{"id":"https://openalex.org/W2113444807","doi":"https://doi.org/10.2307/2274181","title":"Partial degrees and the density problem. Part 2: The enumeration degrees of the <i>\u03a3</i><sub>2</sub> sets are dense","display_name":"Partial degrees and the density problem. Part 2: The enumeration degrees of the <i>\u03a3</i><sub>2</sub> sets are dense","publication_year":1984,"publication_date":"1984-06-01","ids":{"openalex":"https://openalex.org/W2113444807","doi":"https://doi.org/10.2307/2274181","mag":"2113444807"},"language":"en","primary_location":{"id":"doi:10.2307/2274181","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2274181","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, England","(University of Leeds"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Leeds, Leeds, England","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":0.5371,"has_fulltext":false,"cited_by_count":71,"citation_normalized_percentile":{"value":0.67761256,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"49","issue":"2","first_page":"503","last_page":"513"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9994999766349792,"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":0.9994999766349792,"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":0.9927999973297119,"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":0.9900000095367432,"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/enumeration","display_name":"Enumeration","score":0.8708160519599915},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5946418046951294},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5778534412384033},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.5438573360443115},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.48696932196617126},{"id":"https://openalex.org/keywords/partial-function","display_name":"Partial function","score":0.47961413860321045},{"id":"https://openalex.org/keywords/recursively-enumerable-language","display_name":"Recursively enumerable language","score":0.47350579500198364},{"id":"https://openalex.org/keywords/notation","display_name":"Notation","score":0.4553225338459015},{"id":"https://openalex.org/keywords/computability-theory","display_name":"Computability theory","score":0.44336575269699097},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.14674407243728638}],"concepts":[{"id":"https://openalex.org/C156340839","wikidata":"https://www.wikidata.org/wiki/Q2704791","display_name":"Enumeration","level":2,"score":0.8708160519599915},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5946418046951294},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5778534412384033},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.5438573360443115},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.48696932196617126},{"id":"https://openalex.org/C182359184","wikidata":"https://www.wikidata.org/wiki/Q1756942","display_name":"Partial function","level":2,"score":0.47961413860321045},{"id":"https://openalex.org/C203702658","wikidata":"https://www.wikidata.org/wiki/Q1073063","display_name":"Recursively enumerable language","level":2,"score":0.47350579500198364},{"id":"https://openalex.org/C45357846","wikidata":"https://www.wikidata.org/wiki/Q2001982","display_name":"Notation","level":2,"score":0.4553225338459015},{"id":"https://openalex.org/C111142201","wikidata":"https://www.wikidata.org/wiki/Q818930","display_name":"Computability theory","level":2,"score":0.44336575269699097},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.14674407243728638},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.2307/2274181","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2274181","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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":5,"referenced_works":["https://openalex.org/W1483777631","https://openalex.org/W2110883968","https://openalex.org/W2478803253","https://openalex.org/W2530006810","https://openalex.org/W2905135108"],"related_works":["https://openalex.org/W2158059264","https://openalex.org/W2001962035","https://openalex.org/W1975592766","https://openalex.org/W2089947806","https://openalex.org/W95138543","https://openalex.org/W2070295070","https://openalex.org/W4242488152","https://openalex.org/W2186864846","https://openalex.org/W2185077833","https://openalex.org/W2012561888"],"abstract_inverted_index":{"As":[0],"in":[1,35,158,202],"Rogers":[2],"[3],":[3],"we":[4,73],"treat":[5],"the":[6,13,18,27,53,60,64,76,85,92,97,112,121,153,164,167,176,185,239],"partial":[7,19,42,54,61,67],"degrees":[8,15,55,62,87],"as":[9,201,236],"notational":[10],"variants":[11],"of":[12,21,30,46,63,102,133,163,214,241,261],"enumeration":[14,28,86,188],"(that":[16,58],"is,":[17,59],"degree":[20,29],"a":[22,159],"function":[23],"is":[24,49,173,223],"identified":[25],"with":[26,227],"its":[31],"graph).":[32],"We":[33,229],"showed":[34],"[1]":[36],"that":[37,52,75,89],"there":[38,222],"are":[39,69,104,129,198,210],"no":[40],"minimal":[41],"degrees.":[43,99,125],"The":[44,100],"purpose":[45],"this":[47,72],"paper":[48],"to":[50,88,107,110,131,237,263],"show":[51],"below":[56],"0\u2032":[57],"\u03a3":[65,77,211],"2":[66,78,212],"functions)":[68],"dense.":[70],"From":[71],"see":[74],"sets":[79,95,213],"play":[80],"an":[81,224,231],"analagous":[82],"role":[83],"within":[84,96],"played":[90],"by":[91],"recursively":[93,122],"enumerable":[94,123],"Turing":[98,124],"techniques,":[101],"course,":[103],"very":[105],"different":[106],"those":[108,132],"required":[109],"prove":[111],"Sacks":[113],"Density":[114],"Theorem":[115,204],"(see":[116],"[4,":[117],"p.":[118],"20])":[119],"for":[120,152,179,267,269],"Notation":[126],"and":[127,184,208,216,289],"terminology":[128],"similar":[130],"[1].":[134,203],"In":[135],"particular,":[136],"W":[137,192],"e":[138,148,154,186,193,219,232],",":[139,142,175,272],"D":[140],"x":[141,174],"\u3008":[143],"m,":[144,181],"n":[145,182],"\u3009,":[146],"\u03c8":[147],"are,":[149],"respectively,":[150,266],"notations":[151],"th":[155,187],"r.e.":[156,165],"set":[157,169],"given":[160],"standard":[161],"listing":[162],"sets,":[166],"finite":[168],"whose":[170],"canonical":[171],"index":[172],"recursive":[177,259],"code":[178],"(":[180],")":[183,288],"operator":[189],"(derived":[190],"from":[191],").":[194,304],"Recursive":[195],"approximations":[196,262],"etc.":[197],"also":[199],"defined":[200],"1.":[205],"If":[206],"B":[207,217,245,275,286],"C":[209,252,265,292,302],"numbers,":[215],"\u2270":[218],"C,":[220],"then":[221],"e-operator":[225],"\u0398":[226,234],"Proof.":[228],"enumerate":[230],"-operator":[233],"so":[235],"satisfy":[238],"list":[240],"conditions:":[242],"Let":[243],"{":[244,251],"s":[246,248,253,255,278,280,282,287,295,298,303],"\u2223":[247,254],"\u2265":[249,256,281,297],"0},":[250],"0}":[257],"be":[258],"sequences":[260],"B,":[264],"which,":[268],"each":[270],"\u0445":[271,273,284,290,300],"\u2208":[274,285,291,301],"\u21d4":[276,293],"(\u2203":[277,294],"*)(\u2200":[279],"*)(":[283,299],"*)(\u2200s":[296]},"counts_by_year":[{"year":2025,"cited_by_count":3},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":1},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":5},{"year":2015,"cited_by_count":4},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":5}],"updated_date":"2026-08-12T21:12:35.861297","created_date":"2025-10-10T00:00:00"}