{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T05:11:04Z","timestamp":1777439464784,"version":"3.51.4"},"reference-count":20,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2023,7,11]],"date-time":"2023-07-11T00:00:00Z","timestamp":1689033600000},"content-version":"unspecified","delay-in-days":10,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Theory and Practice of Logic Programming"],"published-print":{"date-parts":[[2023,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Approximation fixpoint theory (AFT) is an abstract and general algebraic framework for studying the semantics of non-monotonic logics. In recent work, AFT was generalized to non-deterministic operators, that is, operators whose range are sets of elements rather than single elements. In this paper, we make three further contributions to non-deterministic AFT: (1) we define and study ultimate approximations of non-deterministic operators, (2) we give an algebraic formulation of the semi-equilibrium semantics by Amendola <jats:italic>et al<\/jats:italic>., and (3) we generalize the characterizations of disjunctive logic programs to disjunctive logic programs with aggregates.<\/jats:p>","DOI":"10.1017\/s1471068423000236","type":"journal-article","created":{"date-parts":[[2023,7,11]],"date-time":"2023-07-11T07:26:51Z","timestamp":1689060411000},"page":"632-647","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":2,"title":["Non-deterministic Approximation Operators: Ultimate Operators, Semi-equilibrium Semantics, and Aggregates"],"prefix":"10.1017","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3825-4052","authenticated-orcid":false,"given":"JESSE","family":"HEYNINCK","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3460-4251","authenticated-orcid":false,"given":"BART","family":"BOGAERTS","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2023,7,11]]},"reference":[{"key":"S1471068423000236_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/j.artint.2019.04.004"},{"key":"S1471068423000236_ref12","unstructured":"Heyninck, J. and Bogaerts, B. 2023. Non-deterministic approximation operators: Ultimate operators, semi-equilibrium semantics and aggregates (full version). arXiv preprint arXiv:2305.10846."},{"key":"S1471068423000236_ref11","doi-asserted-by":"publisher","DOI":"10.24963\/kr.2021\/32"},{"key":"S1471068423000236_ref16","doi-asserted-by":"publisher","DOI":"10.1017\/S1471068406002973"},{"key":"S1471068423000236_ref15","doi-asserted-by":"publisher","DOI":"10.1007\/s10472-006-9028-z"},{"key":"S1471068423000236_ref9","doi-asserted-by":"publisher","DOI":"10.1016\/0743-1066(94)00106-G"},{"key":"S1471068423000236_ref1","doi-asserted-by":"crossref","unstructured":"Alc\u00e2ntara, J. , Dam\u00e1sio, C. V. and Pereira, L. M. 2005. A well-founded semantics with disjunction. In Proceedings of ICLP\u201905. Springer, 341\u2013355.","DOI":"10.1007\/11562931_26"},{"key":"S1471068423000236_ref13","doi-asserted-by":"publisher","DOI":"10.1017\/S147106840700302X"},{"key":"S1471068423000236_ref4","unstructured":"Bogaerts, B. 2015. Groundedness in Logics with a Fixpoint Semantics. Ph.D. thesis, Informatics Section, Department of Computer Science, Faculty of Engineering Science."},{"key":"S1471068423000236_ref5","doi-asserted-by":"crossref","unstructured":"Bogaerts, B. 2019. Weighted abstract dialectical frameworks through the lens of approximation fixpoint theory. In Proceedings of AAAI\u201919, vol. 33, 2686\u20132693.","DOI":"10.1609\/aaai.v33i01.33012686"},{"key":"S1471068423000236_ref2","doi-asserted-by":"publisher","DOI":"10.1017\/S1471068422000047"},{"key":"S1471068423000236_ref3","doi-asserted-by":"publisher","DOI":"10.1016\/j.artint.2016.01.011"},{"key":"S1471068423000236_ref8","doi-asserted-by":"crossref","unstructured":"Faber, W. , Leone, N. and Pfeifer, G. 2004. Recursive aggregates in disjunctive logic programs: Semantics and complexity. In Proceedings of JELIA\u201904. LNCS, vol. 3229. Springer, 200\u2013212.","DOI":"10.1007\/978-3-540-30227-8_19"},{"key":"S1471068423000236_ref17","unstructured":"Pelov, N. and Truszczynski, M. 2004. Semantics of disjunctive programs with monotone aggregates: An operator-based approach. In Proceedings of NMR\u201904, 327\u2013334."},{"key":"S1471068423000236_ref19","doi-asserted-by":"publisher","DOI":"10.1145\/321978.321991"},{"key":"S1471068423000236_ref6","doi-asserted-by":"crossref","unstructured":"Denecker, M. , Marek, V. and Truszczy\u0144ski, M. 2000. Approximations, stable operators, well-founded fixpoints and applications in nonmonotonic reasoning. In Logic-based Artificial Intelligence. Engineering and Computer Science, vol. 597. Springer, 127\u2013144.","DOI":"10.1007\/978-1-4615-1567-8_6"},{"key":"S1471068423000236_ref7","unstructured":"Denecker, M. , Marek, V. W. and Truszczynski, M. 2002. Ultimate approximations in nonmonotonic knowledge representation systems. In Proceedings of KR\u201902, 177\u2013190."},{"key":"S1471068423000236_ref14","doi-asserted-by":"crossref","unstructured":"Marek, V. W. and Remmel, J. B. 2004. Set constraints in logic programming. In Logic Programming and Nonmonotonic Reasoning: 7th International Conference, LPNMR 2004. 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