{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T17:27:58Z","timestamp":1778520478627,"version":"3.51.4"},"reference-count":32,"publisher":"Association for Computing Machinery (ACM)","issue":"4","license":[{"start":{"date-parts":[[2006,10,1]],"date-time":"2006-10-01T00:00:00Z","timestamp":1159660800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Comput. Logic"],"published-print":{"date-parts":[[2006,10]]},"abstract":"<jats:p>\n            It is well known that, under certain conditions, it is possible to\n            <jats:italic>split<\/jats:italic>\n            logic programs under stable model semantics, that is, to divide such a program into a number of different \u201clevels\u201d, such that the models of the entire program can be constructed by incrementally constructing models for each level. Similar results exist for other nonmonotonic formalisms, such as auto-epistemic logic and default logic. In this work, we present a general, algebraic splitting theory for logics with a fixpoint semantics. Together with the framework of\n            <jats:italic>approximation theory<\/jats:italic>\n            , a general fixpoint theory for arbitrary operators, this gives us a uniform and powerful way of deriving splitting results for each logic with a fixpoint semantics. We demonstrate the usefulness of these results, by generalizing existing results for logic programming, auto-epistemic logic and default logic.\n          <\/jats:p>","DOI":"10.1145\/1183278.1183284","type":"journal-article","created":{"date-parts":[[2007,1,16]],"date-time":"2007-01-16T19:38:29Z","timestamp":1168976309000},"page":"765-797","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":20,"title":["Splitting an operator"],"prefix":"10.1145","volume":"7","author":[{"given":"Joost","family":"Vennekens","sequence":"first","affiliation":[{"name":"K.U. Leuven, Leuven, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"David","family":"Gilis","sequence":"additional","affiliation":[{"name":"K.U. Leuven, Leuven, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marc","family":"Denecker","sequence":"additional","affiliation":[{"name":"K.U. Leuven, Leuven, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2006,10]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"crossref","unstructured":"Apt K. Blair H. and Walker A. 1988. Towards a theory of declarative knowledge. In Foundations of Deductive Databases and Logic Programming J. Minker Ed. Morgan-Kaufmann Los Alamitos CA.]]   Apt K. Blair H. and Walker A. 1988. Towards a theory of declarative knowledge. In Foundations of Deductive Databases and Logic Programming J. Minker Ed. Morgan-Kaufmann Los Alamitos CA.]]","DOI":"10.1016\/B978-0-934613-40-8.50006-3"},{"key":"e_1_2_1_2_1","volume-title":"Proceedings of the International Workshop on Logic Programming and Nonmonotonic Reasoning (Washington DC), A. Nerode, W. Marek, and V. Subrahmanian, Eds. MIT Press","author":"Baral C."},{"key":"e_1_2_1_3_1","volume-title":"Proceedings of the 1st International Conference on Computational Logic (CL2000)","volume":"1861","author":"Denecker M.","year":"2000"},{"key":"e_1_2_1_4_1","volume-title":"Proceedings of the 15th National Conference on Artificial Intelligence. MIT Press\/AAAI-Press, 840--845","author":"Denecker M."},{"key":"e_1_2_1_5_1","doi-asserted-by":"crossref","unstructured":"Denecker M. Marek V. and Truszczynski M. 2000. Approximating operators stable operators well-founded fixpoints and applications in non-monotonic reasoning. In Logic-Based Artificial Intelligence. The Kluwer International Series in Engineering and Computer Science. Kluwer Academic Publishers Boston MA 127--144.]]   Denecker M. Marek V. and Truszczynski M. 2000. Approximating operators stable operators well-founded fixpoints and applications in non-monotonic reasoning. In Logic-Based Artificial Intelligence. The Kluwer International Series in Engineering and Computer Science. Kluwer Academic Publishers Boston MA 127--144.]]","DOI":"10.1007\/978-1-4615-1567-8_6"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0004-3702(02)00293-X"},{"key":"e_1_2_1_7_1","doi-asserted-by":"crossref","unstructured":"Denecker M. and Ternovska E. 2004. A logic of non-monotone inductive definitions and its modularity properties. In LPNMR. 47--60.]]  Denecker M. and Ternovska E. 2004. A logic of non-monotone inductive definitions and its modularity properties. In LPNMR. 47--60.]]","DOI":"10.1007\/978-3-540-24609-1_7"},{"key":"e_1_2_1_8_1","first-page":"257","article-title":"A classification theory of semantics normal logic programs: II. weak properties","author":"Dix J.","year":"1995","journal-title":"Fund. Inf."},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1145\/261124.261126"},{"key":"e_1_2_1_10_1","volume-title":"Proc. Logic Programming and Non Monotonic Reasoning, LPNMR'04","volume":"2923","author":"Erdo\u011fan S."},{"key":"e_1_2_1_11_1","volume-title":"Reasoning with incomplete information. Research notes in Artificial Intelligence. Morgan-Kaufmann","author":"Etherington D."},{"key":"e_1_2_1_12_1","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1016\/S0743-1066(85)80005-4","article-title":"A Kripke-Kleene semantics for logic programs","volume":"2","author":"Fitting M.","year":"1985","journal-title":"J. Logic Prog."},{"key":"e_1_2_1_13_1","doi-asserted-by":"publisher","DOI":"10.1016\/0743-1066(91)90014-G"},{"key":"e_1_2_1_14_1","volume-title":"Proceedings of AAAI87","author":"Gelfond M.","year":"1987"},{"key":"e_1_2_1_15_1","first-page":"59","article-title":"On consistency and completeness of autoepistemic theories","volume":"16","author":"Gelfond M.","year":"1992","journal-title":"Fund. Inf."},{"key":"e_1_2_1_16_1","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1111\/j.1467-8640.1988.tb00280.x","article-title":"Multivalued logics: A uniform approach to reasoning in artificial intelligence","volume":"4","author":"Ginsberg M.","year":"1988","journal-title":"Computat. Intell."},{"key":"e_1_2_1_17_1","volume-title":"Readings in Nonmonotonic Reasoning","author":"Konolige K."},{"key":"e_1_2_1_18_1","volume-title":"Proceedings of the International Logic Programming Symposium (ILPS '95)","author":"Leone N."},{"key":"e_1_2_1_19_1","volume-title":"Proceedings of the 11th International Conference on Logic Programming. MIT Press","author":"Lifschitz V."},{"key":"e_1_2_1_20_1","volume-title":"Foundations of Logic Programming","author":"Lloyd J."},{"key":"e_1_2_1_21_1","volume-title":"Epistemic Logic for Computer Science and Artificial Intelligence","author":"Meyer J.-J.Ch."},{"key":"e_1_2_1_22_1","volume-title":"Proceedings of the Non-Monotonic Reasoning Workshop. AAAI","author":"Moore R. C.","year":"1984"},{"key":"e_1_2_1_23_1","volume-title":"Proceedings of the 10th International Workshop on Non-Monotonic Reasoning (NMR","author":"Pelov N.","year":"2004"},{"key":"e_1_2_1_24_1","first-page":"2","article-title":"On the impact of stratification on the complexity of nonmonotonic reasoning","volume":"4","author":"Niemel\u00e4 I.","year":"1994","journal-title":"J. Appl. Non-Classical Log."},{"key":"e_1_2_1_25_1","volume-title":"Proceedings of the 8th Symposium on Principles of Database Systems (PODS). 11--21","author":"Przymusinski T.","year":"1998"},{"key":"e_1_2_1_26_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0004-3702(80)90014-4","article-title":"A logic for default reasoning","volume":"13","author":"Reiter R.","year":"1980","journal-title":"Artif. Intell."},{"key":"e_1_2_1_27_1","doi-asserted-by":"crossref","first-page":"285","DOI":"10.2140\/pjm.1955.5.285","article-title":"Lattice-theoretic fixpoint theorem and its applications","volume":"5","author":"Tarski A.","year":"1955","journal-title":"Pac. J. Math."},{"key":"e_1_2_1_28_1","volume-title":"Proceedings of the 13th National Conference on Artificial Intelligence and the 8th Innovative Applications of Artificial Intelligence Conference. AAAI Press, 645--651","author":"Turner H.","year":"1996"},{"key":"e_1_2_1_29_1","doi-asserted-by":"publisher","DOI":"10.1145\/116825.116838"},{"key":"e_1_2_1_30_1","volume-title":"Proceedings of International Workshop on Non-Monotonic Reasoning, Whistler, B.C., Canada.]]","author":"Vennekens J."},{"key":"e_1_2_1_31_1","volume-title":"Proceedings of the 20th International Conference on Logic Programming (ICLP","volume":"3132","author":"Vennekens J.","year":"2004"},{"key":"e_1_2_1_32_1","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1016\/S0743-1066(99)00003-5","article-title":"Compositionality of normal open logic programs","volume":"41","author":"Verbaeten S.","year":"2000","journal-title":"J. Logic Prog."}],"container-title":["ACM Transactions on Computational Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1183278.1183284","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/1183278.1183284","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T15:06:37Z","timestamp":1750259197000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1183278.1183284"}},"subtitle":["Algebraic modularity results for logics with fixpoint semantics"],"short-title":[],"issued":{"date-parts":[[2006,10]]},"references-count":32,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2006,10]]}},"alternative-id":["10.1145\/1183278.1183284"],"URL":"https:\/\/doi.org\/10.1145\/1183278.1183284","relation":{},"ISSN":["1529-3785","1557-945X"],"issn-type":[{"value":"1529-3785","type":"print"},{"value":"1557-945X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,10]]},"assertion":[{"value":"2006-10-01","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}