCrossRefGoogle Scholar
  29. [SS02]
    Schmidt-Schauß, M.: A decision algorithm for stratified context unification. Journal of Logic and Computation 12(6), 929–953 (2002)MATHCrossRefMathSciNetGoogle Scholar
  30. [SS05]
    Schmidt-Schauß, M.: Polynomial equality testing for terms with shared substructures. Frank report 21, Institut für Informatik. FB Informatik und Mathematik. J. W. Goethe-Universität Frankfurt am Main (November 2005)Google Scholar
  31. [SSS98]
    Schmidt-Schauß, M., Schulz, K.U.: On the exponent of periodicity of minimal solutions of context equations. In: Nipkow, T. (ed.) RTA 1998. LNCS, vol. 1379, pp. 61–75. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  32. [SSS04]
    Schmidt-Schauß, M., Stuber, J.: On the complexity of linear and stratified context matching problems. Theory of Computing Systems 37, 717–740 (2004)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Jordi Levy
    • 1
  • Manfred Schmidt-Schauß
    • 2
  • Mateu Villaret
    • 3
  1. 1.IIIA, CSICBarcelonaSpain
  2. 2.Institut für Informatik, FB Informatik und MathematikJohann Wolfgang Goethe-UniversitätFrankfurtGermany
  3. 3.IMA, UdGGironaSpain

Personalised recommendations