{"id":"https://openalex.org/W2110883968","doi":"https://doi.org/10.2307/2273104","title":"Partial degrees and the density problem","display_name":"Partial degrees and the density problem","publication_year":1982,"publication_date":"1982-12-01","ids":{"openalex":"https://openalex.org/W2110883968","doi":"https://doi.org/10.2307/2273104","mag":"2110883968"},"language":"en","primary_location":{"id":"doi:10.2307/2273104","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273104","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":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5076862599","display_name":"S. Barry Cooper","orcid":"https://orcid.org/0000-0001-5103-0400"},"institutions":[{"id":"https://openalex.org/I130828816","display_name":"University of Leeds","ror":"https://ror.org/024mrxd33","country_code":"GB","type":"education","lineage":["https://openalex.org/I130828816"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"S. B. Cooper","raw_affiliation_strings":["University of Leeds, Leeds, Great Britain","(University of Leeds"],"affiliations":[{"raw_affiliation_string":"University of Leeds, Leeds, Great Britain","institution_ids":["https://openalex.org/I130828816"]},{"raw_affiliation_string":"(University of Leeds","institution_ids":["https://openalex.org/I130828816"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5076862599"],"corresponding_institution_ids":["https://openalex.org/I130828816"],"apc_list":null,"apc_paid":null,"fwci":1.6491,"has_fulltext":false,"cited_by_count":56,"citation_normalized_percentile":{"value":0.85210824,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"47","issue":"4","first_page":"854","last_page":"859"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9998999834060669,"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":0.9998999834060669,"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/T11567","display_name":"semigroups and automata theory","score":0.9940999746322632,"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/T11151","display_name":"Advanced Topology and Set Theory","score":0.9825999736785889,"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/partial-function","display_name":"Partial function","score":0.6880954504013062},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6391529440879822},{"id":"https://openalex.org/keywords/computability-theory","display_name":"Computability theory","score":0.6190093755722046},{"id":"https://openalex.org/keywords/partial-evaluation","display_name":"Partial evaluation","score":0.5459988117218018},{"id":"https://openalex.org/keywords/partial-derivative","display_name":"Partial derivative","score":0.4664705693721771},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.46583500504493713},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.41939017176628113},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.36232250928878784},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3222130835056305},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.15776219964027405},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.12734413146972656},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.12179186940193176}],"concepts":[{"id":"https://openalex.org/C182359184","wikidata":"https://www.wikidata.org/wiki/Q1756942","display_name":"Partial function","level":2,"score":0.6880954504013062},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6391529440879822},{"id":"https://openalex.org/C111142201","wikidata":"https://www.wikidata.org/wiki/Q818930","display_name":"Computability theory","level":2,"score":0.6190093755722046},{"id":"https://openalex.org/C195701839","wikidata":"https://www.wikidata.org/wiki/Q4008398","display_name":"Partial evaluation","level":2,"score":0.5459988117218018},{"id":"https://openalex.org/C53846429","wikidata":"https://www.wikidata.org/wiki/Q186475","display_name":"Partial derivative","level":2,"score":0.4664705693721771},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.46583500504493713},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.41939017176628113},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.36232250928878784},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3222130835056305},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.15776219964027405},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.12734413146972656},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.12179186940193176},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.2307/2273104","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273104","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:CULeuclid:euclid.jsl/1183741144","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1183741144","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":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":7,"referenced_works":["https://openalex.org/W1483568252","https://openalex.org/W1547216710","https://openalex.org/W2055606729","https://openalex.org/W2069624526","https://openalex.org/W2087882664","https://openalex.org/W2530006810","https://openalex.org/W2991196623"],"related_works":["https://openalex.org/W561052599","https://openalex.org/W4242488152","https://openalex.org/W2159591963","https://openalex.org/W2184512187","https://openalex.org/W2162114838","https://openalex.org/W2116465509","https://openalex.org/W2110883968","https://openalex.org/W1572830206","https://openalex.org/W2128992690","https://openalex.org/W2363876014"],"abstract_inverted_index":{"A":[0],"notion":[1,116,132],"of":[2,41,55,73,75,79,82,117,133,161,168,171,180,186],"relative":[3],"reducibility":[4,12,134],"for":[5,150,195],"partial":[6,57,95,100,118,136,199],"functions,":[7,16],"which":[8,120],"coincides":[9],"with":[10,102],"Turing":[11,124],"on":[13],"the":[14,56,65,80,123,146,192,196],"total":[15,105],"was":[17,32,84,148],"first":[18],"given":[19],"by":[20],"S.C.":[21],"Kleene":[22],"in":[23,36,141,188],"Introduction":[24],"to":[25],"metamathematics":[26],"[4].":[27],"Following":[28],"Myhill":[29],"[7],":[30],"this":[31,76],"made":[33],"more":[34,63,197],"explicit":[35],"Hartley":[37],"Rogers,":[38],"Jr.,":[39],"Theory":[40],"recursive":[42],"functions":[43,137],"and":[44],"effective":[45],"computability":[46],"[8,":[47],"pp.":[48],"146,":[49],"279],":[50],"where":[51],"some":[52],"basic":[53],"properties":[54],"degrees":[58,67,81,96,101,125],"or":[59],"(equivalent,":[60],"but":[61],"notationally":[62],"convenient)":[64],"enumeration":[66],",":[68],"were":[69],"derived.":[70],"The":[71],"question":[72],"density":[74],"proper":[77,128],"extension":[78],"unsolvability":[83],"left":[85,190],"open,":[86],"although":[87],"Medvedev's":[88],"result":[89],"[6]":[90],"that":[91,142],"there":[92],"are":[93],"quasi-minimal":[94],"(that":[97],"is,":[98],"nonrecursive":[99,104],"no":[103],"predecessors)":[106],"is":[107],"proved.":[108],"In":[109],"1971,":[110],"Sasso":[111,176],"[9]":[112],"introduced":[113],"a":[114,127,155,166,172],"finer":[115],"degree,":[119],"also":[121],"contained":[122],"as":[126,162],"substructure":[129],"(intuitively,":[130],"Sasso's":[131],"between":[135],"differed":[138],"from":[139],"Rogers'":[140],"computations":[143],"terminated":[144],"when":[145],"oracle":[147],"asked":[149],"an":[151],"undefined":[152],"value,":[153],"whereas":[154],"Rogers":[156],"computation":[157,174],"could":[158],"be":[159],"thought":[160],"proceeding":[163],"simultaneously":[164],"along":[165],"number":[167],"different":[169],"branches":[170],"\u2018consistent\u2019":[173],"tree\u2014cf.":[175],"[10]).":[177],"His":[178],"construction":[179],"minimal":[181],"\u2018partial":[182],"degrees\u2019":[183],"[11],":[184],"while":[185],"interest":[187],"itself,":[189],"open":[191],"analogous":[193],"problem":[194],"standard":[198],"degree":[200],"structure.":[201]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":3}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
