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Well-founded and stable semantics of logic programs with aggregates

Published online by Cambridge University Press:  01 May 2007

NIKOLAY PELOV ,
MARC DENECKER  and
MAURICE BRUYNOOGHE
NIKOLAY PELOV
Affiliation:
Department of Computer Science, Katholieke Universiteit Leuven, Belgium (e-mail: pelov@cs.keleuven.be, marcd@cs.keleuven.be, maurice@cs.keleuven.be)
MARC DENECKER
Affiliation:
Department of Computer Science, Katholieke Universiteit Leuven, Belgium (e-mail: pelov@cs.keleuven.be, marcd@cs.keleuven.be, maurice@cs.keleuven.be)
MAURICE BRUYNOOGHE
Affiliation:
Department of Computer Science, Katholieke Universiteit Leuven, Belgium (e-mail: pelov@cs.keleuven.be, marcd@cs.keleuven.be, maurice@cs.keleuven.be)
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Abstract

In this paper, we present a framework for the semantics and the computation of aggregates in the context of logic programming. In our study, an aggregate can be an arbitrary interpreted second order predicate or function. We define extensions of the Kripke-Kleene, the well-founded and the stable semantics for aggregate programs. The semantics is based on the concept of a three-valued immediate consequence operator of an aggregate program. Such an operator approximates the standard two-valued immediate consequence operator of the program, and induces a unique Kripke-Kleene model, a unique well-founded model and a collection of stable models. We study different ways of defining such operators and thus obtain a framework of semantics, offering different trade-offs between precision and tractability. In particular, we investigate conditions on the operator that guarantee that the computation of the three types of semantics remains on the same level as for logic programs without aggregates. Other results show that, in practice, even efficient three-valued immediate consequence operators which are very low in the precision hierarchy, still provide optimal precision.

Keywords

logic programmingaggregates
Type
Regular Papers
Information
Theory and Practice of Logic Programming , Volume 7 , Issue 3 , May 2007 , pp. 301 - 353
Copyright
Copyright © Cambridge University Press 2007

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References

Apt, K. R., Blair, H. A. and Walker, A. 1988. Towards a theory of declarative knowledge. In Foundations of Deductive Databases and Logic Programming, Minker, J., Ed. Morgan Kaufmann, Chapter 2, 89–148.CrossRefGoogle Scholar
Davey, B. A. and Pristley, H. A. 1990. Introduction to Lattices and Order. Cambridge University Press.Google Scholar
Dell'Armi, T., Faber, W., Ielpa, G., Leone, N. and Pfeifer, G. 2003. Aggregate functions in disjunctive logic programming: Semantics, complexity, and implementation in DLV. In 18th International Joint Conference on Artificial Intelligence. Morgan Kaufmann, 847–852.Google Scholar
Denecker, M. 2000. Extending classical logic with inductive definitions. In 1st International Conference on Computational Logic. Lecture Notes in Artificial Intelligence, vol. 1861. Springer, 703–717.Google Scholar
Denecker, M., Marek, V. and Truszczyński, M. 2000. Approximating operators, stable operators, well-founded fixpoints and applications in non-monotonic reasoning. In Logic-based Artificial Intelligence, Minker, J., Ed. Kluwer Academic Publishers, 127–144.CrossRefGoogle Scholar
Denecker, M., Marek, V. and Truszczyński, M. 2003. Uniform semantic treatment of default and autoepistemic logics. Artificial Intelligence 143, 1, 79–122.CrossRefGoogle Scholar
Denecker, M., Marek, V. and Truszczyński, M. 2004. Ultimate approximations and its application in nonmonotonic knowledge representation. Information and Computation 192, 1, 84–121.CrossRefGoogle Scholar
Denecker, M., Pelov, N. and Bruynooghe, M. 2001. Ultimate well-founded and stable model semantics for logic programs with aggregates. In 17th International Conference on Logic Programming. LNCS, vol. 2237. Springer, 212–226.Google Scholar
Elkabani, I., Pontelli, E. and Son, T. C. 2004. Smodels with CLP and its applications: A simple and effective approach to aggregates in ASP. In International Conference on Logic Programming. 73–89.CrossRefGoogle Scholar
Faber, W., Leone, N. and Pfeifer, G. 2004. Recursive aggregates in disjunctive logic programs: Semantics and complexity. In 9th European Conference on Artificial Intelligence (JELIA). LNCS, vol. 3229. Springer, 200–212.Google Scholar
Ferraris, P. 2005. Answer sets for propositional theories. In 8th International Conference on Logic Programming and Nonmonotonic Reasoning. LNCS, vol. 3662. Springer, 119–131.Google Scholar
Fitting, M. 1985. A Kripke-Kleene semantics for logic programs. Journal of Logic Programming 2, 4, 295–312.CrossRefGoogle Scholar
Gelfond, M. 2002. Representing knowledge in A-Prolog. In Computational Logic: Logic Programming and Beyond, Essays in Honour of Robert A. Kowalski, Part II, Kakas, A. C. and Sadri, F., Eds. LNCS, vol. 2408. Springer, 413–451.CrossRefGoogle Scholar
Gelfond, M. and Lifschitz, V. 1988. The stable model semantics for logic programming. In Logic Programming, Proc. of the 5th International Conference and Symposium. MIT Press, 1070–1080.Google Scholar
Kemp, D. B. and Stuckey, P. J. 1991. Semantics of logic programs with aggregates. In International Logic Programming Symposium. MIT Press, 387–401.Google Scholar
Lifschitz, V., Pearce, D. and Valverde, A. 2001. Strongly equivalent logic programs. ACM Transactions on Computational Logic 2, 4, 526–541.CrossRefGoogle Scholar
Lindström, P. 1966. First order predicate logic with generalized quantifiers. Theoria 32, 186–195.CrossRefGoogle Scholar
Marek, V., Niemelä, I. and Truszczyński, M. 2004. Logic programs with monotone cardinality atoms. In 7th International Conference on Logic Programming and Nonmonotonic Reasoning. LNCS, vol. 2923. Springer, 155–166.Google Scholar
Marek, V. and Remmel, J. 2004. Set constraints in logic programming. In 7th International Conference on Logic Programming and Nonmonotonic Reasoning. LNCS, vol. 2923. Springer, 167–179.Google Scholar
Marek, V. W. and Truszczyński, M. 2004. Logic programs with abstract constraint atoms. In National Conference on Artificial Intelligence. AAAI Press/The MIT Press, 86–91.Google Scholar
Mitchell, D. and Ternovska, E. 2005. A framework for representing and solving NP-search problems. In Proc. of the National Conference on Artificial Intelligence. 430–435.Google Scholar
Mumick, I. S., Pirahesh, H. and Ramakrishnan, R. 1990. The magic of duplicates and aggregates. In 16th International Conference on Very Large Data Bases. Morgan Kaufmann, 264–277.Google Scholar
Pelov, N. 2004. Semantics of logic programs with aggregates. Ph.D. thesis, K.U.Leuven.Google Scholar
Pelov, N., Denecker, M. and Bruynooghe, M. 2004. Partial stable semantics for logic programs with aggregates. In 7th International Conference on Logic Programming and Nonmonotonic Reasoning. LNCS, vol. 2923. Springer, 207–219.Google Scholar
Ross, K. A. and Sagiv, Y. 1997. Monotonic aggregation in deductive databases. Journal of Computer and System Sciences 54, 1, 79–97.CrossRefGoogle Scholar
Simons, P., Niemelä, I. and Soininen, T. 2002. Extending and implementing the stable model semantics. Artificial Intelligence 138, 1–2, 181–234.CrossRefGoogle Scholar
Son, T. C. and Pontelli, E. 2007. A constructive semantic characterization of aggregates in answer set programming. Theory and Practice of Logic Programming. Accepted as a Technical Note.CrossRefGoogle Scholar
van Eijck, J. 1996. Quantifiers and partiallity. In Quantifiers, Logic, and Language, van der Does, J. and van Eijck, J., Eds. CSLI, 105–144.Google Scholar
van Emden, M. H. and Kowalski, R. A. 1976. The semantics of predicate logic as a programming language. Journal of the ACM 23, 4, 733–742.CrossRefGoogle Scholar
Van Gelder, A. 1992. The well-founded semantics of aggregation. In 11th ACM Symposium on Principles of Database Systems. ACM Press, 127–138.Google Scholar
Van Gelder, A., Ross, K. A. and Schlipf, J. S. 1991. The well-founded semantics for general logic programs. Journal of the ACM 38, 3, 620–650.CrossRefGoogle Scholar
Van Nuffelen, B. and Denecker, M. 2000. Problem solving in ID-logic with aggregates: some experiments. In 8th International Workshop on Nonmonotonic Reasoning, special track on Abductive Reasoning.Google Scholar
Vennekens, J., Gilis, D. and Denecker, M. 2006. Splitting an operator: Algrebraic modularity results for logics with fixpoint semantics. ACM Transactions on Computational Logic. Accepted.CrossRefGoogle Scholar