A General Form of Relative Recursion

A General Form of Relative Recursion

The purpose of this note is to observe a generalization of the concept "computable in..." to arbitrary partial combinatory algebras. For every partial combinatory algebra (pca) A and every partial endofunction on A, a pca A[f] is constructed such that in A[f], the function f is representab...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Notre Dame journal of formal logic 2006, Vol.47 (3), p.311-318
1. Verfasser: van Oosten, Jaap
Format: Artikel
Sprache:eng
Schlagworte:
03B40
68N18
partial combinatory algebras
realizability
relative recursion
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
container_end_page 318
container_issue 3
container_start_page 311
container_title Notre Dame journal of formal logic
container_volume 47
creator van Oosten, Jaap
description The purpose of this note is to observe a generalization of the concept "computable in..." to arbitrary partial combinatory algebras. For every partial combinatory algebra (pca) A and every partial endofunction on A, a pca A[f] is constructed such that in A[f], the function f is representable by an element; a universal property of the construction is formulated in terms of Longley's 2-category of pcas and decidable applicative morphisms. It is proved that there is always a geometric inclusion from the realizability topos on A[f] into the one on A and that there is a meaningful preorder on the partial endofunctions on A which generalizes Turing reducibility.
doi_str_mv 10.1305/ndjfl/1163775438
format Article
fullrecord <record><control><sourceid>crossref_proje</sourceid><recordid>TN_cdi_projecteuclid_primary_oai_CULeuclid_euclid_ndjfl_1163775438</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>10_1305_ndjfl_1163775438</sourcerecordid><originalsourceid>FETCH-LOGICAL-c2598-b468785877cdaabba2d8ad608244009960703470f95909babcaf3de9e882347b3</originalsourceid><addsrcrecordid>eNplkE1Lw0AYhBdRMFbvHnPwGvvuV3b3WIOtQkAQew77CQlptmxawX9vtMEePM0wMA_MIHSP4RFT4MvBdaFfYlxSITij8gJlWFFVgCDlJcoAiCoYJ-Ia3YxjB4AZVyxDD6t84wefdJ-vY9rlMeTvvteH9tNPxh7T2MbhFl0F3Y_-btYF2q6fP6qXon7bvFarurCEK1kYVkohuRTCOq2N0cRJ7UqQhDEApUoQQJmAoLgCZbSxOlDnlZeSTLmhC_R04u5T7Lw9-KPtW9fsU7vT6auJum2qbT2ns_zubs67JwicIDbFcUw-_PUxND9X_a98A6zpXDA</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>A General Form of Relative Recursion</title><source>EZB-FREE-00999 freely available EZB journals</source><source>Project Euclid Complete</source><creator>van Oosten, Jaap</creator><creatorcontrib>van Oosten, Jaap</creatorcontrib><description>The purpose of this note is to observe a generalization of the concept "computable in..." to arbitrary partial combinatory algebras. For every partial combinatory algebra (pca) A and every partial endofunction on A, a pca A[f] is constructed such that in A[f], the function f is representable by an element; a universal property of the construction is formulated in terms of Longley's 2-category of pcas and decidable applicative morphisms. It is proved that there is always a geometric inclusion from the realizability topos on A[f] into the one on A and that there is a meaningful preorder on the partial endofunctions on A which generalizes Turing reducibility.</description><identifier>ISSN: 0029-4527</identifier><identifier>EISSN: 1939-0726</identifier><identifier>DOI: 10.1305/ndjfl/1163775438</identifier><language>eng</language><publisher>Duke University Press</publisher><subject>03B40 ; 68N18 ; partial combinatory algebras ; realizability ; relative recursion</subject><ispartof>Notre Dame journal of formal logic, 2006, Vol.47 (3), p.311-318</ispartof><rights>Copyright 2006 University of Notre Dame</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c2598-b468785877cdaabba2d8ad608244009960703470f95909babcaf3de9e882347b3</citedby><cites>FETCH-LOGICAL-c2598-b468785877cdaabba2d8ad608244009960703470f95909babcaf3de9e882347b3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>230,314,780,784,885,926,4024,27923,27924,27925</link.rule.ids></links><search><creatorcontrib>van Oosten, Jaap</creatorcontrib><title>A General Form of Relative Recursion</title><title>Notre Dame journal of formal logic</title><description>The purpose of this note is to observe a generalization of the concept "computable in..." to arbitrary partial combinatory algebras. For every partial combinatory algebra (pca) A and every partial endofunction on A, a pca A[f] is constructed such that in A[f], the function f is representable by an element; a universal property of the construction is formulated in terms of Longley's 2-category of pcas and decidable applicative morphisms. It is proved that there is always a geometric inclusion from the realizability topos on A[f] into the one on A and that there is a meaningful preorder on the partial endofunctions on A which generalizes Turing reducibility.</description><subject>03B40</subject><subject>68N18</subject><subject>partial combinatory algebras</subject><subject>realizability</subject><subject>relative recursion</subject><issn>0029-4527</issn><issn>1939-0726</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2006</creationdate><recordtype>article</recordtype><recordid>eNplkE1Lw0AYhBdRMFbvHnPwGvvuV3b3WIOtQkAQew77CQlptmxawX9vtMEePM0wMA_MIHSP4RFT4MvBdaFfYlxSITij8gJlWFFVgCDlJcoAiCoYJ-Ia3YxjB4AZVyxDD6t84wefdJ-vY9rlMeTvvteH9tNPxh7T2MbhFl0F3Y_-btYF2q6fP6qXon7bvFarurCEK1kYVkohuRTCOq2N0cRJ7UqQhDEApUoQQJmAoLgCZbSxOlDnlZeSTLmhC_R04u5T7Lw9-KPtW9fsU7vT6auJum2qbT2ns_zubs67JwicIDbFcUw-_PUxND9X_a98A6zpXDA</recordid><startdate>2006</startdate><enddate>2006</enddate><creator>van Oosten, Jaap</creator><general>Duke University Press</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>2006</creationdate><title>A General Form of Relative Recursion</title><author>van Oosten, Jaap</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c2598-b468785877cdaabba2d8ad608244009960703470f95909babcaf3de9e882347b3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2006</creationdate><topic>03B40</topic><topic>68N18</topic><topic>partial combinatory algebras</topic><topic>realizability</topic><topic>relative recursion</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>van Oosten, Jaap</creatorcontrib><collection>CrossRef</collection><jtitle>Notre Dame journal of formal logic</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>van Oosten, Jaap</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A General Form of Relative Recursion</atitle><jtitle>Notre Dame journal of formal logic</jtitle><date>2006</date><risdate>2006</risdate><volume>47</volume><issue>3</issue><spage>311</spage><epage>318</epage><pages>311-318</pages><issn>0029-4527</issn><eissn>1939-0726</eissn><abstract>The purpose of this note is to observe a generalization of the concept "computable in..." to arbitrary partial combinatory algebras. For every partial combinatory algebra (pca) A and every partial endofunction on A, a pca A[f] is constructed such that in A[f], the function f is representable by an element; a universal property of the construction is formulated in terms of Longley's 2-category of pcas and decidable applicative morphisms. It is proved that there is always a geometric inclusion from the realizability topos on A[f] into the one on A and that there is a meaningful preorder on the partial endofunctions on A which generalizes Turing reducibility.</abstract><pub>Duke University Press</pub><doi>10.1305/ndjfl/1163775438</doi><tpages>8</tpages><oa>free_for_read</oa></addata></record>
fulltext fulltext
identifier ISSN: 0029-4527
ispartof Notre Dame journal of formal logic, 2006, Vol.47 (3), p.311-318
issn 0029-4527
1939-0726
language eng
recordid cdi_projecteuclid_primary_oai_CULeuclid_euclid_ndjfl_1163775438
source EZB-FREE-00999 freely available EZB journals; Project Euclid Complete
subjects 03B40
68N18
partial combinatory algebras
realizability
relative recursion
title A General Form of Relative Recursion
url https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-02T19%3A05%3A42IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-crossref_proje&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=A%20General%20Form%20of%20Relative%20Recursion&rft.jtitle=Notre%20Dame%20journal%20of%20formal%20logic&rft.au=van%20Oosten,%20Jaap&rft.date=2006&rft.volume=47&rft.issue=3&rft.spage=311&rft.epage=318&rft.pages=311-318&rft.issn=0029-4527&rft.eissn=1939-0726&rft_id=info:doi/10.1305/ndjfl/1163775438&rft_dat=%3Ccrossref_proje%3E10_1305_ndjfl_1163775438%3C/crossref_proje%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_id=info:pmid/&rfr_iscdi=true