Advertisement

Extending Classical Logic with Inductive Definitions

  • Marc Denecker
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1861)

Abstract

The goal of this paper is to extend classical logic with a generalized notion of inductive definition supporting positive and negative induction, to investigate the properties of this logic, its relationships to other logics in the area of non-monotonic reasoning, logic programming and deductive databases, and to show its application for knowledge representation by giving a typology of definitional knowledge.

Keywords

Logic Program Knowledge Representation Logic Programming Classical Logic Nonmonotonic Reasoning 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    P. Aczel. An Introduction to Inductive Definitions. In J. Barwise, editor, Handbook of Mathematical Logic, pages 739–782. North-Holland Publishing Company, 1977.Google Scholar
  2. 2.
    Gianni Amati, Luigia Carlucci Aiello, and Fiora Pirri. Definability and commonsense reasoning. Artificial Intelligence Journal, 93:1–30, 1997. Abstract of this paper appeared also in Third Symposium on Logical Formalization of Commonsense Reasoning, Stanford, USA, 96.CrossRefGoogle Scholar
  3. 3.
    K.R. Apt and M. Bezem. Acyclic programs. In Proc. of the International Conference on Logic Programming, pages 579–597. MIT press, 1990.Google Scholar
  4. 4.
    R. J. Brachman and H.J. Levesque. Competence in Knowledge Representation. In Proc. of the National Conference on Artificial Intelligence, pages 189–192, 1982.Google Scholar
  5. 5.
    W. Buchholz, S. Feferman, and W. Pohlers W. Sieg. Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies. Springer-Verlag, Lecture Notes in Mathematics 897, 1981.Google Scholar
  6. 6.
    K.L. Clark. Negation as failure. In H. Gallaire and J. Minker, editors, Logic and Databases, pages 293–322. Plenum Press, 1978.Google Scholar
  7. 7.
    M. Denecker. A Terminological Interpretation of (Abductive) Logic Programming. In V.W. Marek, A. Nerode, and M. Truszczynski, editors, International Conference on Logic Programming and Nonmonotonic Reasoning, Lecture notes in Artificial Intelligence 928, pages 15–29. Springer, 1995.Google Scholar
  8. 8.
    M. Denecker, D. Theseider Dupré, and K. VanBelleghem. An inductive definition approach to ramifications. Linköping Electronic Articles in Computer and Information Science, 3(7):1–43, 1998. URL: http://www.ep.liu.se/ea/cis/1998/007/.Google Scholar
  9. 9.
    Marc Denecker. The well-founded semantics is the principle of inductive definition. In J. Dix, L. Fari nas del Cerro, and U. Furbach, editors, Logics in Artificial Intelligence, pages 1–16, Schloss Daghstull, October 12–15 1998. Springer-Verlag, Lecture notes in Artificial Intelligence 1489.Google Scholar
  10. 10.
    Marc Denecker and Bert VanNuffelen. Experiments for integration CLP and abduction. In Krysztof R. Apt, Antonios C. Kakas, Eric Monfroy, and Francesca Rossi, editors, Workshop on Constraints, pages 1–15. ERCIM/COMPULOG, October 25–27 1999.Google Scholar
  11. 11.
    S. Feferman. Formal theories for transfinite iterations of generalised inductive definitions and some subsystems of analysis. In A. Kino, J. Myhill, and R.E. Vesley, editors, Intuitionism and Proof theory, pages 303–326. North Holland, 1970.Google Scholar
  12. 12.
    M. Fitting. A Kripke-Kleene Semantics for Logic Programs. Journal of Logic Programming, 2(4):295–312, 1985.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    A. VanGelder. The Alternating Fixpoint of Logic Programs with Negation. Journal of computer and system sciences, 47:185–221, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    M. Gelfond and V. Lifschitz. The stable model semantics for logic programming. In Proc. of the International Joint Conference and Symposium on Logic Programming, pages 1070–1080. IEEE, 1988.Google Scholar
  15. 15.
    M. Gelfond and V. Lifschitz. Classical negation in logic programs and disjunctive databases. New Generation Computing, pages 365–387, 1991.Google Scholar
  16. 16.
    A. C. Kakas, R.A. Kowalski, and F. Toni. Abductive Logic Programming. Journal of Logic and Computation, 2(6):719–770, 1993.CrossRefMathSciNetGoogle Scholar
  17. 17.
    V.W. Marek and M. Truszczyński. Nonmonotonic Logic Context-Dependent Reasoning. Springer-Verlag, 1993.Google Scholar
  18. 18.
    J. McCarthy. Circumscription-a form of nonmonotonic reasoning. Artifical Intelligence, 13:27–39, 1980.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    John McCarthy. Elaboration tolerance. In COMMON SENSE 98, Symposium On Logical Formalizations Of Commonsense Reasoning, January 1998.Google Scholar
  20. 20.
    Y. N. Moschovakis. Elementary Induction on Abstract Structures. North-Holland Publishing Company, Amsterdam-New York, 1974.zbMATHGoogle Scholar
  21. 21.
    T.C. Przymusinski. On the semantics of Stratified Databases. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming. Morgan Kaufman, 1988.Google Scholar
  22. 22.
    T.C. Przymusinski. Extended Stable Semantics for Normal and Disjunctive Programs. In D.H.D. Warren and P. Szeredi, editors, Proc. of the seventh international conference on logic programming, pages 459–477. MIT press, 1990.Google Scholar
  23. 23.
    T.C. Przymusinski. Well founded semantics coincides with three valued Stable Models. Fundamenta Informaticae, 13:445–463, 1990.zbMATHMathSciNetGoogle Scholar
  24. 24.
    H. Reichgelt. Knowledge Representation: an AI Perspecitive. Ablex Publishing Corporation, 1991.Google Scholar
  25. 25.
    R. Reiter. Equality and domain closure in first-order databases. JACM, 27:235–249, 1980.zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    R. Reiter. The Frame Problem in the Situation Calculus: A simple Solution (Sometimes) and a Completeness Result for Goal Regression. In V. Lifschitz, editor, Artificial Intelligence and Mathematical Theory of Computation: Papers in Honour of John McCarthy, pages 359–380. Academic Press, 1991.Google Scholar
  27. 27.
    R. Reiter. Nonmonotonic Reasoning: Compiled vs Interpreted Theories. Distributed at the conference of Nonmonotonic Reasoning NMR96 as considerations for the panel discussion, 1996.Google Scholar
  28. 28.
    A. Tarski. Lattice-theoretic fixpoint theorem and its applications. Pacific journal of Mathematics, 5:285–309, 1955.zbMATHMathSciNetGoogle Scholar
  29. 29.
    E. Ternovskaia. Inductive Definability and the Situation Calculus. In Burkhard Freitag, Hendrik Decker, Michael Kifer, and Andrei Voronkov, editors, Transactions and Change in Logic Databases, volume 1472 of LNCS. Springer-Verlag, Berlin, 1998.CrossRefGoogle Scholar
  30. 30.
    M. van Emden and R.A Kowalski. The semantics of Predicate Logic as a Programming Language. Journal of the ACM, 4(4):733–742, 1976.CrossRefGoogle Scholar
  31. 31.
    A. VanGelder, K.A. Ross, and J.S. Schlipf. The Well-Founded Semantics for General Logic Programs. Journal of the ACM, 38(3):620–650, 1991.zbMATHCrossRefGoogle Scholar
  32. 32.
    S. Verbaeten, M. Denecker, and D. DeSchreye. Compositionality of normal open logic programs. Journal of Logic Programming, 41(3):151–183, March 2000.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Marc Denecker
    • 1
  1. 1.Department of Computer ScienceK.U.LeuvenHeverleeBelgium

Personalised recommendations