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Complete sets of reductions modulo associativity, commutativity and identity

  • Timothy B. Baird
  • Gerald E Peterson
  • Ralph W. Wilkerson
Regular Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 355)

Abstract

We describe the theory and implementation of a process which finds complete sets of reductions modulo equational theories which contain one or more associative and commutative operators with identity (ACI theories). We emphasize those features which distinguish this process from the similar one which works modulo associativity and commutativity. A primary difference is that for some rules in ACI complete sets, restrictions are required on the substitutions allowed when the rules are applied. Without these restrictions, termination cannot be guaranteed. We exhibit six examples of ACI complete sets that were generated by an implementation.

Keywords

Equational Theory Unification Algorithm Critical Pair Unification Mechanism Completion Process 
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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References

  1. [BD87]
    L. Bachmair and N. Dershowitz, “Completion for rewriting modulo a congruence,” Rewriting Techniques and Applications, Lecture Notes in Computer Science 256, Springer-Verlag (1987), pp. 192–203.Google Scholar
  2. [Ba88]
    T. Baird, “Complete sets of reductions modulo a class of equational theories which generate infinite congruence classes,” Ph.D. Dissertation, University of Missouri—Rolla, Rolla, MO, (1988).Google Scholar
  3. [FG84]
    R. Forgaard and J. V. Guttag, “A term rewriting system generator with failure-resistant Knuth-Bendix," Technical Report, MIT Laboratory for Computer Science, Massachussets Institute of Technology, Cambridge, MA (1984).Google Scholar
  4. [JK86]
    J.-P. Jouannaud and H. Kirchner, “Completion of a set of rules modulo a set of equations,” SIAM Journal of Computing, 15 (1986), pp. 1155–1194.Google Scholar
  5. [KB70]
    D. Knuth and P. Bendix, “Simple word problems in universal algebras,” Computational Problems in Abstract Algebras, J. Leech, ed., Pergamon Press, Oxford, England, (1970), pp. 263–297.Google Scholar
  6. [LB77]
    D. Lankford and A. Ballantyne, “Decision procedures for simple equational theories with commutative-associative axioms: complete sets of commutative-associative reductions,” Memo ATP-39, Dept. of Mathematics and Computer Science, University of Texas, Austin, Texas (1977).Google Scholar
  7. [Ma88]
    B. Mayfield, “The role of term symmetry in equational unification and completion procedures,”, Ph.D. Dissertation, University of Missouri—Rolla, Rolla, MO, 1988.Google Scholar
  8. [PS81]
    G. Peterson and M. Stickel, “Complete sets of reductions for some equational theories,” J. ACM, 28 (1981), pp. 233–264.Google Scholar
  9. [St81]
    M. Stickel, “A unification algorithm for associative-commutative functions,” J. ACM, 28 (1981), pp. 423–434.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Timothy B. Baird
    • 1
  • Gerald E Peterson
    • 2
  • Ralph W. Wilkerson
    • 3
  1. 1.Department of Mathematics and Computer ScienceHarding UniversitySearcy
  2. 2.McDonnell Douglas Corporation, W400/105/2/206St. Louis
  3. 3.Department of Computer ScienceUniversity of Missouri—RollaRolla

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