{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T21:18:14Z","timestamp":1649193494684},"reference-count":9,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2015,7,29]],"date-time":"2015-07-29T00:00:00Z","timestamp":1438128000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[2017,3]]},"abstract":"<jats:p>A natural question in the \u03bb-calculus asks what is the possible number of fixed points of a combinator (closed term). A complete answer to this question is still missing (Problem 25 of TLCA Open Problems List) and we investigate the related question about the number of fixed points of a combinator in \u03bb-theories. We show the existence of a recursively enumerable lambda theory where the number is always one or infinite. We also show that there are \u03bb-theories such that some terms have only two fixed points. In a first example, this is obtained by means of a non-constructive (more precisely non-r.e.) \u03bb-theory where the range property is violated. A second, more complex example of a non-r.e. \u03bb-theory (with a higher unsolvability degree) shows that some terms can have only two fixed points while the range property holds for every term.<\/jats:p>","DOI":"10.1017\/s0960129515000419","type":"journal-article","created":{"date-parts":[[2015,7,29]],"date-time":"2015-07-29T06:12:03Z","timestamp":1438150323000},"page":"405-427","source":"Crossref","is-referenced-by-count":0,"title":["Lambda theories allowing terms with a finite number of fixed points"],"prefix":"10.1017","volume":"27","author":[{"given":"BENEDETTO","family":"INTRIGILA","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"RICHARD","family":"STATMAN","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2015,7,29]]},"reference":[{"key":"S0960129515000419_ref1","volume-title":"The Lambda Calculus. Its Syntax and Semantics","author":"Barendregt","year":"1984"},{"key":"S0960129515000419_ref6","doi-asserted-by":"publisher","DOI":"10.2178\/jsl.7704080"},{"key":"S0960129515000419_ref2","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(93)90083-6"},{"key":"S0960129515000419_ref8","first-page":"178","volume-title":"Proceedings of the 12th International Workshop on Computer Science Logic","author":"Statman","year":"1999"},{"key":"S0960129515000419_ref3","volume-title":"Term Rewriting Systems","author":"Bezem","year":"2003"},{"key":"S0960129515000419_ref7","volume-title":"Theory of Recursive Functions and Effective Computability","author":"Rogers","year":"1967"},{"key":"S0960129515000419_ref5","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2003.05.001"},{"key":"S0960129515000419_ref4","doi-asserted-by":"publisher","DOI":"10.1017\/S0960129500003091"},{"key":"S0960129515000419_ref9","unstructured":"TLCA List of Open Problems (2014). Available at http:\/\/tlca.di.unito.it\/opltlca\/."}],"container-title":["Mathematical Structures in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0960129515000419","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,18]],"date-time":"2019-04-18T02:08:29Z","timestamp":1555553309000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0960129515000419\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,7,29]]},"references-count":9,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2017,3]]}},"alternative-id":["S0960129515000419"],"URL":"https:\/\/doi.org\/10.1017\/s0960129515000419","relation":{},"ISSN":["0960-1295","1469-8072"],"issn-type":[{"value":"0960-1295","type":"print"},{"value":"1469-8072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,7,29]]}}}