{"id":"https://openalex.org/W2964235990","doi":"https://doi.org/10.1017/jsl.2017.33","title":"A DIRECT PROOF OF SCHWICHTENBERG\u2019S BAR RECURSION CLOSURE THEOREM","display_name":"A DIRECT PROOF OF SCHWICHTENBERG\u2019S BAR RECURSION CLOSURE THEOREM","publication_year":2018,"publication_date":"2018-02-12","ids":{"openalex":"https://openalex.org/W2964235990","doi":"https://doi.org/10.1017/jsl.2017.33","mag":"2964235990"},"language":"en","primary_location":{"is_oa":false,"landing_page_url":"https://doi.org/10.1017/jsl.2017.33","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_indexed_in_scopus":true,"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":null,"is_accepted":false,"is_published":false},"type":"article","type_crossref":"journal-article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://boris.unibe.ch/112774/1/os17.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5084236771","display_name":"Paulo Oliva","orcid":"https://orcid.org/0000-0002-0492-4855"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"PAULO OLIVA","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5088949609","display_name":"Silvia Steila","orcid":"https://orcid.org/0000-0003-0394-1204"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"SILVIA STEILA","raw_affiliation_strings":[],"affiliations":[]}],"institution_assertions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.605,"has_fulltext":true,"fulltext_origin":"pdf","cited_by_count":10,"citation_normalized_percentile":{"value":0.733805,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":85,"max":86},"biblio":{"volume":"83","issue":"1","first_page":"70","last_page":"83"},"is_retracted":false,"is_paratext":false,"primary_topic":{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.9989,"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"}},"topics":[{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.9989,"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":0.9967,"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/T10304","display_name":"Geometric and Algebraic Topology","score":0.9905,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/closure","display_name":"Closure (psychology)","score":0.66498744},{"id":"https://openalex.org/keywords/bar","display_name":"Bar (unit)","score":0.5443967},{"id":"https://openalex.org/keywords/sigma","display_name":"Sigma","score":0.509027}],"concepts":[{"id":"https://openalex.org/C168773036","wikidata":"https://www.wikidata.org/wiki/Q264164","display_name":"Recursion (computer science)","level":2,"score":0.82194877},{"id":"https://openalex.org/C146834321","wikidata":"https://www.wikidata.org/wiki/Q2979672","display_name":"Closure (psychology)","level":2,"score":0.66498744},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.55699927},{"id":"https://openalex.org/C188721877","wikidata":"https://www.wikidata.org/wiki/Q103510","display_name":"Bar (unit)","level":2,"score":0.5443967},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.51096785},{"id":"https://openalex.org/C2778049214","wikidata":"https://www.wikidata.org/wiki/Q7512234","display_name":"Sigma","level":2,"score":0.509027},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.48138645},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.45084858},{"id":"https://openalex.org/C48103436","wikidata":"https://www.wikidata.org/wiki/Q599031","display_name":"State (computer science)","level":2,"score":0.42648557},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.37791255},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1607323},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.13806799},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C153294291","wikidata":"https://www.wikidata.org/wiki/Q25261","display_name":"Meteorology","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C34447519","wikidata":"https://www.wikidata.org/wiki/Q179522","display_name":"Market economy","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"is_oa":false,"landing_page_url":"https://doi.org/10.1017/jsl.2017.33","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_indexed_in_scopus":true,"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":null,"is_accepted":false,"is_published":false},{"is_oa":true,"landing_page_url":"https://boris.unibe.ch/112774/","pdf_url":"https://boris.unibe.ch/112774/1/os17.pdf","source":{"id":"https://openalex.org/S4306401747","display_name":"BORIS (University Library Bern)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_indexed_in_scopus":false,"is_core":false,"host_organization":"https://openalex.org/I4210103840","host_organization_name":"Nakuru County Council","host_organization_lineage":["https://openalex.org/I4210103840"],"host_organization_lineage_names":["Nakuru County Council"],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"publishedVersion","is_accepted":true,"is_published":true},{"is_oa":true,"landing_page_url":"http://qmro.qmul.ac.uk/xmlui/handle/123456789/22422","pdf_url":"https://qmro.qmul.ac.uk/xmlui/bitstream/123456789/22422/1/Borges%20A%20direct%20proof%20of%20Schwichtenberg%27s%202017%20Accepted.pdf","source":{"id":"https://openalex.org/S4306400530","display_name":"Queen Mary Research Online (Queen Mary University of London)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_indexed_in_scopus":false,"is_core":false,"host_organization":"https://openalex.org/I166337079","host_organization_name":"Queen Mary University of London","host_organization_lineage":["https://openalex.org/I166337079"],"host_organization_lineage_names":["Queen Mary University of London"],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false},{"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1607.05237","pdf_url":"http://arxiv.org/pdf/1607.05237","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_indexed_in_scopus":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":["Cornell University"],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false}],"best_oa_location":{"is_oa":true,"landing_page_url":"https://boris.unibe.ch/112774/","pdf_url":"https://boris.unibe.ch/112774/1/os17.pdf","source":{"id":"https://openalex.org/S4306401747","display_name":"BORIS (University Library Bern)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_indexed_in_scopus":false,"is_core":false,"host_organization":"https://openalex.org/I4210103840","host_organization_name":"Nakuru County Council","host_organization_lineage":["https://openalex.org/I4210103840"],"host_organization_lineage_names":["Nakuru County Council"],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"publishedVersion","is_accepted":true,"is_published":true},"sustainable_development_goals":[],"grants":[],"datasets":[],"versions":[],"referenced_works_count":13,"referenced_works":["https://openalex.org/W1525064220","https://openalex.org/W1864285868","https://openalex.org/W2037300268","https://openalex.org/W2078456795","https://openalex.org/W2090521688","https://openalex.org/W2095689360","https://openalex.org/W2147234020","https://openalex.org/W2170105957","https://openalex.org/W2293708017","https://openalex.org/W2563354002","https://openalex.org/W2734815139","https://openalex.org/W4236347697","https://openalex.org/W4248689402"],"related_works":["https://openalex.org/W4391212601","https://openalex.org/W4294611394","https://openalex.org/W4281614748","https://openalex.org/W3202311233","https://openalex.org/W315394019","https://openalex.org/W30954630","https://openalex.org/W2765883196","https://openalex.org/W2377519369","https://openalex.org/W2034826746","https://openalex.org/W1976679120"],"abstract_inverted_index":{"Abstract":[0],"In":[1,109],"[12],":[2],"Schwichtenberg":[3],"showed":[4],"that":[5],"the":[6,30,35,45,71,101,153,160,175,193],"System":[7],"T":[8,98,104,177],"definable":[9,191],"functionals":[10],"are":[11,190],"closed":[12],"under":[13,152],"a":[14,64,129,142,168],"rule-like":[15],"version":[16],"Spector\u2019s":[17,39],"bar":[18,40,47,147],"recursion":[19,48,75,83],"of":[20,38,49,155],"lowest":[21],"type":[22,50],"levels":[23,51],"0":[24,52],"and":[25,53,70,81],"1.":[26],"More":[27],"precisely,":[28],"if":[29,171],"functional":[31],"Y":[32,172,189],"which":[33,124],"controls":[34],"stopping":[36],"condition":[37],"recursor":[41],"is":[42,55,173],"T-definable,":[43],"then":[44,179],"corresponding":[46,102],"1":[54],"already":[56],"T-definable.":[57],"Schwichtenberg\u2019s":[58,156],"original":[59],"proof,":[60],"however,":[61],"relies":[62],"on":[63,94,120],"detour":[65,88],"through":[66,136],"Tait\u2019s":[67],"infinitary":[68],"terms":[69,180],"correspondence":[72],"between":[73],"ordinal":[74],"for":[76,145,186],"$\\alpha":[77],"<":[78],"{\\varepsilon":[79],"_0}$":[80],"primitive":[82],"over":[84],"finite":[85],"types.":[86],"This":[87],"makes":[89],"it":[90],"hard":[91],"to":[92],"calculate":[93],"given":[95],"concrete":[96],"system":[97,103],"input,":[99],"what":[100],"output":[105],"would":[106],"look":[107],"like.":[108],"this":[110,140,187],"paper":[111],"we":[112,125,163],"present":[113],"an":[114,121,137],"alternative":[115],"(more":[116],"direct)":[117],"proof":[118],"based":[119],"explicit":[122,161],"construction":[123,162],"prove":[126],"correct":[127],"via":[128],"suitably":[130],"defined":[131],"logical":[132],"relation.":[133],"We":[134],"show":[135],"example":[138],"how":[139],"gives":[141],"straightforward":[143],"mechanism":[144],"converting":[146],"recursive":[148],"definitions":[149],"into":[150],"T-definitions":[151],"conditions":[154],"theorem.":[157],"Finally,":[158],"with":[159],"can":[164],"also":[165],"easily":[166],"state":[167],"sharper":[169],"result:":[170],"in":[174,192],"fragment":[176,194],"i":[178],"built":[181],"from":[182],"$BR^{\\mathbb{N},\\sigma":[183],"}":[184],"$":[185],"particular":[188],"${T_{i":[195],"+":[196,202],"{\\rm{max}}\\left\\{":[197],"{1,{\\rm{level}}\\left(":[198],"\\sigma":[199],"\\right)}":[200],"\\right\\}":[201],"2}}$":[203],".":[204]},"cited_by_api_url":"https://api.openalex.org/works?filter=cites:W2964235990","counts_by_year":[{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":7},{"year":2018,"cited_by_count":1}],"updated_date":"2025-01-20T16:37:39.523042","created_date":"2019-07-30"}