On modularity in term rewriting and narrowing

  • Christian Prehofer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 845)


We introduce a modular property of equational proofs, called modularity of normalization, for the union of term rewrite systems with shared symbols. The idea is, that every normalization with R=R1+R2 may be obtained by first normalizing with R1 followed by an R2 normalization.

We develop criteria for this that cover non-convergent TRS R, where, as the main restriction, R1 is required to be left-linear and convergent. As interesting applications we consider solving equations modulo a theory given by a TRS. Here we present a modular narrowing strategy that can be combined with nearly all common narrowing strategies. Furthermore, we also prove some modularity results for decidability of unification and matching (via termination of narrowing).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Antoy, R. Echahed, and M. Hanus. A needed narrowing strategy. In Proc. 21st ACM Symposium on Principles of Programming Languages, pages 268–279, Portland, 1994.Google Scholar
  2. 2.
    L. Bachmair and N. Dershowitz. Commutation, transformation, and termination. In Joerg H. Siekmann, editor, Proc. 8th Int. Conf. Automated Deduction. LNCS 607, 1986.Google Scholar
  3. 3.
    A. Bockmayr. Narrowing with inductively defined functions. Technical report, Univ. Kaiserslautern, 1986. SEKI Memo 25/86.Google Scholar
  4. 4.
    Alexander Bockmayr, Stefan Krischer, and Andreas Werner. An optimal narrowing strategy for general canonical systems. In Michaël Rusinowitch and Jean-Luc Rémy, editors, Conditional Term Rewriting Systems, Third International Workshop, LNCS 656, pages 483–497, Pont-à-Mousson, France, July 8–10, 1992. Springer-Verlag.Google Scholar
  5. 5.
    P. G. Bosco, E. Giovanetti, and C. Moiso. Narrowing vs. SLD-resolution. Theoretical Computer Science, 59:3–23, 1988.Google Scholar
  6. 6.
    Jim Christian. Some termination criteria for narrowing and E-narrowing. In Kapur [14], pages 582–588.Google Scholar
  7. 7.
    N. Dershowitz and J.-P. Jouannaud. Rewrite systems. In Jan Van Leeuwen, editor, Handbook of Theoretical Computer Science Volume B: Formal Models and Semantics, pages 243–320. Elsevier, 1990.Google Scholar
  8. 8.
    Nachum Dershowitz, Subrata Mitra, and G. Sivakumar. Decidable matching for convergent systems (preliminary version). In Kapur [14], pages 589–602.Google Scholar
  9. 9.
    A. Geser. Relative Termination. PhD thesis, Univ. Passau, 1990.Google Scholar
  10. 10.
    B. Gramlich. Generalized sufficient conditions for modular termination of rewriting. In H. Kirchner and G. Levi, editors, Algebraic and Logic Programming: Proc. of the Third International Conference, pages 53–68. Springer, Berlin, Heidelberg, 1992.Google Scholar
  11. 11.
    M. Hanus. The integration of functions into logic programming: A survey. 1994. To appear in Journal of Logic Programming.Google Scholar
  12. 12.
    M. Hanus. Lazy unification with simplification. In Proc. 5th European Symposium on Programming, pages 272–286. Springer LNCS 788, 1994.Google Scholar
  13. 13.
    Jean-Marie Hullot. Canonical forms and unification. In W. Bibel and R. Kowalski, editors, Proceedings of 5th Conference on Automated Deduction, pages 318–334. Springer Verlag, LNCS, 1980.Google Scholar
  14. 14.
    Deepak Kapur, editor. 11th International Conference on Automated Deduction, LNAI 607, Saratoga Springs, New York, USA, June 15–18, 1992. Springer-Verlag.Google Scholar
  15. 15.
    Aart Middeldorp. Modular aspects of properties of term rewriting systems related to normal forms. In Proc. 3rd Int. Conf. Rewriting Techniques and Applications, pages 263–277. LNCS 355, 1989.Google Scholar
  16. 16.
    Aart Middeldorp. Modular Properties of Term Rewriting Systems. PhD thesis, Free University Amsterdam, 1990.Google Scholar
  17. 17.
    Aart Middeldorp and Y. Toyama. Completeness of combinations of constructor systems. In Proc. 4th Int. Conf. Rewriting Techniques and Applications. LNCS 488, 1991.Google Scholar
  18. 18.
    T. Nipkow and G. Weikum. Operationelle Semantik axiomatisch spezifizierter Abstrakter Datentypen. Master's thesis, TH Darmstadt, 1982. In German.Google Scholar
  19. 19.
    W. Nutt and P. Réty and. Basic narrowing revisited. In C. Kirchner, editor, Unification. Academic Press, 1990.Google Scholar
  20. 20.
    M. R. K. Krisna Rao. Completeness of hierarchical combinations of term rewriting system. In R.K. Shyamasundar, editor, Foundations of Software Technology and Theoretical Computer Science, pages 125–138. LNCS 761, 1993.Google Scholar
  21. 21.
    J. Raoult and J. Vuillemin. Operational and semantic equivalences between recursive Programms. J. of th ACM, 27:772–796, 1980.Google Scholar
  22. 22.
    U. S. Reddy. Narrowing as the operational semantics of functional languages. In Symposium on Logic Programming, pages 138–151. IEEE Computer Society, Technical Committee on Computer Languages, The Computer Society Press, July 1985.Google Scholar
  23. 23.
    K. Stroetmann. The union of rewrite systems. unpublished, 1992.Google Scholar
  24. 24.
    Y. Toyama. On the Church-Rosser property for the direct sum of term rewriting systems. Journal of the ACM, 34(1):128–143, 1987.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Christian Prehofer
    • 1
  1. 1.Fakultät für InformatikTechnische Universität MünchenMünchenGermany

Personalised recommendations