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Confluent and coherent equational term rewriting systems application to proofs in abstract data types

  • Jean-Pierre Jouannaud
Contributed Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 159)

Abstract

The well known Knuth and Bendix completion procedure computes a convergent term rewriting system from a given set of equational axioms. We describe here an abstract model of computation to handle the case where some axioms cannot be treated as rewrite rules without loosing the required termination property. We call Equational Term Rewriting Systems such mixted sets of rules and equations. We show that two abstract properties, namely E-confluence and E-coherence are both necessary and sufficient ones to compute with these models. These abstract properties can be checked on critical pairs yielding Huet's classical results on “confluence modulo” as well as a more general version of Peterson and Stickel's without any linearity hypothesis on the equations. These results are finally used to check consitency of an algebraic specification of data type.

Keywords

Normal Form Equational Theory Critical Pair Abstract Property Linearity Hypothesis 
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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9. Bibliography

  1. [DER,/9&82]
    DERSHOWITZ N.: “Orderings for term-rewriting systems” Proc 20th FOCS, pp 123–131 (1979) and TCS 17-3 (1982)Google Scholar
  2. [DER,82]
    DERSHOWITZ N.: “Computing with term rewriting systems” to be publishedGoogle Scholar
  3. [FAY,/9]
    FAY M.: “First order unification in an equational theory” Proc. 4th CADE, Austin Texas (19/9)Google Scholar
  4. [GOG,80]
    GOGUEN J.A.: “How to prove algebraic inductive hypotheses without induction, with application to the correctness of data type implementation” Proc. 5th CADE, les Arcs (1980)Google Scholar
  5. [H&D,82]
    HSIANG J. DERSHOWITZ N.: “Using rewrites methods for clausal and non clausal theorem proving” Proc. 10th ICALP (1983)Google Scholar
  6. [H&H,80]
    HUET G. HULLOT J.M.: “Proofs by induction in equational theories with constructors” Proc. 21th FOCS (1980) and JCSS 25-2 (1982)Google Scholar
  7. [H&P,82]
    HSIANG J. PLAISTED D.A.: “A deductive program generation system” to be publishedGoogle Scholar
  8. [HUE,77&80]
    HUET G.: “Confluent reductions: abstract properties and applications to term rewriting systems” Proc. 18th. FOCS (19//) and JACM 2/–4 pp /9/–821 (1980)Google Scholar
  9. [HUE,81]
    HUET G.: “A complete proof of correctness of the Knuth and Bendix completion algorithm” JCSS 23, pp 11–21 (1981)Google Scholar
  10. [HUL,80]
    HULLOT J.M.: “Canonical forms and unification” Proc. 5th CADE, Les Arcs (1980)Google Scholar
  11. [JEA,80]
    JEANROND H.J.: “Deciding unique termination of permutative rewriting systems: choose your term algebra carefully” Proc. 5th CADE, Les Arcs (1980)Google Scholar
  12. [JKK,82]
    JOUANNAUD J.P. KIRCHNER C. KIRCHNER H.: “Incremental construction of unification algorithms in equationnal theories” Proc. 10th ICALP, Barcelonna (1983).Google Scholar
  13. [JKR,83]
    JOUANNAUD J.P. KIRCHNER H. REMY J.L.: “Churh-Rosser properties of equational term rewriting systems: new results” to be published.Google Scholar
  14. [JLR,82]
    JOUANNAUD J.P. LESCANNE P. REINIG F.: “Recursive decomposition ordering” in “Formal description of programming concepts 2” Ed. BJORNER D., North Holland (1982)Google Scholar
  15. [K&B,/0]
    KNUTH D. BEND IX P.: “Simple word problems in universal algebras” in “Computational problems in abstract algebra” Leech J. ed. Pergamon Press, pp 263–297 (1970)Google Scholar
  16. [K&L,80]
    KAMIN S. LEVY J.J.: “Attempts for generalizing the recursive path ordering” unpublished notes (1980)Google Scholar
  17. [LAN,81]
    LANKFORD D.S.: “A simple explanation of inductionless induction” Louisiana Tech. University, Math. Dept. Rep MTP-14 (1981)Google Scholar
  18. [L&B,//a]
    LANKFORD D.S. BALLANTYNE A.M.: “Decision procedures for simple equational theories with permutative axioms: complete sets of permutative reductions” Rep. ATP-3/, Dpt. of Comp. Sc., Univ. of Texas at AustinGoogle Scholar
  19. [L&B,7/b]
    LANKFORD D.S. BALLANTYNE A.M.: “Decision procedures for simple equational theories with commutative-associative axioms: complete sets of commutative-associative reductions” Rep. ATP-39, Dpt. of Comp. Sc., Univ. of Texas at AustinGoogle Scholar
  20. [LES,82]
    LESCANNE P.: “Computer experiments with the REVE term rewriting system generator” Proc. 10th POPL conference (1983)Google Scholar
  21. [MUS,80b]
    MUSSER D.R.: “On proving inductive properties of abstract data types” Proc. 7th POPL Conference, Las Vegas (1980)Google Scholar
  22. [PAD,82]
    PADAWITZ P.: “Equational data type specification and recursive program scheme” in “Formal Description of Programming Concepts 2” Ed. BJORNER D., North Holland (1982)Google Scholar
  23. [P&S,81]
    PETERSON G.E. STICKEL M.E.: “Complete sets of reductions for equational theories with complete unification algorithms” J.ACM 28, no.2, pp 233–264 (1981)CrossRefGoogle Scholar
  24. [STI,81]
    STICKEL M.E.: “A unification algorithm for associative-commutative functions” J.ACM 28-3, pp 423–434 (1981)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1983

Authors and Affiliations

  • Jean-Pierre Jouannaud
    • 1
  1. 1.Centre de Recherche en Informatique de Nancy et Greco de Programmation Campus ScientifiqueVandoeuvre-les-Nancy Cedex

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