Elimination of Negation in a Logical Framework

  • Alberto Momigliano
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1862)

Abstract

Logical frameworks with a logic programming interpretation such as hereditary Harrop formulae (HHF) [12] cannot express directly negative information, although negation is a useful specification tool. Since negation-as-failure does not fit well in a logical framework, especially one endowed with hypothetical and parametric judgments, we adapt the idea of elimination of negation introduced in [17] for Horn logic to a fragment of higher-order HHF. This entails finding a middle ground between the Closed World Assumption usually associated with negation and the Open World Assumption typical of logical frameworks; the main technical idea is to isolate a set of programs where static and dynamic clauses do not overlap.

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References

  1. [1]
    D. P. A. Brogi, P. Mancarella and F. Turini. Universal quantification by case analysis. In Proc. ECAI-90, pages 111–116, 1990.Google Scholar
  2. [2]
    K. Apt and R. Bol. Logic programming and negation. Journal of Logic Programming, 19/20:9–72, May/July 1994.Google Scholar
  3. [3]
    R. Barbuti, P. Mancarella, D. Pedreschi, and F. Turini. A transformational approach to negation in logic programming. Journal of Logic Programming, 8:201–228, 1990.CrossRefMATHMathSciNetGoogle Scholar
  4. [4]
    A. Bonner. Hypothetical reasoning with intuitionistic logic. In R. Demolombe and T. Imielinski, editors, Non-Standard Queries and Answers, volume 306 of Studies in Logic and Computation, pages 187–219. Oxford University Press, 1994.Google Scholar
  5. [5]
    K. L. Clark. Negation as failure. In H. Gallaire and J. Minker, editors, Logic and Databases, pages 293–322. Plenum Press, New York, 1978.Google Scholar
  6. [6]
    D. M. Gabbay. N-Prolog: An extension of Prolog with hypothetical implications II. Logical foundations and negation as failure. Journal of Logic Programming, 2(4):251–283, Dec. 1985.Google Scholar
  7. [7]
    L. Giordano and N. Olivetti. Negation as failure and embedded implication. Journal of Logic Programming, 36(2):91–147, August 1998.Google Scholar
  8. [8]
    J. Harland. On Hereditary Harrop Formulae as a Basis for Logic Programming. PhD thesis, Edinburgh, Jan. 1991.Google Scholar
  9. [9]
    R. Harper, F. Honsell, and G. Plotkin. A framework for defining logics. Journal of the Association for Computing Machinery, 40(1):143–184, Jan. 1993.Google Scholar
  10. [10]
    J.-L. Lassez and K. Marriot. Explicit representation of terms defined by counter examples. Journal of Automated Reasoning, 3(3):301–318, Sept. 1987.Google Scholar
  11. [11]
    R. McDowell and D. Miller. A logic for reasoning with higher-order abstract syntax: An extended abstract. In G. Winskel, editor, Proceedings of the Twelfth Annual Symposium on Logic in Computer Science, pages 434–445, Warsaw, Poland, June 1997.Google Scholar
  12. [12]
    D. Miller, G. Nadathur, F. Pfenning, and A. Scedrov. Uniform proofs as a foundation for logic programming. Annals of Pure and Applied Logic, 51:125–157, 1991.CrossRefMATHMathSciNetGoogle Scholar
  13. [13]
    A. Momigliano. Elimination of Negation in a Logical Framework. PhD thesis, Carnegie Mellon University, 2000. Forthcoming.Google Scholar
  14. [14]
    A. Momigliano and F. Pfenning. The relative complement problem for higher-order patterns. In D. D. Schreye, editor, Proceedings of the 1999 International Conference on Logic Programming (ICLP’99), pages 389–395, La Cruces, New Mexico, 1999. MIT Press.Google Scholar
  15. [15]
    G. Nadathur and D. Miller. An overview of λProlog. In K. A. Bowen and R. A Kowalski, editors, Fifth International Logic Programming Conference, pages 810–827, Seattle, Washington, Aug. 1988. MIT Press.Google Scholar
  16. [16]
    F. Pfenning. Logical frameworks. In A Robinson and A. Voronkov, editors, Handbook of Automated Reasoning. Elsevier Science Publishers, 2000. In preparation.Google Scholar
  17. [17]
    T. Sato and H. Tamaki. Transformational logic program synthesis. In International Conference on Fifth Generation Computer Systems, 1984.Google Scholar
  18. [18]
    C. Schürmann. Automating the Meta-Theory of Deductive Systems. PhD thesis, Carnegie-Mellon University, 2000. forthcoming.Google Scholar
  19. [19]
    C. Schürmann and F. Pfenning. Automated theorem proving in a simple meta-logic for LF. In C. Kirchner and H. Kirchner, editors, Proceedings of the 15th International Conference on Automated Deduction (CADE-15), pages 286–300, Lindau, Germany, July 1998. Springer-Verlag LNCS 1421.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Alberto Momigliano
    • 1
  1. 1.Department of PhilosophyCarnegie Mellon UniversityPittsburghUSA

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