Elimination of Negation in a Logical Framework

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


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.


Logic Program Logic Programming Operational Semantic Deductive System Logical Framework 
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. [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.CrossRefzbMATHMathSciNetGoogle 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.CrossRefzbMATHMathSciNetGoogle 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

Personalised recommendations