New Algorithms for Unification Modulo One-Sided Distributivity and Its Variants

  • Andrew M. Marshall
  • Paliath Narendran
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7364)

Abstract

An algorithm for unification modulo one-sided distributivity is an early result by Tiden and Arnborg [14]. Unfortunately the algorithm presented in the paper, although correct, has recently been shown not to be polynomial time bounded as claimed [11]. In addition, for some instances, there exist most general unifiers that are exponentially large with respect to the input size. In this paper we first present a new polynomial time algorithm that solves the decision problem for a non-trivial subcase, based on a typed theory, of unification modulo one-sided distributivity. Next we present a new polynomial algorithm that solves the decision problem for unification modulo one-sided distributivity. A construction, employing string compression, is used to achieve the polynomial bound.

Keywords

Equivalence Class Standard Form Inference Rule Sink Node Transitive Closure 
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.

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References

  1. 1.
    Baader, F., Nipkow, T.: Term rewriting and all that. Cambridge University Press, New York (1998)Google Scholar
  2. 2.
    Baader, F., Snyder, W.: Unification theory. In: Alan Robinson, J., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 445–532. Elsevier, MIT Press (2001)Google Scholar
  3. 3.
    Gascón, A., Godoy, G., Schmidt-Schauß, M.: Unification with Singleton Tree Grammars. In: Treinen, R. (ed.) RTA 2009. LNCS, vol. 5595, pp. 365–379. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  4. 4.
    Gascòn, A., Godoy, G., Schmidt-Schauß, M.: Unification and matching on compressed terms. ACM TOCL 12(4) (2011)Google Scholar
  5. 5.
    Jouannaud, J.-P., Kirchner, C.: Solving equations in abstract algebras: A rule-based survey of unification. In: Computational Logic - Essays in Honor of Alan Robinson, pp. 257–321 (1991)Google Scholar
  6. 6.
    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
  7. 7.
    Levy, J., Schmidt-Schauß, M., Villaret, M.: The complexity of monadic second-order unification. SIAM J. Computation 38(3), 1113–1140 (2008)MATHCrossRefGoogle Scholar
  8. 8.
    Lifshits, Y.: Processing Compressed Texts: A Tractability Border. In: Ma, B., Zhang, K. (eds.) CPM 2007. LNCS, vol. 4580, pp. 228–240. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. 9.
    Marshall, A.M., Narendran, P.: New algorithms for unification modulo one-sided distributivity and its variants. Technical Report SUNYA-CS-12-02, University at Albany-SUNY (2012), http://www.cs.albany.edu/~ncstrl/treports/Data/README.html
  10. 10.
    Miyazaki, M., Shinohara, A., Takeda, M.: An Improved Pattern Matching Algorithm for Strings in Terms of Straight-line Programs. In: Hein, J., Apostolico, A. (eds.) CPM 1997. LNCS, vol. 1264, pp. 1–11. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  11. 11.
    Narendran, P., Marshall, A.M., Mahapatra, B.: On the complexity of the Tiden-Arnborg algorithm for unification modulo one-sided distributivity. In: UNIF 24. EPTCS, vol. 42, pp. 54–63 (2010)Google Scholar
  12. 12.
    Plandowski, W.: Testing Equivalence of Morphisms on Context-free Languages. In: van Leeuwen, J. (ed.) ESA 1994. LNCS, vol. 855, pp. 460–470. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  13. 13.
    Rytter, W.: Application of Lempel-Ziv factorization to the approximation of grammar-based compression. Theoretical Computer Science 302, 211–222 (2003)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Tidén, E., Arnborg, S.: Unification problems with one-sided distributivity. J. Symb. Comput. 3(1/2), 183–202 (1987)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Andrew M. Marshall
    • 1
  • Paliath Narendran
    • 1
  1. 1.College of Computing and Information, Computer Science DepartmentUniversity at Albany–SUNYUSA

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