Termination of Rule-Based Calculi for Uniform Semi-Unification

  • Takahito Aoto
  • Munehiro Iwami
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7810)


Uniform semi-unification is a generalization of unification; its efficient algorithms have been extensively studied in (Kapur et al., 1994) and (Oliart&Snyder, 2004). For (uniform) semi-unification, several variants of rule-based calculi are known. But, some of these calculi in the literature lack the termination property, i.e. not all derivations are terminating. We revisit symbolic semi-unification whose solvability coincides with that of uniform semi-unification. We give an abstract criterion of the strategy on which a general rule-based calculus for symbolic semi-unification terminates. Based on this, we give an alternative and robust correctness proof of a rule-based uniform semi-unification algorithm.


Semi-Unification Rule-Based Calculi Termination 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press (1998)Google Scholar
  2. 2.
    Dörre, J., Rounds, W.C.: On subsumption and semiunification in feature algebras. In: Proc. of LICS 1990, pp. 300–310. IEEE (1990)Google Scholar
  3. 3.
    Giesl, J., Thiemann, R., Schneider-Kamp, P.: Proving and Disproving Termination of Higher-Order Functions. In: Gramlich, B. (ed.) FroCos 2005. LNCS (LNAI), vol. 3717, pp. 216–231. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  4. 4.
    Henglein, F.: Algebraic properties of semi-unification. In: Proc. of LFP 1988. ACM Press (1988)Google Scholar
  5. 5.
    Henglein, F.: Polymorphic Type Inference and Semi-Unification. Ph.D. thesis, Rutgers University (1989)Google Scholar
  6. 6.
    Henglein, F.: Fast Left-Linear Semi-Unification. In: Akl, S.G., Koczkodaj, W.W., Fiala, F. (eds.) ICCI 1990. LNCS, vol. 468, pp. 82–91. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  7. 7.
    Henglein, F.: Type inference with polymorphic recursion. TOPLAS 15(2), 253–289 (1993)CrossRefGoogle Scholar
  8. 8.
    Kapur, D., Musser, D., Narendran, P., Stillman, J.: Semi-unification. TCS 81(2), 169–187 (1991)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Kfoury, A.J., Tiuryn, J., Urzyczyn, P.: Computational consequences and partial solutions of a generalized unification problem. In: Proc. of LICS 1989, pp. 98–104. IEEE (1989)Google Scholar
  10. 10.
    Kfoury, A.J., Tiuryn, J., Urzyczyn, P.: The undecidability of the semi-unification problem. IC 102(1), 83–101 (1993)MathSciNetMATHGoogle Scholar
  11. 11.
    Kfoury, A.J., Tiuryn, J., Urzyczyn, P.: An analysis of ML typability. JACM 41(2), 368–398 (1994)MATHCrossRefGoogle Scholar
  12. 12.
    Leiß, H.: Polymorphic Recursion and Semi-Unification. In: Börger, E., Kleine Büning, H., Richter, M.M. (eds.) CSL 1989. LNCS, vol. 440, pp. 211–224. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  13. 13.
    Leiß, H., Henglein, F.: A Decidable Case of the Semi-Unification Problem. In: Tarlecki, A. (ed.) MFCS 1991. LNCS, vol. 520, pp. 318–327. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  14. 14.
    Lushman, B., Cormack, G.V.: A larger decidable semiunification problem. In: Proc. of PPDP 2007, pp. 143–152. ACM Press (2007)Google Scholar
  15. 15.
    Oliart, A., Snyder, W.: Fast algorithms for uniform semi-unification. JSC 37(4), 455–484 (2004)MathSciNetMATHGoogle Scholar
  16. 16.
    Payet, É.: Loop detection in term rewriting using the eliminating unfoldings. TCS 403(2-3), 307–327 (2008)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Pudlák, P.: On a unification problem related to Kreisel’s conjecture. Commentationes Mathematicae Universitatis Carolinae 29(3), 551–556 (1988)MathSciNetMATHGoogle Scholar
  18. 18.
    Purdom Jr., P.W.: Detecting Looping Simplifications. In: Lescanne, P. (ed.) RTA 1987. LNCS, vol. 256, pp. 54–61. Springer, Heidelberg (1987)CrossRefGoogle Scholar
  19. 19.
    Ružička, P.: An Efficient Decision Algorithm for the Uniform Semi-Unification Problem. In: Tarlecki, A. (ed.) MFCS 1991. LNCS, vol. 520, pp. 415–425. Springer, Heidelberg (1991)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Takahito Aoto
    • 1
  • Munehiro Iwami
    • 2
  1. 1.RIECTohoku UniversityJapan
  2. 2.Dept. of Mathematics and Computer Science, Interdisciplinary Faculty of Science and EngineeringShimane UniversityJapan

Personalised recommendations