Computational representations of herbrand models using grammars

  • Robert Matzinger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1258)


Finding computationally valuable representations of models of predicate logic formulas is an important subtask in many fields related to automated theorem proving, e.g. automated model building or semantic resolution. In this article we investigate the use of context-free languages for representing single Herbrand models, which appear to be a natural extension of “linear atomic representations” already known from the literature. We focus on their expressive power (which we find out to be exactly the finite models) and on algorithmic issues like clause evaluation and equivalence test (which we solve by using a resolution theorem prover), thus proving our approach to be an interesting base for investigating connections between formal language theory and automated theorem proving and model building.


Function Symbol Predicate Symbol Ground Term Ground Atom Finite Model 
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. [BCP94]
    C. Bourely, R. Caferra, and N. Peltier. A method for building models automatically. Experiments with an extension of otter. In Alan Bundy, editor, 12th International Conference on Automated Deduction, pages 72–86, Nancy/France, June 1994. Springer Verlag, LNAI 814.Google Scholar
  2. [BJ74]
    G. Boolos and R. Jeffrey. Computability and Logic. Cambridge University Press, 1974.Google Scholar
  3. [CHK90]
    H. Chen, J. Hsiang, and H.-C. Kong. On finite representations of infinite sequences of terms. In S. Kaplan and M. Okada, editors, Proceedings 2nd International Workshop on Conditional and Typed Rewriting Systems, Montreal (Canada), volume 516 of Lecture Notes in Computer Science, pages 100–114. Springer, June 1990.Google Scholar
  4. [CL73]
    C.-L. Chang and R. C. T. Lee. Symbolic Logic an Mechanical Theorem Proving. Academic Press, New York, 1973.Google Scholar
  5. [Com95]
    H. Comon. On unification of terms with integer exponents. Mathematical Systems Theory, 28(1):67–88, 1995.Google Scholar
  6. [CZ92]
    R. Caferra and N. Zabel. A method for simultanous search for refutations and models by equational constraint solving. Journal of Symbolic Computation, 13(6):613–641, June 1992.Google Scholar
  7. [Fer90]
    C. Fermüller. Deciding some Horn clause sets by resolution. In Yearbook of the Kurt-Gödel-Society 1989, pages 60–73, Vienna, 1990.Google Scholar
  8. [FL93]
    C. Fermüller and A. Leitsch. Model building by resolution. In Computer Science Logic (CSL'92), pages 134–148, San Miniato, Italy, 1993. Springer Verlag. LNCS 702.Google Scholar
  9. [FL96]
    C. Fermüller and A. Leitsch. Hyperresolution and automated model building. J. of Logic and Computation, 6(2):173–203, 1996.Google Scholar
  10. [FLTZ93]
    C. Fermüller, A. Leitsch, T. Tammet, and N. Zamov. Resolution Methods for the Decision Problem. Springer Verlag, 1993. LNAI 679.Google Scholar
  11. [GS84]
    F. Gécseg and M. Steinby. Tree Automata. Akadémiai Kiadó, Budapest, 1984.Google Scholar
  12. [HG97]
    M. Hermann and R. Galbavý. Unification of infinite sets of terms schematized by primal grammars. Theoretical Computer Science, 176, April 1997. To appear.Google Scholar
  13. [Joy76]
    W.H. Joyner. Resolution strategies as decision procedures. Journal of ACM, 23(1):398–417, 1976.Google Scholar
  14. [Lov78]
    D. Loveland. Automated Theorem Proving — A Logical Basis. North Holland, 1978.Google Scholar
  15. [McA92]
    D. McAllester. Grammar rewriting. In D. Kapur, editor, Proceedings 11th International Conference on Automated Deduction, Saratoga Springs, (New York, USA), volume 607 of Lecture Notes in Computer Science (in Art. Intelligence). Springer, 1992.Google Scholar
  16. [RS92]
    G. Rozenberg and A. Salomaa, editors. Lindenmayer Systems. Springer, 1992.Google Scholar
  17. [Sal73]
    A. Salomaa. Formal languages. Academic Press, Orlando, 1973.Google Scholar
  18. [Sal92]
    G. Salzer. The unification of infinite sets of terms and its applications. In A. Voronkov, editor, Proceedings Logic Programming and Automated Reasoning, St. Petersburg (Russia), volume 624 of Lecture Notes in Computer Science (in Art. Intelligence). Springer, 1992.Google Scholar
  19. [Sla92]
    J. Slaney. FINDER (finite domain enumerator): Notes and guide. Technical Report TR-ARP-1/92, Australien National University Automated Reasoning Project, Canberra, 1992.Google Scholar
  20. [Sla93]
    J. Slaney. SCOTT: A model-guided theorem prover. In Proceedings of the 13th international joint conference on artificial intelligence (IJCAI'93), volume 1, pages 109–114. Morgan Kaufmann Publishers, 1993.Google Scholar
  21. [Tam91]
    T. Tammet. Using resolution for deciding solvable classes and building finite models. In Baltic Computer Science, pages 33–64. Springer Verlag, 1991. LNCS 502.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Robert Matzinger
    • 1
  1. 1.Technische Universität WienAustria

Personalised recommendations