Skip to main content
SpringerLink
Log in
Book cover

European Workshop on Logics in Artificial Intelligence

JELIA 1990: Logics in AI pp 180–195Cite as

The optimal model of a program with negation

  • Selected Papers
  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 478))

Abstract

The aim of this paper is to provide another denotational semantics than the least fixpoint semantics for a logic program with negation.

It takes place in a more general work about three-valued (partial) logic and logic programming. We have developped a logical and algebrical theory which seems well-suited to logic programs with negation. In this theory, we have extended the "consequence" operator of Van Emden and Kowalski associated with a program without negation to a "consequence" operator for programs with negation taking inconsistent programs in account. As the models of a program are exactly the post-fixpoints of this operator, the first provided denotational semantics of a program is the least fixpoint semantics.

As an intersection of three-valued Herbrand models of a program is still a model of this program, this least fixpoint is also the set of all the ground literals of a program which are three-valued logical consequences of this program. Even if the least model is thus unique and defined for every consistent program, it is a partial model and may contain only few literals. A second denotational semantics is given by considering the three-valued Herbrand interpretations as partial functions from the set of ground atoms to the set of the truth values {T, F}; the theory of Manna and Shamir can be adapted in two ways:

  • The first one is the notion of optimal fixpoint of the operator.

  • The second one seems more interesting for our purpose, since the models of a logic program with negation are exactly the post-fixpoints of the consequence operator. It is the notion of optimal post-fixpoint of our "consequence" operator, which is the optimal model of our program. This notion is about to strengthen the three-valued logic. This optimal model is still unique and defined for every consistent program. It is still a partial model but generally strictly contains the least model, and thus really provides another denotational semantics than the least fixpoint semantics for a program with negation. It may even be more interesting since it contains more informations on the program itself.

This is a preview of subscription content, log in via an institution.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. K. Apt, H. Blair and A. Walker, Towards a theory of declarative knowledge, in: J. Minker, ed, Foundations of Deductive Databases and Logic Programming (Morgan Kaufman, Los Altos, 1988), 89–142.

    Google Scholar 

  2. K. Apt and M. Van Emden, Contribution to the theory of logic programming, J. ACM 29-3 (1982) 841–842.

    Google Scholar 

  3. M. Ben Jacob and M. Fitting, Stratified and Three-valued Logic Programming Semantics, in: R.A. Kowalski and A. Bowen, ed., Proceedings of the Fifth International Conference and symposium on Logic Programming (MIT Press, 1988) 1055–1069.

    Google Scholar 

  4. N. Bidoit et C. Froidevaux, Variations sur la contrainte de stratification: Stratification, Stratification locale, Sup-stratification: Journées Bases de données avancées, 1988.

    Google Scholar 

  5. N. Bidoit et C. Froidevaux, General logic databases and programs: Default logic semantics and stratification,To appear in J. of Information and computation.

    Google Scholar 

  6. N. Bidoit et C. Froidevaux, Negation by default and Unstratifiable Logic Programs,To appear in TCS.

    Google Scholar 

  7. N. Bidoit, Negation in Rule-Based Database Language: A Survey, To appear in TCS.

    Google Scholar 

  8. K.L. Clark, Negation as failure, in: Gallaire and Minker ed., Logic and databases (Plenum Press, New York,1978).293–324.

    Google Scholar 

  9. J.P. Delahaye: Chainage avant et calcul booléen et trivalué, Journées internationales sur les systèmes experts et leurs applications, Avignon 1987, 1341–1360.

    Google Scholar 

  10. J.P. Delahaye: Programmation en logique trivaluée, rapport de recherche I.T no 115, Université des Sciences et Techniques de Lille, 1987.

    Google Scholar 

  11. J.P. Delahaye, P. Matthieu: Logique partielle et Prolog, séminaire de programmation logique de Tregastel, 1989.

    Google Scholar 

  12. M. Fitting, A Kripke-Kleene semantic for logic programs, J. Logic Programming 4 (1985) 295–312.

    Google Scholar 

  13. M. Fitting, Notes on the mathematical aspects of Kripke's theory of truth, Notre Dame Journal of Formal Logic, vol.27, no 1 (1986) 75–88.

    Google Scholar 

  14. M. Fitting, Partial models and logic programming, Theoretical Computer Science 48 (1986) 229–255.

    Google Scholar 

  15. K. Kunen, Negation in Logic Programming, Journal of Logic Programming (1987) 289–308.

    Google Scholar 

  16. K. Kunen, Some remarks on the completed databases, in: R. Kowalski and K. Bowen, eds, Proceedings of the fifth Logic Programming symposium (Association for Logic Programming, MIT Press, Cambridge, Mass, 1988) 978–992.

    Google Scholar 

  17. K. Kunen, Signed data dependencies in logic programming, Journal of Logic Programming (1989) 231–245.

    Google Scholar 

  18. J.L. Lassez et M.J. Maher, Optimal fixpoints of logic programs, Theoretical Computer Science 39 (1985) 15–25.

    Google Scholar 

  19. J.W. Lloyd, Foundations of Logic Programming (second edition, Springer, Berlin, 1987).

    Google Scholar 

  20. Z. Manna and A. Shamir, The theoretical aspects of the optimal fixpoints, Siam J. comput., vol 5, no 3 (1976). 414–426.

    Google Scholar 

  21. Z. Manna and A. Shamir, The optimal approach to recursive programs, ACM vol 20, no 11 (1977) 824–831

    Google Scholar 

  22. A. Mycroft, Logic programs and many-valued logic,in: STACS 84, Lecture Notes in Computer Science 166 (Springer, Berlin, 1984) 274–286.

    Google Scholar 

  23. T. Przymunsinski and H. Przymunsinski, Weakly perfect model semantics for Logic programs, Conference and Symp. on Logic Programming, (1988) 1106–1120.

    Google Scholar 

  24. T. Przymunsinski, Every Logic Program has a natural Stratification and an iterated fixed point model, Proc. ACM Symp. Principles of Databases Systems, (1988) 11–21.

    Google Scholar 

  25. T. Przymunsinski, Non monotonic formalisms and Logic Programming, ICLP, (1989) 655–674.

    Google Scholar 

  26. T. Przymunsinski, On the declarative semantics of deductive databases and logic programs, in: J. Minker, ed, Foundations of Deductive Databases and Logic Programming (Morgan Kaufman, Los Altos, 1988).

    Google Scholar 

  27. J.C. Shepherdson, Negation in Logic Programming, in: J. Minker, ed, Foundations of Deductive Databases and Logic Programming (Morgan Kaufman, Los Altos, 1988) 19–87.

    Google Scholar 

  28. V. Thibau, Thèse de doctorat: Une logique trivaluée appliquée à la programmation logique, Université des Sciences et Techniques de Lille, 1990.

    Google Scholar 

  29. M.H. Van Emden, R.A. Kowalski, The semantics of predicate logic as a programming language, J. ACM, 23 (1976), 733–742.

    Google Scholar 

  30. A. Van Gelder, The well-founded semantics for general logic programs, Proc. ACM Symp. Principles of Databases Systems, (March 1988) 221–230.

    Google Scholar 

  31. A. Van Gelder, The alternative fixpoint of logic programs with negation, Proc. ACM Symp. Principles of Databases Systems, (March 1989) 1–10.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. L.I.F.L. U.A.-C.N.R.S. no 369 BAT.M3, Université des Sciences et Techniques de Lille Flandres Artois, 59655, Villeneuve d'Ascq Cedex, France

    J. P. Delahaye & V. Thibau

Authors
  1. J. P. Delahaye
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. Thibau
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

J. van Eijck

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Delahaye, J.P., Thibau, V. (1991). The optimal model of a program with negation. In: van Eijck, J. (eds) Logics in AI. JELIA 1990. Lecture Notes in Computer Science, vol 478. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0018441

Download citation

  • DOI: https://doi.org/10.1007/BFb0018441

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-53686-4

  • Online ISBN: 978-3-540-46982-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation