How to prove equivalence of term rewriting systems without induction

  • Yoshihito Toyama
Term Rewriting Systems
Part of the Lecture Notes in Computer Science book series (LNCS, volume 230)


A simple method is proposed for testing equivalence in a restricted domain of two given term rewriting systems. By using the Church-Rosser property and the reachability of term rewriting systems, the method allows us to prove equivalence of these systems without the explicit use of induction; this proof usually requires some kind of induction. The method proposed is a general extension of inductionless induction methods developed by Musser, Goguen, Huet and Hullot, and allows us to extend inductionless induction concepts to not only term rewriting systems with the termination property, but also various reduction systems: term rewriting systems without the termination property, string rewriting systems, graph rewriting systems, combinatory reduction systems, and resolution systems. This method is applied to test equivalence of term rewriting systems, to prove the inductive theorems, and to derive a new term rewriting system from a given system by using equivalence transformation rules.


Normal Form Function Symbol Reduction System Equivalence Transformation Termination Property 
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. [1]
    Barendregt,H.P. “The lambda calculus, its syntax and semantics”, North-Holland (1981).Google Scholar
  2. [2]
    Book, R. “Confluent and other types of Thue systems”, J.ACM, Vol.29 (1982), pp.171–182.CrossRefGoogle Scholar
  3. [3]
    Burstall, R.M. and Darlington, J. “A transformation system for developing recursive programs”, J.ACM, Vol.24 (1977), pp.44–67.CrossRefGoogle Scholar
  4. [4]
    Goguen,J.A. “How to prove algebraic inductive hypotheses without induction, with applications to the correctness of data type implementation”, Lecture Notes in Comput. Sci., Vol.87, Springer-Verlag (1980), pp.356–373.Google Scholar
  5. [5]
    Huet, G. “Confluent reductions: abstract properties and applications to term rewriting systems”, J.ACM, Vol.27 (1980), pp.797–821.CrossRefGoogle Scholar
  6. [6]
    Huet,G. and Oppen,D.C. “Equations and rewrite rules: a survey”, Formal languages: perspectives and open problems, Ed.Book,R., Academic Press (1980), pp.349–393.Google Scholar
  7. [7]
    Huet, G. and Hullot, J.M. “Proofs by induction in equational theories with constructors”, J. Comput. and Syst.Sci., Vol.25 (1982), pp.239–266.CrossRefGoogle Scholar
  8. [8]
    Kapur,D. and Narendran,P. “An equational approach to theorem proving in first-order predicate calculus”, General Electric Corporate Research Development Report, No.84CRD322, (1985).Google Scholar
  9. [9]
    Kirchner, H. “A general inductive completion algorithm and application to abstract data types”, Lecture Notes in Comput. Sci., Vol.170, Springer-Verlag (1985), pp.282–302.Google Scholar
  10. [10]
    Klop,J.W. “Combinatory reduction systems”, Dissertation, Univ. of Utrecht (1980).Google Scholar
  11. [11]
    Knuth,D.E. and Bendix,P.G. “Simple word problems in universal algebras”, Computational problems in abstract algebra, Ed.Leech,J., Pergamon Press (1970), pp.263–297.Google Scholar
  12. [12]
    Kounalis, E. “Completeness in data type specifications”, Lecture Notes in Comput. Sci., Vol.204, Springer-Verlag (1985), pp.348–362.Google Scholar
  13. [13]
    Musser,D.R. “On proving inductive properties of abstract data types”, Proc. 7th ACM Sympo. Principles of programming languages (1980), pp.154–162.Google Scholar
  14. [14]
    Nipkow, T. and Weikum, G. “A decidability results about sufficient-completeness of axiomatically specified abstract data type”, Lecture Notes in Comput. Sci., Vol.145, Springer-Verlag (1983), pp.257–267.Google Scholar
  15. [15]
    Paul,E. “Proof by induction in equational theories with relations between constructors”, 9th Colloquium on trees in algebra and programming, Ed. Courcelle,B., Cambridge University Press (1984), pp.211–225.Google Scholar
  16. [16]
    Paul, E. “On solving the equality problem in theories defined by Horn clauses”, Lecture Notes in Comput. Sci., Vol.204, Springer-Verlag (1985), pp.363–377.Google Scholar
  17. [17]
    Raoult, J.C. “On graph rewriting”, Theoretical Comput. Sci. Vol.32 (1984), pp.1–24.CrossRefGoogle Scholar
  18. [18]
    Thiel,J.J. “Stop losing sleep over incomplete data type specifications”, Proc. 11th ACM Sympo. Principles of programming languages (1984), pp.76–82.Google Scholar
  19. [19]
    Toyama,Y. “On commutativity of term rewriting systems”, Trans. IECE Japan, J66-D, 12, pp.1370–1375 (1983), in Japanese.Google Scholar
  20. [20]
    Toyama, Y. “On equivalence transformations for term rewriting systems”, RIMS Symposia on Software Science and Engineering, Kyoto (1984), Lecture Notes in Comput. Sci., Vol.220, Springer-Verlag (1986), pp.44–61.Google Scholar
  21. [21]
    Toyama,Y. “On the Church-Rosser property for the direct sum of term rewriting systems”, to appear in J.ACM.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Yoshihito Toyama
    • 1
  1. 1.NTT Electrical Communications LaboratoriesMusashino-shi, TokyoJapan

Personalised recommendations