Extending resolution for model construction

  • Ricardo Caferra
  • Nicolas Zabel
Selected Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 478)


A method is proposed to systematize the simultaneous search for a refutation and Herbrand models of a given conjecture. It is based on an extension of resolution using equational problems and the inference system included in the method is proved to be sound and refutational complete. For some classes of formulae the method is indeed a decision procedure. Some examples of model construction — including one for which other resolution based decision procedures fail to detect satisfiability — are developed in detail.

Models are built by constructing relations on Herbrand universe. The relationship between these models and finite ones is established. The class of these constructible relations is precisely characterized. Some of the rules introduced, in order to extend resolution, are essentially new. Their necessity in constructing models is proved. A brief comparison with existing methods which bear similarity with ours, either in the use of constraints (a particular case of equational problems) or in the search of a model, shows the originality of our proposal.


Theorem Proving Model Construction Equational Problems Unification Disunification Decision Procedures 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    W. Ackermann. Solvable cases for the decision problem. North-Holland, 1954.Google Scholar
  2. [2]
    W. W. Bledsoe and D. W. Loveland, editors. Automated theorem proving after 25 years, volume 29. Contemporary Mathematics, 1984.Google Scholar
  3. [3]
    T. Boy de la Tour, R. Caferra, and G. Chaminade. Some tools for an inference laboratory (atinf). In Proc. of CADE 9, pages 744–745. Springer-Verlag LNCS 310, 1988.Google Scholar
  4. [4]
    H. Bürckert. Solving disequations in equational theories. In Proc. of CADE 9, pages 517–526. Springer-Verlag LNCS 310, 1988.Google Scholar
  5. [5]
    R. Caferra and N. Zabel. Une extension des tableaux sémantiques utilisant les problèmes équationnels pour guider la recherche de modèles et de réfutations. In Actes 3ème Journées Nationales PRC-GRD Intelligence Artificielle, pages 213–222. Éd. Hèrmes, 1990.Google Scholar
  6. [6]
    H. Comon. Disunification: a survey. Technical Report 540, L.R.I, Université Paris-Sud, 1990. to appear in “Festschrift for Robinson” (J.L. Lassez and G.Plotkin, eds.) MIT Press 1990.Google Scholar
  7. [7]
    H. Comon and P. Lescanne. Equational problem and disunification. JSC, 7:371–425, 1989.Google Scholar
  8. [8]
    V. J. Digricoli and M. C. Harrison. Equality based binary resolution. JACM, 33(2):253–289, 1986.Google Scholar
  9. [9]
    B. Dreben and W. D. Goldfarb. The decision problem — Solvable Classes of Quantificational Formulas. Addison-Wesley, 1979.Google Scholar
  10. [10]
    J. H. Gallier. Logic for Computer Science. Harper & Row, 1986.Google Scholar
  11. [11]
    J. Jaffar and J.-L. Lassez. Constraint logic programming. In Proc. of 14th ACM Symp. on Princ. of Prog. Lang., pages 111–119, Munich, 1987.Google Scholar
  12. [12]
    W. H. Joyner. Resolution strategies as decision procedures. JACM, 23(3):253–417, 1976.Google Scholar
  13. [13]
    C. Kirchner and H. Kirchner. Constrained equational reasoning. In Proc of the ACM SIGSAM — Int. Symp. Symbolic and Algebraic Computation, pages 382–389, Portland, U.S.A, 1989.Google Scholar
  14. [14]
    C. Kirchner and P. Lescanne. Solving disequations. Technical Report 686, INRIA, 1987.Google Scholar
  15. [15]
    D. W. Loveland. Automatic Theorem Proving: a logical basis. North Holland, 1978.Google Scholar
  16. [16]
    M. Rusinowitch. Démonstration automatique par des techniques de réécriture. Inter Editions, Paris, 1989.Google Scholar
  17. [17]
    M. E. Stickel. Automated deduction by theory resolution. JAR, 1(4):333–355, 1985.Google Scholar
  18. [18]
    S. Winker. Generation and verification of finite models and counter-examples using an automated theorem prover answering two open questions. JACM, 29(2):273–284, 1982.Google Scholar
  19. [19]
    L. Wos. Automated reasoning: 33 basic research problems. Prentice-Hall, 1988.Google Scholar
  20. [20]
    L. Wos and S. Winker. Open questions solved with the assistance of AURA, 1984. in [2].Google Scholar
  21. [21]
    N. Zamov. On a bound for a complexity of terms in the resolution method. In Proc. Steklov of Inst. Math. 121, pages 1–10, 1972.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Ricardo Caferra
    • 1
  • Nicolas Zabel
    • 1
  1. 1.Laboratoire d'Informatique Fondamentale et d'Intelligence ArtificielleImag-CnrsGrenobleFrance

Personalised recommendations