{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T20:53:19Z","timestamp":1776891199724,"version":"3.51.2"},"reference-count":13,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","issue":"3","license":[{"start":{"date-parts":[[2008,11,7]],"date-time":"2008-11-07T00:00:00Z","timestamp":1226016000000},"content-version":"unspecified","delay-in-days":5973,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Funct. Prog."],"published-print":{"date-parts":[[1992,7]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Let \u03c8 be a partial recursive function (of one argument) with \u03bb-defining term\n                    <jats:italic>F<\/jats:italic>\n                    \u2208\u039b\u00b0. This means\n                    <jats:disp-formula>\n                      <jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0956796800000447eqnU1\"\/>\n                    <\/jats:disp-formula>\n                    There are several proposals for what\n                    <jats:italic>F<\/jats:italic>\n                    \u231c\n                    <jats:italic>n<\/jats:italic>\n                    \u231d should be in case \u03c8(\n                    <jats:italic>n<\/jats:italic>\n                    ) is undefined: (1) a term without a normal form (Church); (2) an unsolvable term (Barendregt); (3) an easy term (Visser); (4) a term of order 0 (Statman).\n                  <\/jats:p>\n                  <jats:p>These four possibilities will be covered by one \u2018master\u2019 result of Statman which is based on the \u2018Anti Diagonal Normalization Theorem\u2019 of Visser (1980). That ingenious theorem about precomplete numerations of Ershov is a powerful tool with applications in recursion theory, metamathematics of arithmetic and lambda calculus.<\/jats:p>","DOI":"10.1017\/s0956796800000447","type":"journal-article","created":{"date-parts":[[2008,11,7]],"date-time":"2008-11-07T11:14:19Z","timestamp":1226056459000},"page":"367-374","source":"Crossref","is-referenced-by-count":19,"title":["Theoretical Pearls:\n                    <i>Representing \u2018undefined\u2019 in lambda calculus<\/i>"],"prefix":"10.46298","volume":"2","author":[{"given":"Henk","family":"Barendregt","sequence":"first","affiliation":[]}],"member":"25203","published-online":{"date-parts":[[2008,11,7]]},"reference":[{"key":"S0956796800000447_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-7288-6"},{"key":"S0956796800000447_ref008","volume-title":"Introduction to metamathematics","author":"Kleene","year":"1952"},{"key":"S0956796800000447_ref006","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19730191901"},{"key":"S0956796800000447_ref012","volume-title":"The Visser Fixed Point Theorem","author":"Statman","year":"1990"},{"key":"S0956796800000447_ref013","first-page":"259","volume-title":"To H. B. Curry: Essays on Combinatory Logic, Lambda-Calculus and Formalism","author":"Visser","year":"1980"},{"key":"S0956796800000447_ref007","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-36-00227-2"},{"key":"S0956796800000447_ref001","unstructured":"Barendregt H. P. 1971. Some extensional term models for lambda calculi and combinatory logics. PhD Thesis, Utrecht University."},{"key":"S0956796800000447_ref004","doi-asserted-by":"publisher","DOI":"10.2307\/2273443"},{"key":"S0956796800000447_ref010","volume-title":"Theory of recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S0956796800000447_ref002","volume-title":"The lambda calculus: its syntax and semantics","author":"Barendregt","year":"1984"},{"key":"S0956796800000447_ref011","volume-title":"Two recursion theoretic problems in lambda calculus","author":"Statman","year":"1987"},{"key":"S0956796800000447_ref003","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1017\/S0956796800020062","article-title":"Theoretical pearls: Self-interpretation in \u03bb-calculus","volume":"1","author":"Barendregt","year":"1991","journal-title":"J. Functional Program"},{"key":"S0956796800000447_ref005","volume-title":"The calculi of lambda conversion","author":"Church","year":"1941"}],"container-title":["Journal of Functional Programming"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0956796800000447","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T20:17:52Z","timestamp":1776889072000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0956796800000447\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1992,7]]},"references-count":13,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1992,7]]}},"alternative-id":["S0956796800000447"],"URL":"https:\/\/doi.org\/10.1017\/s0956796800000447","relation":{},"ISSN":["0956-7968","1469-7653"],"issn-type":[{"value":"0956-7968","type":"print"},{"value":"1469-7653","type":"electronic"}],"subject":[],"published":{"date-parts":[[1992,7]]}}}