Abstract
We give a purely model-theoretic (denotational) characterization of the semantics of logic programs with negation allowed in clause bodies. In our semantics (the first of its kind) the meaning of a program is, as in the classical case, the unique minimum model in a programindependent ordering. We use an expanded truth domain that has an uncountable linearly ordered set of truth values between False (the minimum element) and True (the maximum), with a Zero element in the middle. The truth values below Zero are ordered like the countable ordinals. The values above Zero have exactly the reverse order. Negation is interpreted as refiection about Zero followed by a step towards Zero; the only truth value that remains unaffected by negation is Zero. We show that every program has a unique minimum model M P, and that this model can be constructed with a T P iteration which proceeds through the countable ordinals. Furthermore, collapsing the true and false values of the infinite-valued model M P to (the classical) True and False, gives a three-valued model identical to the well-founded one.
This work has been partially supported by the University of Athens under the project “Extensions of the Logic Programming Paradigm”.
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References
K. Apt and R. Bol. Logic Programming and Negation: A Survey. Journal of Logic Programming, 19, 20:9–71, 1994.
K.R. Apt, H.A. Blair, and A. Walker. Towards a Theory of Declarative Knowledge. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming, pages 89–148. Morgan Kaufmann, Los Altos, CA, 1988.
P. Cholak and H.A. Blair. The Complexity of Local Stratification. Fundamenta Informaticae, 21(4):333–344, 1994.
M. Fitting. Fixpoint Semantics for Logic Programming: A Survey. Theoretical Computer Science. (to appear).
M. Gelfond and V. Lifschitz. The Stable Model Semantics for Logic Programming. In Proceedings of the Fifth Logic Programming Symposium, pages 1070–1080. MIT Press, 1988.
G. Lako. and R. Nunez. Where Mathematics comes from. Basic Books, 2000.
J. Lloyd. Foundations of Logic Programming. Springer-Verlag, 1987.
C. Nomikos, P. Rondogiannis, and M. Gergatsoulis. A Stratification Test for Temporal Logic Programs. In Proceedings of the 3rd Panhellenic Logic Symposium, July 17–21 2001. Crete, Greece.
H. Przymusinska and T. Przymusinski. Semantic Issues in Deductive Databases and Logic Programs. In R. Banerji, editor, Formal Techniques in Artificial Intelligence, pages 321–367. North Holland, 1990.
T. Przymusinski. On the Declarative Semantics of Deductive Databases and Logic Programs. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming, pages 193–216. Morgan Kaufmann, Los Altos, CA, 1988.
T.C. Przymusinski. Every Logic Program has a Natural Stratification and an Iterated Fixed Point Model. In Proceedings of the 8th Symposium on Principles of Database Systems, pages 11–21. ACM SIGACT-SIGMOD, 1989.
P. Rondogiannis. Stratified Negation in Temporal Logic Programming and the Cycle-Sum Test. Theoretical Computer Science, 254(1–2):663–676, 2001.
A. van Gelder. The Alternating Fixpoint of Logic Programs with Negation. In Proceedings of the 8th Symposium on Principles of Database Systems, pages 11–21. ACM SIGACT-SIGMOD, 1989.
A. van Gelder. The Alternating Fixpoint of Logic Programs with Negation. Journal of Computer and System Sciences, 47(1):185–221, 1993.
A. van Gelder, K. A. Ross, and J. S. Schlipf. The Well-Founded Semantics for General Logic Programs. Journal of the ACM, 38(3):620–650, 1991.
C. Zaniolo, N. Arni, and K. Ong. Negation and Aggregates in Recursive Rules: the LDL + + Approach. In Proceedings of DOOD-93, pages 204–221, 1993.
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Rondogiannis, P., Wadge, W.W. (2002). An Infinite-Valued Semantics for Logic Programs with Negation. In: Flesca, S., Greco, S., Ianni, G., Leone, N. (eds) Logics in Artificial Intelligence. JELIA 2002. Lecture Notes in Computer Science(), vol 2424. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45757-7_38
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