{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,1,27]],"date-time":"2024-01-27T16:08:01Z","timestamp":1706371681963},"reference-count":15,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":3206,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2005,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Symmetic combinatory logic with the symmetric analogue of a combinatorially complete base (in the form of symmetric \u03bb-calculus) is known to lack the Church-Rosser property. We prove a much<jats:italic>stronger<\/jats:italic>theorem that no symmetric combinatory logic that contains<jats:italic>at least two proper symmetric combinatory<\/jats:italic>has the Church-Rosser property. Although the statement of the result looks similar to an earlier one concerning dual combinatory logic,<jats:italic>the proof is different<\/jats:italic>because symmetric combinators may form redexes in both left and right associated terms. Perhaps surprisingly, we are also able to show that certain symmetric combinatory logics that include just<jats:italic>one particular constant<\/jats:italic>are not confluent. This result (beyond other differences) clearly sets apart symmetric combinatory logic from dual combinatory logic, since all dual combinatory systems with a single combinator or a single dual combinator are Church-Rosser. Lastly, we prove that a symmetric combinatory logic that contains the fixed point and the one-place identity combinator has the Church-Rosser property.<\/jats:p>","DOI":"10.2178\/jsl\/1120224727","type":"journal-article","created":{"date-parts":[[2005,7,1]],"date-time":"2005-07-01T15:04:33Z","timestamp":1120230273000},"page":"536-556","source":"Crossref","is-referenced-by-count":2,"title":["The Church-Rosser property in symmetric combinatory logic"],"prefix":"10.1017","volume":"70","author":[{"given":"Katalin","family":"Bimb\u00f3","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200007064_ref009","first-page":"545","volume":"34","author":"Hindley","year":"1969","journal-title":"An abstract form of the Church-Rosser theorem, I"},{"key":"S0022481200007064_ref002","doi-asserted-by":"publisher","DOI":"10.1023\/A:1005252431462"},{"key":"S0022481200007064_ref014","first-page":"127","article-title":"A mathematical logic without variables","volume":"2","author":"Rosser","year":"1936","journal-title":"Annals of Mathematics"},{"key":"S0022481200007064_ref005","volume-title":"The calculi of lambda-conversion","author":"Church","year":"1941"},{"key":"S0022481200007064_ref015","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198534501.001.0001","volume-title":"Diagonalization and self-reference","author":"Smullyan","year":"1994"},{"key":"S0022481200007064_ref013","doi-asserted-by":"publisher","DOI":"10.1145\/321738.321750"},{"key":"S0022481200007064_ref008","doi-asserted-by":"publisher","DOI":"10.1093\/jigpal\/5.4.505"},{"key":"S0022481200007064_ref010","volume-title":"Introduction to combinators and \u03bb-calculus","author":"Hindley","year":"1986"},{"key":"S0022481200007064_ref003","first-page":"132","volume":"68","author":"Bimb\u00f3","year":"2003","journal-title":"The Church-Rosser property in dual combinatory logic"},{"key":"S0022481200007064_ref011","doi-asserted-by":"publisher","DOI":"10.2307\/1968749"},{"key":"S0022481200007064_ref006","volume-title":"Combinatory logic","volume":"I","author":"Curry","year":"1958"},{"key":"S0022481200007064_ref001","doi-asserted-by":"publisher","DOI":"10.1006\/inco.1996.0025"},{"key":"S0022481200007064_ref007","volume-title":"Combinatory logic","volume":"II","author":"Curry","year":"1972"},{"key":"S0022481200007064_ref012","first-page":"463","article-title":"Dual combinators bite the dust, (abstract)","volume":"4","author":"Meyer","year":"1998","journal-title":"The Bulletin of Symbolic Logic"},{"key":"S0022481200007064_ref004","doi-asserted-by":"publisher","DOI":"10.1023\/B:LOGI.0000021709.73522.34"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200007064","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,1,27]],"date-time":"2024-01-27T15:21:16Z","timestamp":1706368876000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200007064\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,6]]},"references-count":15,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2005,6]]}},"alternative-id":["S0022481200007064"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1120224727","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2005,6]]}}}