{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T15:36:50Z","timestamp":1753889810199,"version":"3.41.2"},"reference-count":1,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2015,10,1]],"date-time":"2015-10-01T00:00:00Z","timestamp":1443657600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:p>Ellipses are a meta-linguistic notation for denoting terms the size of which\nare specified by a meta-variable that ranges over the natural numbers. In this\nwork, we present a systematic approach for encoding such meta-expressions in\nthe \\^I-calculus, without ellipses: Terms that are parameterized by\nmeta-variables are replaced with corresponding \\^I-abstractions over actual\nvariables. We call such \\^I-terms arity-generic. Concrete terms, for particular\nchoices of the parameterizing variable are obtained by applying an\narity-generic \\^I-term to the corresponding numeral, obviating the need to use\nellipses. For example, to find the multiple fixed points of n equations, n\ndifferent \\^I-terms are needed, every one of which is indexed by two\nmeta-variables, and defined using three levels of ellipses. A single\narity-generic \\^I-abstraction that takes two Church numerals, one for the\nnumber of fixed-point equations, and one for their arity, replaces all these\nmultiple fixed-point combinators. We show how to define arity-generic\ngeneralizations of two historical fixed-point combinators, the first by Curry,\nand the second by Turing, for defining multiple fixed points. These historical\nfixed-point combinators are related by a construction due to B\\~Ahm: We show\nthat likewise, their arity-generic generalizations are related by an\narity-generic generalization of B\\~Ahm's construction. We further demonstrate\nthis approach to arity-generic \\^I-definability with additional \\^I-terms that\ncreate, project, extend, reverse, and map over ordered n-tuples, as well as an\narity-generic generator for one-point bases.<\/jats:p>","DOI":"10.2168\/lmcs-11(3:25)2015","type":"journal-article","created":{"date-parts":[[2016,11,21]],"date-time":"2016-11-21T13:46:02Z","timestamp":1479735962000},"source":"Crossref","is-referenced-by-count":0,"title":["Ellipses and Lambda Definability"],"prefix":"10.46298","volume":"Volume 11, Issue 3","author":[{"given":"Mayer","family":"Goldberg","sequence":"first","affiliation":[]}],"member":"25203","published-online":{"date-parts":[[2015,10,1]]},"reference":[{"key":"1051:not-found"}],"container-title":["Logical Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/lmcs.episciences.org\/1601\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/lmcs.episciences.org\/1601\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,11]],"date-time":"2023-04-11T20:07:35Z","timestamp":1681243655000},"score":1,"resource":{"primary":{"URL":"https:\/\/lmcs.episciences.org\/1601"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,10,1]]},"references-count":1,"URL":"https:\/\/doi.org\/10.2168\/lmcs-11(3:25)2015","relation":{"is-same-as":[{"id-type":"arxiv","id":"1508.02864","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1508.02864","asserted-by":"subject"}]},"ISSN":["1860-5974"],"issn-type":[{"type":"electronic","value":"1860-5974"}],"subject":[],"published":{"date-parts":[[2015,10,1]]},"article-number":"1601"}}