{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,23]],"date-time":"2026-08-23T15:19:11Z","timestamp":1787498351065,"version":"build-2736575974"},"reference-count":33,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2007,5,1]],"date-time":"2007-05-01T00:00:00Z","timestamp":1177977600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Theory and Practice of Logic Programming"],"published-print":{"date-parts":[[2007,5]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    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\n                    <jats:italic>immediate consequence operator<\/jats:italic>\n                    of an aggregate program. Such an operator\n                    <jats:italic>approximates<\/jats:italic>\n                    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\n                    <jats:italic>precision<\/jats:italic>\n                    and\n                    <jats:italic>tractability<\/jats:italic>\n                    . 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.\n                  <\/jats:p>","DOI":"10.1017\/s1471068406002973","type":"journal-article","created":{"date-parts":[[2007,3,14]],"date-time":"2007-03-14T05:10:41Z","timestamp":1173849041000},"page":"301-353","source":"Crossref","is-referenced-by-count":78,"title":["Well-founded and stable semantics of logic programs with aggregates"],"prefix":"10.1017","volume":"7","author":[{"given":"NIKOLAY","family":"PELOV","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"MARC","family":"DENECKER","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"MAURICE","family":"BRUYNOOGHE","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2007,5,1]]},"reference":[{"key":"S1471068406002973_N104C4","doi-asserted-by":"publisher","DOI":"10.1145\/116825.116838"},{"key":"S1471068406002973_N104B2","doi-asserted-by":"crossref","unstructured":"Van Gelder A. 1992. The well-founded semantics of aggregation. In 11th ACM Symposium on Principles of Database Systems. ACM Press, 127\u2013138.","DOI":"10.1145\/137097.137854"},{"key":"S1471068406002973_N1048C","doi-asserted-by":"publisher","DOI":"10.1145\/321978.321991"},{"key":"S1471068406002973_N10445","doi-asserted-by":"crossref","unstructured":"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.","DOI":"10.1017\/S1471068406002936"},{"key":"S1471068406002973_N103F3","doi-asserted-by":"publisher","DOI":"10.1006\/jcss.1997.1453"},{"key":"S1471068406002973_N103A9","unstructured":"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\u2013277."},{"key":"S1471068406002973_N10379","unstructured":"Marek V. W. and Truszczy\u0144ski M. 2004. Logic programs with abstract constraint atoms. In National Conference on Artificial Intelligence. AAAI Press\/The MIT Press, 86\u201391."},{"key":"S1471068406002973_N10326","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1111\/j.1755-2567.1966.tb00600.x","article-title":"First order predicate logic with generalized quantifiers","volume":"32","author":"Lindstr\u00f6m","year":"1966","journal-title":"Theoria"},{"key":"S1471068406002973_N102FA","doi-asserted-by":"publisher","DOI":"10.1145\/383779.383783"},{"key":"S1471068406002973_N10508","doi-asserted-by":"crossref","unstructured":"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.","DOI":"10.1145\/1183278.1183284"},{"key":"S1471068406002973_N102E2","unstructured":"Kemp D. B. and Stuckey P. J. 1991. Semantics of logic programs with aggregates. In International Logic Programming Symposium. MIT Press, 387\u2013401."},{"key":"S1471068406002973_N102CA","unstructured":"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\u20131080."},{"key":"S1471068406002973_N1027B","doi-asserted-by":"publisher","DOI":"10.1016\/S0743-1066(85)80005-4"},{"key":"S1471068406002973_N101E3","doi-asserted-by":"publisher","DOI":"10.1016\/j.ic.2004.02.004"},{"key":"S1471068406002973_N103D5","unstructured":"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\u2013219."},{"key":"S1471068406002973_N1012F","volume-title":"Introduction to Lattices and Order","author":"Davey","year":"1990"},{"key":"S1471068406002973_N10343","unstructured":"Marek V. , Niemel\u00e4 I. and Truszczy\u0144ski M. 2004. 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