{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T16:50:42Z","timestamp":1762102242037,"version":"3.37.3"},"reference-count":25,"publisher":"Cambridge University Press (CUP)","issue":"3-4","license":[{"start":{"date-parts":[[2018,8,10]],"date-time":"2018-08-10T00:00:00Z","timestamp":1533859200000},"content-version":"unspecified","delay-in-days":40,"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":[[2018,7]]},"abstract":"Abstract<\/jats:title>In this paper, we propose a variant of Answer Set Programming (ASP) with evaluable functions that extends their application to sets of objects, something that allows a fully logical treatment of aggregates. Formally, we start from the syntax of First Order Logic with equality and the semantics of Quantified Equilibrium Logic with evaluable functions (${\\rm QEL}^=_{\\cal F}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>). Then, we proceed to incorporate a new kind of logical term,intensional set<\/jats:italic>(a construct commonly used to denote the set of objects characterised by a given formula), and to extend${\\rm QEL}^=_{\\cal F}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>semantics for this new type of expression. In our extended approach, intensional sets can be arbitrarily used as predicate or function arguments or even nested inside other intensional sets, just as regular first-order logical terms. As a result, aggregates can be naturally formed by the application of some evaluable function (count<\/jats:monospace>,sum<\/jats:monospace>,maximum<\/jats:monospace>, etc) to a set of objects expressed as an intensional set. This approach has several advantages. First, while other semantics for aggregates depend on some syntactic transformation (either via a reduct or a formula translation), the${\\rm QEL}^=_{\\cal F}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>interpretation treats them as regular evaluable functions, providing a compositional semantics and avoiding any kind of syntactic restriction. Second, aggregates can be explicitly defined now within the logical language by the simple addition of formulas that fix their meaning in terms of multiple applications of some (commutative and associative) binary operation. For instance, we can use recursive rules to definesum<\/jats:monospace>in terms of integer addition. Last, but not least, we prove that the semantics we obtain for aggregates coincides with the one defined by Gelfond and Zhang for the${\\cal A}\\mathit{log}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>language, when we restrict to that syntactic fragment.<\/jats:p>","DOI":"10.1017\/s1471068418000169","type":"journal-article","created":{"date-parts":[[2018,8,10]],"date-time":"2018-08-10T05:44:17Z","timestamp":1533879857000},"page":"390-405","source":"Crossref","is-referenced-by-count":11,"title":["Functional ASP with Intensional Sets: Application to Gelfond-Zhang Aggregates"],"prefix":"10.1017","volume":"18","author":[{"given":"PEDRO","family":"CABALAR","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3917-8717","authenticated-orcid":false,"given":"JORGE","family":"FANDINNO","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"LUIS","family":"FARI\u00d1AS DEL CERRO","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"DAVID","family":"PEARCE","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2018,8,10]]},"reference":[{"key":"S1471068418000169_ref9","unstructured":"Cabalar P. , Kaminski R. , Ostrowski M. , and Schaub T. 2016. An ASP semantics for default reasoning with constraints. In Proc. of the 25th Intl. Joint Conference on Artificial Intelligence (IJCAI'16), New York, USA, July 9-15th, 2016, S. Kambhampati , Ed. 1015\u20131021."},{"key":"S1471068418000169_ref24","doi-asserted-by":"publisher","DOI":"10.1016\/S0004-3702(02)00187-X"},{"key":"S1471068418000169_ref1","first-page":"547","article-title":"ASP with non-herbrand partial functions: a language and system for practical use","volume":"13","author":"Balduccini","year":"2013","journal-title":"TPLP"},{"key":"S1471068418000169_ref4","doi-asserted-by":"publisher","DOI":"10.1016\/0743-1066(91)90036-O"},{"key":"S1471068418000169_ref6","unstructured":"Cabalar P. 2013. Setting the stage for ASP functions. ALP Newsletter."},{"key":"S1471068418000169_ref16","first-page":"855","article-title":"Program completion in the input language of GRINGO","volume":"17","author":"Harrison","year":"2017","journal-title":"TPLP"},{"key":"S1471068418000169_ref18","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0000(90)90033-H"},{"key":"S1471068418000169_ref10","unstructured":"Dovier A. , Omodeo E. G. , Pontelli E. , and Rossi G. 1991. {log}: A logic programming language with finite sets. In Logic Programming, Proceedings of the Eigth International Conference, Paris, France, June 24-28, 1991, K. Furukawa , Ed. MIT Press, 111\u2013124."},{"key":"S1471068418000169_ref2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511543357"},{"key":"S1471068418000169_ref19","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-04238-6_17"},{"key":"S1471068418000169_ref14","unstructured":"Ferraris P. and Lifschitz V. 2010. On the stable model semantics of first-order formulas with aggregates. In Proceedings of the 2010 Workshop on Nonmonotonic Reasoning."},{"key":"S1471068418000169_ref13","doi-asserted-by":"publisher","DOI":"10.1145\/1970398.1970401"},{"key":"S1471068418000169_ref8","doi-asserted-by":"crossref","unstructured":"Cabalar P. , Fari\u00f1as del Cerro L. , Pearce D. , and Valverde A. 2014. A free logic for stable models with partial intensional functions. In Proc. of the 14th European Conference on Logics in Artificial Intelligence (JELIA'14). Lecture Notes in Artificial Intelligence, vol. 8761. 340\u2013354.","DOI":"10.1007\/978-3-319-11558-0_24"},{"key":"S1471068418000169_ref12","doi-asserted-by":"publisher","DOI":"10.1016\/j.artint.2010.04.002"},{"key":"S1471068418000169_ref11","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1007\/978-3-540-24599-5_20","volume-title":"Logic Programming, 19th International Conference, ICLP 2003, Mumbai, India, December 9-13, 2003, Proceedings","author":"Dovier","year":"2003"},{"key":"S1471068418000169_ref22","doi-asserted-by":"crossref","unstructured":"Pearce D. and Valverde A. 2004. Towards a first order equilibrium logic for nonmonotonic reasoning. In Logics in Artificial Intelligence, 9th European Conference, JELIA 2004, Lisbon, Portugal, September 27-30, 2004, Proceedings, J. J. Alferes and J. A. Leite , Eds. Lecture Notes in Computer Science, vol. 3229. Springer, 147\u2013160.","DOI":"10.1007\/978-3-540-30227-8_15"},{"key":"S1471068418000169_ref17","unstructured":"Heyting A. 1930. Die formalen Regeln der intuitionistischen Logik. Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse, 42\u201356."},{"volume-title":"Non monotonic extensions of logic programming. Proc. NMELP'96. (LNAI 1216)","year":"1996","author":"Pearce","key":"S1471068418000169_ref21"},{"key":"S1471068418000169_ref5","doi-asserted-by":"publisher","DOI":"10.1017\/S1471068410000517"},{"volume-title":"Principles of Knowledge Representation and Reasoning: Proceedings of the Fourteenth International Conference, KR 2014, Vienna, Austria, July 20-24, 2014","year":"2014","author":"Bartholomew","key":"S1471068418000169_ref3"},{"key":"S1471068418000169_ref20","unstructured":"Lin F. and Wang Y. 2008. Answer set programming with functions. In Proc. of the 11th Intl. 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