M.A. Khamsi, V. Kreinovich, and D. Misane. A new method of proving the existence of answer sets for disjunctive logic programs: A metric fixed-point theorem for multivalued mappings. In C. Baral and M. Gelfond, editors, Proceedings of the Workshop on Logic Programming with Incomplete Information, Vancouver, B.C., Canada, pages 58--73, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... Although there is a broad base of theoretical work on answer set programming ( Lif99, MT99] and deductive databases ( Min97] domain theoretic foundations have, to our knowledge, not yet been studied for these paradigms apart from some investigations concerning fixed point semantics, e.g. [KKM93, HS99, DMT00, Hit01]. Extended disjunctive logic programming as presented in this report may provide an important link. ....

M.A. Khamsi, V. Kreinovich, and D. Misane. A new method of proving the existence of answer sets for disjunctive logic programs: A metric fixed-point theorem for multivalued mappings. In C. Baral and M. Gelfond, editors, Proceedings of the Workshop on Logic Programming with Incomplete Information, Vancouver, B.C., Canada, pages 58--73, 1993.

.... Although there is a broad base of theoretical work on answer set programming ( Lif99, MT99] and deductive databases ( Min97] domain theoretic foundations have, to our knowledge, not yet been studied for these paradigms apart from some investigations concerning xed point semantics, e.g. [KKM93, HS99, DMT00, Hit01]. Extended disjunctive logic programming as presented in this report may provide an important link. 4) Application to reasoning on concept lattices [GW99b, GW99a] Although concept lattices are structurally di erent from algebraic domains, we expect that reasoning techniques developed for the ....

M.A. Khamsi, V. Kreinovich, and D. Misane. A new method of proving the existence of answer sets for disjunctive logic programs: A metric xed-point theorem for multivalued mappings. In C. Baral and M. Gelfond, editors, Proceedings of the Workshop on Logic Programming with Incomplete Information, Vancouver, B.C., Canada, pages 58-73, 1993.

.... Although there is a broad base of theoretical work on answer set programming ( Lif99, MT99] and deductive databases ( Min97] domain theoretic foundations have, to our knowledge, not yet been studied for these paradigms apart from some investigations concerning xed point semantics, e.g. [KKM93, HS99, DMT00, Hit01]. Extended disjunctive logic programming as presented in this report may provide an important link. ....

M.A. Khamsi, V. Kreinovich, and D. Misane. A new method of proving the existence of answer sets for disjunctive logic programs: A metric xed-point theorem for multivalued mappings. In C. Baral and M. Gelfond, editors, Proceedings of the Workshop on Logic Programming with Incomplete Information, Vancouver, B.C., Canada, pages 58-73, 1993.

....Special Session on Topology in Computer Science, New York, August, 1999. Topology Proceedings Volume 24, 1999 Summer Issue. y The first named author acknowledges financial support under grant SC 98 621 from Enterprise Ireland. 1 stable and the perfect model can be viewed in these terms, see [BMP99, KKM93, SH97]. In [HS99a] the usual immediate consequence operator associated with a normal logic program P was extended to disjunctive programs Pi, obtaining a multivalued operator T Pi . This extension is rather satisfactory in that it was further shown in [HS99a] that an interpretation is a supported ....

....The same sort of problem arises, incidentally, when dealing with operators in three valued logic, see [Fit85, HS99b] Thus, the name Knaster Tarski Theorem for Theorem 2. 4 is appropriate in that the iterations involved need not cut off at the first limit ordinal, On the other hand, in [KKM93], Khamsi et al. established a version of the Banach contraction mapping theorem for multivalued mappings, see Corollary 3.6 below, and applied it to obtain the stable model of a countably stratified disjunctive program. In this case, needless to say, the iterates involved do cut off at . Of ....

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M.A. Khamsi, V. Kreinovich and D. Misane, A New Method of Proving the Existence of Answer Sets for Disjunctive Logic Programs: A Metric Fixed Point 16 Theorem for Multivalued Maps. Proceedings of the Workshop on Logic Programming with Incomplete Information, Vancouver, British-Columbia, Canada, 1993, pp. 58--73.

....arguments based on order, and it is this theme which will mainly concern us in this paper, see [5] for a discussion of the r ole of continuous mathematics in this context. Thus, one nds methods based on topology and dynamical systems ( 4, 13, 14, 16, 27, 29, 30] methods based on metrics ([9, 12, 17]) methods based on quasi metrics ( 15, 28] and nally, one nds methods based on ultrametric spaces. In fact, metric and ultrametric methods were introduced to logic programming by Fitting in [9] although all the metrics he considered are actually ultrametrics as is usually the case in ....

....convenient to consider extended disjunctive logic programs and to allow two different kinds of negation. One of these is interpreted as classical negation and the other is interpreted procedurally as negation as failure, see [19] for this notion. We introduce the following terminology following [11, 17] closely. Let L denote a rst order language. By a literal L in L we mean either an atom A or its negation, A, where A is an atom in L; a literal is called ground if it contains no variable symbols. We denote the set of all ground literals in L by Lit. A rule r in L is a universally quantifed ....

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Khamsi, M.A., Kreinovich, V. and Misane, D. A New Method of Proving the Existence of Answer Sets for Disjunctive Logic Programs: A Metric Fixed-Point Theorem For Multivalued Mappings, in: Baral, C. and Gelfond, M. (Eds.) Proc. of the Workshop on Logic Programming with Incomplete Information, Vancouver, B.C., Canada, October 1993, pp. 58-73.

....seem to be quite a useful tool here. The application of metric techniques is being extended in several directions. Seda, by using quasi metric spaces, shows how both the lattice theoretic and the metric approaches can be combined into a single treatment [32] Khamsi, Kreinovich and Misane [18] have shown that more powerful metric fixed point theorems also have natural applications here. 11 Conclusion There are two important topics that are common in articles on semantics but that have not yet been mentioned: higher types, and non determinism. As a matter of fact, both of these have ....

....works with sets of valuations, and this suggests that a powerdomain approach should be appropriate. This is, indeed, the case, though the literature so far uses powerdomain theory implicitly, rather than explicitly. On the other hand, the metric approach of section 10 carries over rather neatly, [18], and shows promise of further extension. We have tried to show that at least some of the kinds of concerns that are important in developing semantic approaches to imperative and functional programs also arise in logic programming. But logic programming contributes its own twists that make the ....

Khamsi, M. A., Kreinovich, V., and Misane, D. A new method of proving the existence of answer sets for disjunctive logic programs: a metric fixed point theorem for multi-valued mappings. Journal of Logic Programming (forthcoming).

....as the limit of the sequence f n (x) for any x 2 X . Note that the proof is constructive, i.e. the fixed point is in fact the limit of any sequence of iterates of f . The Banach theorem has found application to logic programming in [4, 19, 20] and a multivalued version was considered in [10] and will be discussed in Section 4. In fact, quite a lot of work has been done, some of it by the present authors, in applying generalizations of the Banach theorem in which the axioms in the definition of a metric are relaxed, see [14, 19, 20] and we briefly consider this next. 2.2.2 ....

....4.4 Definition Let (M; d) be a metric space. A multivalued mapping T : M 2 M is called a contraction if there exists a real number k 1 such that for every x 2 M , for every y 2 M , and for all a 2 T (x) there exists b 2 T (y) such that d(a; b) kd(x; y) The following result is taken from [10], and depends on an old result of S.B. Nadler. 4.5 Theorem Assume that M is a complete metric space, and that T is a multivalued contraction on M such that the set T (x) is closed and non empty for every x 2 M . Then T has a fixed point. Again, this theorem was established with a specific ....

[Article contains additional citation context not shown here]

Khamsi MA, Kreinovich V, Misane D. A New Method of Proving the Existence of Answer Sets for Disjunctive Logic Programs: A Metric Fixed Point Theorem for Multivalued Maps. Proc. of the Workshop on Logic Programming with Incomplete Information. Vancouver, British-Columbia, Canada, 1993, pp 58-73

....as the limit of the sequence f n (x) for any x 2 X . Note that the proof is constructive, i.e. the fixed point is in fact the limit of any sequence of iterates of f . The Banach theorem has found application to logic programming in [Fit94, SH97, SH98] and a multivalued version was considered in [KKM93] and will be discussed in Section 4. In fact, quite a lot of work has been done, some of it by the present authors, in applying generalizations of the Banach theorem in which the axioms in the definition of a metric are relaxed, see [PR97, SH97, SH98] and we briefly consider this next. 2.2.2 ....

....4.4 Definition Let (M; d) be a metric space. A multivalued mapping T : M 2 M is called a contraction if there exists a real number k 1 such that for every x 2 M , for every y 2 M , and for all a 2 T (x) there exists b 2 T (y) such that d(a; b) kd(x; y) The following result is taken from [KKM93], and depends on an old result of S.B. Nadler. 4.5 Theorem Assume that M is a complete metric space, and that T is a multivalued contraction on M such that, for every x 2 M , the set T (x) is closed and non empty. Then T has a fixed point. Again, this theorem was established with a specific ....

[Article contains additional citation context not shown here]

M.A. Khamsi, V. Kreinovich and D. Misane, A New Method of Proving the Existence of Answer Sets for Disjunctive Logic Programs: A Metric Fixed Point Theorem for Multivalued Maps. Proc. of the Workshop on Logic Programming with Incomplete Information, Vancouver, British-Columbia, Canada, 1993, pp. 58-73.

....i.e. whether or not those programs within it can express transitive closure we do not know. Acknowledgement. The author thanks two anonymous referees for comments which substantially improved the presentation of this paper at certain points. In particular, he thanks one of them for bringing [14, 20] to his attention. These two papers use metric space theory to establish interesting and significant results concerning logic programming which are close in spirit to ours, although there is no actual overlap between our results and those of [14, 20] 2 Topologies on I J L 2.1 Notation and ....

....particular, he thanks one of them for bringing [14, 20] to his attention. These two papers use metric space theory to establish interesting and significant results concerning logic programming which are close in spirit to ours, although there is no actual overlap between our results and those of [14, 20]. 2 Topologies on I J L 2.1 Notation and terminology It will be convenient first to establish and collect together certain pieces of notation and terminology which we will use throughout without further mention. We refer to [26] for all terms and notation relating to logic programming which ....

M. A. Khamsi, V. Kreinovich and D. Misane, A New Method of Proving the Existence of Answer Sets for Disjunctive Logic Programs: A Metric Fixed Point Theorem For Multi-Valued Mappings, Preprint, 1994.

....point which is the least fixed point above x in the order X defined by y X z iff d(y; z) 0. 2. If f is Continuous and contractive, then f has a unique fixed point. Attempts to use the Banach contraction mapping theorem in logic programming have been made with some success in [12] and in [17] where problems arising out of attempts to formalize common sense reasoning are considered. Nevertheless, it is our claim that it is quasi metrics that should be used instead, in conjunction with theorems such as Theorem 5. This point of view is substantiated by the following two observations: ....

M. A. Khamsi, V. Kreinovich and D. Misane, A New Method of Proving the Existence of Answer Sets for Disjunctive Logic Programs: A Metric Fixed Point Theorem For Multi-Valued Mappings, in Proceedings of the Workshop on Logic Programming with Incomplete Information, Vancouver, B.C., Canada, October 1993, pp. 58-73.

.... of BP and hence as a set dynamical system, see [4,18] For databases there are further problems in that the appropriate operator T is multi valued and we want I such that I 2 T (I) a fixed point of T ) We shall not, however, discuss databases as such in detail, but instead refer the reader to [7] where a multi valued version of the contraction mapping theorem can be found, and also an application of it to finding models of disjunctive databases. Returning to programs, various syntactic conditions, see [1,2,12,13,14] have been considered in attempting to find fixed points of non monotonic ....

M.A. Khamsi, V. Kreinovich and D. Misane, A new method of proving the existence of answer sets for disjunctive logic programs: a metric fixed point theorem for multi-valued mappings, in Proceedings of the Workshop on Logic Programming with Incomplete Information, Vancouver, B.C., Canada, October 1993, pp. 58-73.

....specifically for describing such fuzziness in precise mathematical terms, it is natural to use fuzzy logic as a basis for our definition of naturalness. The idea of using fuzzy logic to describe naturalness is not only natural itself, it is also known to be successful: in our previous papers [2, 4], we have shown that the use of fuzzy logic makes a special metric used in logic programming very natural. In this paper, we expand on this result and show that an arbitrary metric can be thus interpreted. 4 Motivations of the Following Definitions What is a natural way to describe the closeness ....

M. A. Khamsi, V. Kreinovich, and D. Misane (1993). A new method of proving the existence of answer sets for disjunctive logic programs, Proc. Workshop on Logic Programming with Incomplete Information, Vancouver, B.C., Canada, October 1993, pp. 58--73.

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