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Inductive proofs by specification transformations

  • Hubert Comon
Regular Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 355)

Abstract

We show how to transform equational specifications with relations between constructors (or without constructors) into order-sorted equational specifications where every function symbol is either a free constructor or a completely defined function.

This method allows to reduce the problem of inductive proofs in equational theories to Huet and Hullot's proofs by consistency [HH82]. In particular, it is no longer necessary to use the socalled “inductive reducibility test” which is the most expensive part of the Jouannaud and Kounalis algorithm [JK86].

Keywords

Function Symbol Equational Theory Ground Term Tree Automaton Inductive Proof 
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. [Bac88]
    L. Bachmair. Proof by consistency in equational theories. In Proc. 3rd IEEE Symp. Logic in Computer Science, Edinburgh, July 1988.Google Scholar
  2. [CL88]
    H. Comon and P. Lescanne. Equational Problems and Disunification. Research Report Lifia 82 Imag 727, Univ. Grenoble, May 1988. To appear in J. Symbolic Computation.Google Scholar
  3. [Com88a]
    H. Comon. An effective method for handling initial algebras. In Proc. 1st Workshop on Algebraic and Logic Programming, Gaussig, 1988.Google Scholar
  4. [Com88b]
    H. Comon. Unification et Disunification: Théorie et Applications. Thèse de Doctorat, I.N.P. de Grenoble, France, 1988.Google Scholar
  5. [CR87]
    H. Comon and J.-L. Remy. How to Characterize the Language of Ground Normal Forms. Research Report 676, INRIA, June 1987.Google Scholar
  6. [Fri86]
    L. Fribourg. A strong restriction of the inductive completion procedure. In Proc. 13th ICALP, Rennes, LNCS 226, pages 105–115, Springer-Verlag, 1986.Google Scholar
  7. [GB85]
    J. H. Gallier and R. V. Book. Reductions in tree replacement systems. Theoretical Computer Science, 37:123–150, 1985.Google Scholar
  8. [GM87]
    J. Goguen and J. Meseguer. Order-Sorted Algebra I: Partial and Overloaded Operators, Errors and Inheritance. Draft, Computer Science Lab., SRI International, 1987.Google Scholar
  9. [Gog80]
    J. A. Goguen. How to prove inductive hypothesis without induction. In Proc. 5th Conf. on Automated Deduction, LNCS 87, 1980.Google Scholar
  10. [HH82]
    G. Huet and J.-M. Hullot. Proofs by induction in equational theories with constructors. Journal of Computer and System Sciences, 25(2), 1982.Google Scholar
  11. [JK86]
    J.-P. Jouannaud and E. Kounalis. Automatic proofs by induction in equational theories without constructors. In Proc. 1st IEEE Symp. Logic in Computer Science, Cambridge, Mass., June 1986.Google Scholar
  12. [KKM88]
    C. Kirchner, H. Kirchner, and J. Meseguer. Operational semantics of obj-3. In Proc. 15th Int. Conf on Automata, Languages and Programming, LNCS 317, Springer-Verlag, July 1988.Google Scholar
  13. [KM86]
    D. Kapur and R. D. Musser. Inductive reasoning with incomplete specifications. In Proc. 1st IEEE Symp. Logic in Computer Science, Cambridge, Mass., June 1986.Google Scholar
  14. [KM87]
    D. Kapur and D. Musser. Proof by consistency. Artificial Intelligence, 31(2), February 1987.Google Scholar
  15. [KNZ85]
    D. Kapur, P. Narendran, and H. Zhang. On Sufficient Completeness and Related Properties of Term Rewriting Systems. Research Report, General Electric Company, October 1985. Preprint.Google Scholar
  16. [KNZ86]
    D. Kapur, P. Narendran, and H. Zhang. Proof by induction using test sets. In Proc. 8th Conf. on Automated Deduction, Oxford, LNCS 230, Springer-Verlag, July 1986.Google Scholar
  17. [Kuc87]
    W. Kuchlin. Inductive Completion by Ground Proofs Transformation. Research Report, University of Delaware, February 1987.Google Scholar
  18. [Lan81]
    D. Lankford. A simple explanation of inductionless induction. Technical Report MTP-14, Mathematics Department, Louisiana Tech. Univ., 1981.Google Scholar
  19. [Mus80]
    D. Musser. Proving inductive properties of abstract data types. In Proc. 7th ACM Symp. Principles of Programming Languages, Las Vegas, 1980.Google Scholar
  20. [Pla85]
    D. Plaisted. Semantic confluence tests and completion methods. Information and Control, 65:182–215, 1985.Google Scholar
  21. [SNGM87]
    G. Smolka, W. Nutt, J. Goguen, and J. Meseguer. Order-Sorted Equational Computation. SEKI Report SR-87-14, Univ. Kaiserslautern, December 1987.Google Scholar
  22. [Tha85]
    S. R. Thatte. On the correspondance between two classes of reductions systems. Information Processing Letters, 20:83–85, February 1985.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Hubert Comon
    • 1
  1. 1.Laboratoire d'Informatique fondamentale et d'Intelligence ArtificielleInstitut IMAGGrenoble cedexFrance

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