{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T17:30:06Z","timestamp":1778520606621,"version":"3.51.4"},"reference-count":28,"publisher":"Association for Computing Machinery (ACM)","issue":"2","license":[{"start":{"date-parts":[[2005,4,1]],"date-time":"2005-04-01T00:00:00Z","timestamp":1112313600000},"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":[[2005,4]]},"abstract":"<jats:p>\n            We give a purely model-theoretic characterization of the semantics of logic programs with negation-as-failure allowed in clause bodies. In our semantics, the meaning of a program is, as in the classical case, the unique\n            <jats:italic>minimum<\/jats:italic>\n            model in a program-independent ordering. We use an expanded truth domain that has an uncountable linearly ordered set of truth values between\n            <jats:italic>False<\/jats:italic>\n            (the minimum element) and\n            <jats:italic>True<\/jats:italic>\n            (the maximum), with a\n            <jats:italic>Zero<\/jats:italic>\n            element in the middle. The truth values below\n            <jats:italic>Zero<\/jats:italic>\n            are ordered like the countable ordinals. The values above\n            <jats:italic>Zero<\/jats:italic>\n            have exactly the reverse order. Negation is interpreted as reflection about\n            <jats:italic>Zero<\/jats:italic>\n            followed by a step towards\n            <jats:italic>Zero<\/jats:italic>\n            ; the only truth value that remains unaffected by negation is\n            <jats:italic>Zero<\/jats:italic>\n            . We show that every program has a unique minimum model\n            <jats:italic>M<\/jats:italic>\n            <jats:sub>P<\/jats:sub>\n            , and that this model can be constructed with a\n            <jats:italic>T<\/jats:italic>\n            <jats:sub>P<\/jats:sub>\n            iteration which proceeds through the countable ordinals. Furthermore, we demonstrate that\n            <jats:italic>M<\/jats:italic>\n            <jats:sub>P<\/jats:sub>\n            can alternatively be obtained through a construction that generalizes the well-known model intersection theorem for classical logic programming. Finally, we show that by collapsing the true and false values of the infinite-valued model\n            <jats:italic>M<\/jats:italic>\n            <jats:sub>P<\/jats:sub>\n            to (the classical)\n            <jats:italic>True<\/jats:italic>\n            and\n            <jats:italic>False<\/jats:italic>\n            , we obtain a three-valued model identical to the well-founded one.\n          <\/jats:p>","DOI":"10.1145\/1055686.1055694","type":"journal-article","created":{"date-parts":[[2005,8,3]],"date-time":"2005-08-03T08:30:55Z","timestamp":1123057855000},"page":"441-467","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":27,"title":["Minimum model semantics for logic programs with negation-as-failure"],"prefix":"10.1145","volume":"6","author":[{"given":"Panos","family":"Rondogiannis","sequence":"first","affiliation":[{"name":"University of Athens, Athens, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"William W.","family":"Wadge","sequence":"additional","affiliation":[{"name":"University of Victoria, BC, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2005,4]]},"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 Altos Calif. 89--148.   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 Altos Calif. 89--148.","DOI":"10.1016\/B978-0-934613-40-8.50006-3"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1016\/0743-1066(94)90024-8"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1145\/357162.357170"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1016\/0743-1066(94)90025-6"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.5555\/2383456.2383460"},{"key":"e_1_2_1_6_1","volume-title":"Negation as failure","author":"Clark K.","unstructured":"Clark , K. 1978. Negation as failure . In Logic and Databases, H. Gallaire and J. Minker, Eds. Plenum Press , New York , 293--322. Clark, K. 1978. Negation as failure. In Logic and Databases, H. Gallaire and J. Minker, Eds. Plenum Press, New York, 293--322."},{"key":"e_1_2_1_7_1","volume-title":"On Nunbers and Games","author":"Conway J.","unstructured":"Conway , J. 1976. On Nunbers and Games . Academic Press , New, York . Conway, J. 1976. On Nunbers and Games. Academic Press, New, York."},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1145\/383779.383789"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0743-1066(85)80005-4"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(00)00330-3"},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0004-3702(02)00207-2"},{"key":"e_1_2_1_12_1","volume-title":"Proceedings of the 5th Logic Programming Symposium. MIT Press","author":"Gelfond M.","unstructured":"Gelfond , M. and Lifschitz , V . 1988. The stable model semantics for logic programming . In Proceedings of the 5th Logic Programming Symposium. MIT Press , Cambridge, Mass. 1070--1080. Gelfond, M. and Lifschitz, V. 1988. The stable model semantics for logic programming. In Proceedings of the 5th Logic Programming Symposium. MIT Press, Cambridge, Mass. 1070--1080."},{"key":"e_1_2_1_13_1","doi-asserted-by":"publisher","DOI":"10.1016\/0743-1066(87)90007-0"},{"key":"e_1_2_1_14_1","unstructured":"Lakoff G. and Nunez R. 2000. Where Mathematics Comes From. Basic Books.  Lakoff G. and Nunez R. 2000. Where Mathematics Comes From. Basic Books."},{"key":"e_1_2_1_15_1","volume-title":"Foundations of Logic Programming","author":"Lloyd J.","unstructured":"Lloyd , J. 1987. Foundations of Logic Programming . Springer-Verlag , New York . Lloyd, J. 1987. Foundations of Logic Programming. Springer-Verlag, New York."},{"key":"e_1_2_1_17_1","doi-asserted-by":"crossref","unstructured":"Marek V. W. and Truszczynski M. 1999. Stable models and an alternative logic programming paradigm. In The Logic Programming Paradigm: A 25-Year Perspective. Springer-Verlag New York 375--398.  Marek V. W. and Truszczynski M. 1999. Stable models and an alternative logic programming paradigm. In The Logic Programming Paradigm: A 25-Year Perspective. Springer-Verlag New York 375--398.","DOI":"10.1007\/978-3-642-60085-2_17"},{"key":"e_1_2_1_18_1","doi-asserted-by":"publisher","DOI":"10.1145\/181911.181920"},{"key":"e_1_2_1_19_1","doi-asserted-by":"publisher","DOI":"10.1016\/0743-1066(92)90055-8"},{"key":"e_1_2_1_20_1","unstructured":"Przymusinska H. and Przymusinski T. 1990. Semantic issues in deductive databases and logic programs. In Formal Techniques in Artificial Intelligence: a Source-Book R. Banerji Ed. North Holland Amsterdam The Netherlands 321--367.  Przymusinska H. and Przymusinski T. 1990. Semantic issues in deductive databases and logic programs. In Formal Techniques in Artificial Intelligence: a Source-Book R. Banerji Ed. 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