{"id":"https://openalex.org/W2170105957","doi":"https://doi.org/10.2307/2273126","title":"On bar recursion of types 0 and 1","display_name":"On bar recursion of types 0 and 1","publication_year":1979,"publication_date":"1979-09-01","ids":{"openalex":"https://openalex.org/W2170105957","doi":"https://doi.org/10.2307/2273126","mag":"2170105957"},"language":"en","primary_location":{"id":"doi:10.2307/2273126","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273126","pdf_url":null,"source":{"id":"https://openalex.org/S177675236","display_name":"Journal of Symbolic Logic","issn_l":"0022-4812","issn":["0022-4812","1943-5886"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Symbolic Logic","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://epub.ub.uni-muenchen.de/4278/","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5112382767","display_name":"Helmut Schwichtenberg","orcid":null},"institutions":[{"id":"https://openalex.org/I8204097","display_name":"Ludwig-Maximilians-Universit\u00e4t M\u00fcnchen","ror":"https://ror.org/05591te55","country_code":"DE","type":"education","lineage":["https://openalex.org/I8204097"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Helmut Schwichtenberg","raw_affiliation_strings":["Mathematisches Institut der Universit\u00e4t, d-8000 M\u00fcnchen 2, Federal Republic of Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Mathematisches Institut der Universit\u00e4t, d-8000 M\u00fcnchen 2, Federal Republic of Germany","institution_ids":["https://openalex.org/I8204097"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5112382767"],"corresponding_institution_ids":["https://openalex.org/I8204097"],"apc_list":null,"apc_paid":null,"fwci":0.6226,"has_fulltext":true,"cited_by_count":27,"citation_normalized_percentile":{"value":0.69502588,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"44","issue":"3","first_page":"325","last_page":"329"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9987000226974487,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9987000226974487,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.9955000281333923,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11567","display_name":"semigroups and automata theory","score":0.9842000007629395,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/recursion","display_name":"Recursion (computer science)","score":0.8310008645057678},{"id":"https://openalex.org/keywords/double-recursion","display_name":"Double recursion","score":0.6609413027763367},{"id":"https://openalex.org/keywords/normalization","display_name":"Normalization (sociology)","score":0.48457467555999756},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4764014780521393},{"id":"https://openalex.org/keywords/reduction","display_name":"Reduction (mathematics)","score":0.4539704918861389},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4475109875202179},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.4413517713546753},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4098365306854248},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.36516815423965454},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.35492396354675293},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.32745933532714844},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.18918746709823608},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1394430696964264},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.12155866622924805}],"concepts":[{"id":"https://openalex.org/C168773036","wikidata":"https://www.wikidata.org/wiki/Q264164","display_name":"Recursion (computer science)","level":2,"score":0.8310008645057678},{"id":"https://openalex.org/C139503435","wikidata":"https://www.wikidata.org/wiki/Q5300083","display_name":"Double recursion","level":3,"score":0.6609413027763367},{"id":"https://openalex.org/C136886441","wikidata":"https://www.wikidata.org/wiki/Q926129","display_name":"Normalization (sociology)","level":2,"score":0.48457467555999756},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4764014780521393},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.4539704918861389},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4475109875202179},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.4413517713546753},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4098365306854248},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.36516815423965454},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.35492396354675293},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.32745933532714844},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.18918746709823608},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1394430696964264},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.12155866622924805},{"id":"https://openalex.org/C19165224","wikidata":"https://www.wikidata.org/wiki/Q23404","display_name":"Anthropology","level":1,"score":0.0},{"id":"https://openalex.org/C144024400","wikidata":"https://www.wikidata.org/wiki/Q21201","display_name":"Sociology","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.2307/2273126","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273126","pdf_url":null,"source":{"id":"https://openalex.org/S177675236","display_name":"Journal of Symbolic Logic","issn_l":"0022-4812","issn":["0022-4812","1943-5886"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Symbolic Logic","raw_type":"journal-article"},{"id":"pmh:oai:epub.ub.uni-muenchen.de:4278","is_oa":true,"landing_page_url":"http://nbn-resolving.de/urn:nbn:de:bvb:19-epub-4278-4","pdf_url":"https://epub.ub.uni-muenchen.de/4278/","source":{"id":"https://openalex.org/S4306401845","display_name":"Open access LMU (Ludwid Maxmilian's Universitat Munchen)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I8204097","host_organization_name":"Ludwig-Maximilians-Universit\u00e4t M\u00fcnchen","host_organization_lineage":["https://openalex.org/I8204097"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"The Journal of Symbolic Logic","raw_type":"doc-type:article"},{"id":"pmh:oai:CULeuclid:euclid.jsl/1183740429","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1183740429","pdf_url":null,"source":{"id":"https://openalex.org/S4306400787","display_name":"Project Euclid (Cornell University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"}],"best_oa_location":{"id":"pmh:oai:epub.ub.uni-muenchen.de:4278","is_oa":true,"landing_page_url":"http://nbn-resolving.de/urn:nbn:de:bvb:19-epub-4278-4","pdf_url":"https://epub.ub.uni-muenchen.de/4278/","source":{"id":"https://openalex.org/S4306401845","display_name":"Open access LMU (Ludwid Maxmilian's Universitat Munchen)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I8204097","host_organization_name":"Ludwig-Maximilians-Universit\u00e4t M\u00fcnchen","host_organization_lineage":["https://openalex.org/I8204097"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"The Journal of Symbolic Logic","raw_type":"doc-type:article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W2170105957.pdf"},"referenced_works_count":9,"referenced_works":["https://openalex.org/W567983113","https://openalex.org/W1566235828","https://openalex.org/W1608607427","https://openalex.org/W1793763201","https://openalex.org/W1864285868","https://openalex.org/W2147234020","https://openalex.org/W2150899916","https://openalex.org/W2734815139","https://openalex.org/W6741054553"],"related_works":["https://openalex.org/W1973821254","https://openalex.org/W4246540581","https://openalex.org/W4229643535","https://openalex.org/W2023101964","https://openalex.org/W2021997287","https://openalex.org/W1582159526","https://openalex.org/W1578002326","https://openalex.org/W2896379815","https://openalex.org/W2006836388","https://openalex.org/W1509114718"],"abstract_inverted_index":{"For":[0],"general":[1],"information":[2],"on":[3,65],"bar":[4,52,165,199,208,226,268],"recursion":[5,53,166,200,209,227,269],"the":[6,10,31,47,66,80,84,129,152,155,189,196,205,223,251,265],"reader":[7],"should":[8],"consult":[9],"papers":[11],"of":[12,33,41,51,54,61,68,79,134,141,154,164,195,198,201,207,210,225,228,231,242,264,267,270],"Spector":[13,42],"[8],":[14],"where":[15],"it":[16],"was":[17],"introduced,":[18],"Howard":[19,193,257,280],"[2]":[20,194],"and":[21,57,92,113,255,281,288],"Tait":[22,74,135],"[11].":[23],"In":[24],"this":[25,106,128],"note":[26],"we":[27,115,157],"shall":[28],"prove":[29],"that":[30,217],"terms":[32,91],"G\u00f6del's":[34],"theory":[35],"T":[36,218],"(in":[37,178],"its":[38],"extensional":[39],"version":[40],"[8])":[43],"are":[44,82,97,174],"closed":[45,221],"under":[46,222],"rule":[48,206,224],"BR":[49,117,243],"0,1":[50,118,244],"types":[55],"0":[56,125],"1.":[58],"Our":[59],"method":[60],"proof":[62,81],"is":[63,108,219],"based":[64],"notion":[67],"an":[69,261],"infinite":[70,90],"term":[71],"introduced":[72,167],"by":[73,132,168,192,250],"[9].":[75],"The":[76],"main":[77],"tools":[78],"(i)":[83,112],"normalization":[85],"theorem":[86],"for":[87,101,284],"(notations":[88],"for)":[89],"(ii)":[93,114],"valuation":[94],"functionals.":[95],"Both":[96],"elaborated":[98],"in":[99,170],"[6];":[100],"brevity":[102],"some":[103],"familiarity":[104],"with":[105,121,247],"paper":[107,156],"assumed":[109],"here.":[110],"Using":[111],"reduce":[116],"to":[119,145,176,181,204,235,245,256,276],"\u03be-recursion":[120,146,246],"\u03be":[122,143,248],"<":[123],"\u03b5":[124],".":[126],"From":[127],"result":[130],"follows":[131],"work":[133],"[10],":[136],"who":[137,183,238,259],"gave":[138,239,260],"a":[139,148,159,229,240],"reduction":[140,190,241],"2":[142],"-recursion":[144],"at":[147],"higher":[149],"type.":[150],"At":[151],"end":[153],"discuss":[158],"perhaps":[160],"more":[161],"natural":[162],"variant":[163],"Kreisel":[169,177,283],"[4].":[171],"Related":[172],"results":[173,185],"due":[175],"his":[179],"appendix":[180],"[8]),":[182],"obtains":[184],"which":[186],"imply,":[187],"using":[188],"given":[191],"constant":[197,266],"type":[202,211,230,271],"\u03c4":[203],"(0":[212],"\u2192":[213,215],"\u03c4)":[214],"\u03c4,":[216],"not":[220],"level":[232],"\u2265":[233],"2,":[234],"Diller":[236],"[1],":[237],"bounded":[249],"least":[252],"\u03c9-critical":[253],"number,":[254],"[3],":[258],"ordinal":[262],"analysis":[263],"0.":[272],"I":[273],"am":[274],"grateful":[275],"H.":[277],"Barendregt,":[278],"W.":[279],"G.":[282],"many":[285],"useful":[286],"comments":[287],"discussions.":[289]},"counts_by_year":[{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":5},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":1},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}