Advertisement

Modular higher-order E-unification

  • Tobias Nipkow
  • Zhenyu Qian
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 488)

Abstract

The combination of higher-order and first-order unification algorithms is studied. We present algorithms to compute a complete set of unifiers of two simply typed λ-terms w.r.t. the union of α, β and η conversion and a first-order equational theory E. The algorithms are extensions of Huet's work and assume that a complete unification algorithm for E is given. Our completeness proofs require E to be at least regular.

Keywords

Normal Form Function Constant Logic Programming Free Variable Equational Theory 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Boudet, A.: Unification in a combination of equational theories: an efficient algorithm. Proc. 10th Int. Conf. Automated Deduction, LNCS 449 (1990), 292–307.Google Scholar
  2. [2]
    Breazu-Tannen, V.: Combing algebra and higher-order types. Proc. 3rd IEEE Symp. Logic in Computer Science (1988), 82–90.Google Scholar
  3. [3]
    Breazu-Tannen, V. and Gallier, J.: Polymorphic rewriting conserves algebraic strong normalization and confluence. Proc. ICALP, LNCS 372 (1989), 137–150.Google Scholar
  4. [4]
    Bürchert, H.-J.: Matching — a special case of unification? J. Symbolic Computation 8 (1989), 523–536.Google Scholar
  5. [5]
    Futatsugi, K., Goguen, J.A., Jouannaud, J.-P., Meseguer, J.: Principles of OBJ2. Proc. 12th ACM Symp. Principles of Programming Languages (1985), 52–66.Google Scholar
  6. [6]
    Gallier, J. and Snyder, W.: Complete sets of transformations for general E-unification. Theoretical Computer Science 67 (1988), 203–260.Google Scholar
  7. [7]
    Garland, S.J., Guttag, J.V.: An Overview of LP, The Larch Prover. Proc. 3rd Int. Conf. Rewriting Techniques and Applications, LNCS 355 (1989), 137–151.Google Scholar
  8. [8]
    Goldfarb, W.: The undecidability of the second-order unification problem. Theoretical Computer Science 13 (1981), 225–230.Google Scholar
  9. [9]
    Herold, A.: Combination of Unification Algorithms. Proc. 8th Int. Conf. Automated Deduction, LNCS 230 (1986), 450–469.Google Scholar
  10. [10]
    Huet, G.: A Unification Algorithm for Typed λ-Calculus. Theoretical Computer Science 1 (1975), 27–57.Google Scholar
  11. [11]
    Huet, G.: Résolution d'equations dans les languages d'ordre 1, 2,...,ω. Thése d'Etat, Université de Paris VII (1976).Google Scholar
  12. [12]
    Kirchner, C.: Méthodes et outils de conception systématique d'algorithmes d'unification dans les théories équationnelles, Thèse d'état de l'Université de Nancy I (1985).Google Scholar
  13. [13]
    Nadathur, G. and Miller, D.: An overview of λProlog. Proc. 5th Int. Conf. Logic Programming, eds. R.A.Kowalski and K.A. Bowen, MIT Press (1988), 810–827.Google Scholar
  14. [14]
    Nelson, G. and Oppen, D.: Simplification by cooperating decision procedures. ACM TOPLAS 1 (1979), 245–257.Google Scholar
  15. [15]
    Nipkow, T.: Combining matching algorithms: the regular case. Proc. 3rd Int. Conf. Rewriting Techniques and Applications, LNCS 355 (1989), 343–358.Google Scholar
  16. [16]
    Paulson, L.C.: Isabelle: The Next 700 Theorem Provers. In P. Odifreddi (editor), Logic and Computer Science, Academic Press (1990), 361–385.Google Scholar
  17. [17]
    Schmidt-Schauß, M.: Unification in a combination of arbitrary disjoint equational theories. J. Symbolic Computation 8 (1989), 51–99.Google Scholar
  18. [18]
    Snyder, W.: Higher-order E-unification. Proc. 10th Int. Conf. Automated Deduction, LNCS 449 (1990), 573–587.Google Scholar
  19. [19]
    Snyder, W and Gallier, J.: Higher-order unification revisited: complete sets of transformations. J. Symbolic Computation 8 (1989), 101–140.Google Scholar
  20. [20]
    Tidén, E.: Unification in Combinations of Collapse-Free Theories with Disjoint Sets of Function Symbols. Proc. 8th Int. Conf. Automated Deduction, LNCS 230 (1986), 431–449.Google Scholar
  21. [21]
    Yelick, K.A.: Unification in combinations of collapse-free regular theories. J. Symbolic Computation 3 (1987), 153–182.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Tobias Nipkow
    • 1
  • Zhenyu Qian
    • 2
  1. 1.Computer LaboratoryUniversity of CambridgeCambridgeEngland
  2. 2.FB InformatikUniversität BremenBremen 33Germany

Personalised recommendations