{"id":"https://openalex.org/W2000784135","doi":"https://doi.org/10.2307/421171","title":"Definability in the Recursively Enumerable Degrees","display_name":"Definability in the Recursively Enumerable Degrees","publication_year":1996,"publication_date":"1996-12-01","ids":{"openalex":"https://openalex.org/W2000784135","doi":"https://doi.org/10.2307/421171","mag":"2000784135"},"language":"en","primary_location":{"id":"doi:10.2307/421171","is_oa":false,"landing_page_url":"https://doi.org/10.2307/421171","pdf_url":null,"source":{"id":"https://openalex.org/S130801046","display_name":"Bulletin of Symbolic Logic","issn_l":"1079-8986","issn":["1079-8986","1943-5894"],"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":"Bulletin 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/A5047690936","display_name":"Andr\u00e9 Nies","orcid":"https://orcid.org/0000-0002-0666-5180"},"institutions":[{"id":"https://openalex.org/I40347166","display_name":"University of Chicago","ror":"https://ror.org/024mw5h28","country_code":"US","type":"education","lineage":["https://openalex.org/I40347166"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Andr\u00e9 Nies","raw_affiliation_strings":["Department of Mathematics, University of Chicago, Chicago, IL 60637, USA.E-mail:","University-of Chicago"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Chicago, Chicago, IL 60637, USA.E-mail:","institution_ids":["https://openalex.org/I40347166"]},{"raw_affiliation_string":"University-of Chicago","institution_ids":["https://openalex.org/I40347166"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5041739806","display_name":"Richard A. Shore","orcid":"https://orcid.org/0000-0003-0381-5259"},"institutions":[{"id":"https://openalex.org/I205783295","display_name":"Cornell University","ror":"https://ror.org/05bnh6r87","country_code":"US","type":"education","lineage":["https://openalex.org/I205783295"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Richard A. Shore","raw_affiliation_strings":["Department of Mathematics, Cornell University, Ithaca, NY 14853, USA. E-mail:","Cornell University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Cornell University, Ithaca, NY 14853, USA. E-mail:","institution_ids":["https://openalex.org/I205783295"]},{"raw_affiliation_string":"Cornell University","institution_ids":["https://openalex.org/I205783295"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5056375116","display_name":"Theodore A. Slaman","orcid":"https://orcid.org/0000-0002-1719-2630"},"institutions":[{"id":"https://openalex.org/I2803209242","display_name":"University of California System","ror":"https://ror.org/00pjdza24","country_code":"US","type":"education","lineage":["https://openalex.org/I2803209242"]},{"id":"https://openalex.org/I95457486","display_name":"University of California, Berkeley","ror":"https://ror.org/01an7q238","country_code":"US","type":"education","lineage":["https://openalex.org/I95457486"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Theodore A. Slaman","raw_affiliation_strings":["Department of Mathematics, University of California, Berkeley, CA 94720, USA. E-mail:","University of California"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of California, Berkeley, CA 94720, USA. E-mail:","institution_ids":["https://openalex.org/I95457486"]},{"raw_affiliation_string":"University of California","institution_ids":["https://openalex.org/I2803209242"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.5168,"has_fulltext":false,"cited_by_count":16,"citation_normalized_percentile":{"value":0.63775418,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"2","issue":"4","first_page":"392","last_page":"404"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":1.0,"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":1.0,"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/T13720","display_name":"Benford\u2019s Law and Fraud Detection","score":0.9830999970436096,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11567","display_name":"semigroups and automata theory","score":0.9790999889373779,"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/mathematics","display_name":"Mathematics","score":0.7059154510498047},{"id":"https://openalex.org/keywords/computability-theory","display_name":"Computability theory","score":0.7020562887191772},{"id":"https://openalex.org/keywords/computable-function","display_name":"Computable function","score":0.6392888426780701},{"id":"https://openalex.org/keywords/halting-problem","display_name":"Halting problem","score":0.6213589310646057},{"id":"https://openalex.org/keywords/recursively-enumerable-language","display_name":"Recursively enumerable language","score":0.62018883228302},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5780956149101257},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5542623996734619},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.5101149678230286},{"id":"https://openalex.org/keywords/recursively-enumerable-set","display_name":"Recursively enumerable set","score":0.50101637840271},{"id":"https://openalex.org/keywords/equivalence-relation","display_name":"Equivalence relation","score":0.44576820731163025},{"id":"https://openalex.org/keywords/reverse-mathematics","display_name":"Reverse mathematics","score":0.44167229533195496},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.44113218784332275},{"id":"https://openalex.org/keywords/countable-set","display_name":"Countable set","score":0.42177626490592957},{"id":"https://openalex.org/keywords/turing-machine","display_name":"Turing machine","score":0.36239808797836304},{"id":"https://openalex.org/keywords/axiom","display_name":"Axiom","score":0.18781313300132751},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.09960433840751648},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.09517449140548706}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7059154510498047},{"id":"https://openalex.org/C111142201","wikidata":"https://www.wikidata.org/wiki/Q818930","display_name":"Computability theory","level":2,"score":0.7020562887191772},{"id":"https://openalex.org/C54271186","wikidata":"https://www.wikidata.org/wiki/Q1148456","display_name":"Computable function","level":2,"score":0.6392888426780701},{"id":"https://openalex.org/C137343475","wikidata":"https://www.wikidata.org/wiki/Q622849","display_name":"Halting problem","level":4,"score":0.6213589310646057},{"id":"https://openalex.org/C203702658","wikidata":"https://www.wikidata.org/wiki/Q1073063","display_name":"Recursively enumerable language","level":2,"score":0.62018883228302},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5780956149101257},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5542623996734619},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.5101149678230286},{"id":"https://openalex.org/C52703633","wikidata":"https://www.wikidata.org/wiki/Q676835","display_name":"Recursively enumerable set","level":3,"score":0.50101637840271},{"id":"https://openalex.org/C156103551","wikidata":"https://www.wikidata.org/wiki/Q130998","display_name":"Equivalence relation","level":2,"score":0.44576820731163025},{"id":"https://openalex.org/C127753061","wikidata":"https://www.wikidata.org/wiki/Q2005236","display_name":"Reverse mathematics","level":3,"score":0.44167229533195496},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.44113218784332275},{"id":"https://openalex.org/C110729354","wikidata":"https://www.wikidata.org/wiki/Q185478","display_name":"Countable set","level":2,"score":0.42177626490592957},{"id":"https://openalex.org/C29248071","wikidata":"https://www.wikidata.org/wiki/Q163310","display_name":"Turing machine","level":3,"score":0.36239808797836304},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.18781313300132751},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.09960433840751648},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.09517449140548706},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"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/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","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":5,"locations":[{"id":"doi:10.2307/421171","is_oa":false,"landing_page_url":"https://doi.org/10.2307/421171","pdf_url":null,"source":{"id":"https://openalex.org/S130801046","display_name":"Bulletin of Symbolic Logic","issn_l":"1079-8986","issn":["1079-8986","1943-5894"],"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":"Bulletin of Symbolic Logic","raw_type":"journal-article"},{"id":"pmh:oai:CULeuclid:euclid.bsl/1182353470","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.bsl/1182353470","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"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.27.2365","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.2365","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.ucla.edu/~asl/bsl/0204/0204-002.ps","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.44.6224","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.44.6224","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://math.cornell.edu/~shore/papers/ps/bsl10.ps","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.47.1875","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.47.1875","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.berkeley.edu/~slaman/papers/NiesShore.ps","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":32,"referenced_works":["https://openalex.org/W111095393","https://openalex.org/W1026721552","https://openalex.org/W1592162066","https://openalex.org/W1964249676","https://openalex.org/W1972685794","https://openalex.org/W1976647442","https://openalex.org/W1980888823","https://openalex.org/W1981509588","https://openalex.org/W1986222532","https://openalex.org/W2006726677","https://openalex.org/W2010076006","https://openalex.org/W2015965037","https://openalex.org/W2031067254","https://openalex.org/W2033957959","https://openalex.org/W2037416374","https://openalex.org/W2045033949","https://openalex.org/W2046907394","https://openalex.org/W2051601785","https://openalex.org/W2056414301","https://openalex.org/W2057381252","https://openalex.org/W2070390550","https://openalex.org/W2082637799","https://openalex.org/W2086881654","https://openalex.org/W2088993859","https://openalex.org/W2213438832","https://openalex.org/W2500837594","https://openalex.org/W2561618607","https://openalex.org/W4232277851","https://openalex.org/W4242335196","https://openalex.org/W4247092801","https://openalex.org/W6658470190","https://openalex.org/W6828899402"],"related_works":["https://openalex.org/W2001962035","https://openalex.org/W2089947806","https://openalex.org/W2010076006","https://openalex.org/W2113069034","https://openalex.org/W2070295070","https://openalex.org/W2467957161","https://openalex.org/W100398054","https://openalex.org/W95138543","https://openalex.org/W2185077833","https://openalex.org/W2000784135"],"abstract_inverted_index":{"\u00a71.":[0],"Introduction":[1],".":[2,49,148,160,216,245,276],"Natural":[3],"sets":[4,18,71,98],"that":[5,182,187,239,270],"can":[6,212],"be":[7,23,91,213],"enumerated":[8],"by":[9,110,167,175],"a":[10,116,229,237,244,260,271],"computable":[11,26,75,135],"function":[12],"(the":[13],"recursively":[14],"enumerable":[15],"or":[16,30,122,141,178,209],"r.e.":[17,46,70,93,97,227,232,258,266],")":[19,29],"always":[20],"seem":[21],"to":[22,37],"either":[24],"actually":[25],"(":[27],"recursive":[28,80],"of":[31,82,103,133,146,159,164,256],"the":[32,41,44,83,92,96,100,106,129,134,144,188],"same":[33,84],"complexity":[34],"(with":[35],"respect":[36],"Turing":[38,85,111],"computability)":[39],"as":[40,87,185],"Halting":[42],"Problem,":[43],"complete":[45],"set":[47],"K":[48,88,147],"The":[50,161],"obvious":[51],"question,":[52],"first":[53],"posed":[54],"in":[55,191],"Post":[56],"[1944]":[57],"and":[58,137,170],"since":[59],"then":[60,65,151],"called":[61],"Post's":[62,149,165],"Problem":[63],"is":[64,115,264],"just":[66],"whether":[67],"there":[68,154,230,263],"are":[69,73,155,231],"which":[72],"neither":[74,79],"nor":[76,81],"complete,":[77],"i.e.,":[78,95],"degree":[86,130,145,228,267],"?":[89],"Let":[90],"degrees,":[94],"modulo":[99],"equivalence":[101],"relation":[102],"equicomputable":[104],"with":[105],"partial":[107,117,207],"order":[108,118,179],"induced":[109],"computability.":[112],"This":[113],"structure":[114,189],"(indeed,":[119],"an":[120,265],"uppersemilattice":[121,211],"usl":[123],")with":[124],"least":[125],"element":[126,139],"0":[127,142],",":[128],"(equivalence":[131],"class)":[132],"sets,":[136],"greatest":[138],"1":[140],"\u2032,":[143],"problem":[150,166],"asks":[152],"if":[153],"any":[156],"other":[157],"elements":[158],"(positive)":[162],"solution":[163],"Friedberg":[168],"[1957]":[169],"Muchnik":[171,201],"[1956]":[172],"was":[173,190],"followed":[174],"various":[176],"algebraic":[177],"theoretic":[180],"results":[181],"were":[183],"interpreted":[184],"saying":[186],"some":[192],"way":[193],"well":[194],"behaved:":[195],"T":[196,217,246],"heorem":[197,218,247],"1.1":[198],"(Embedding":[199],"theorem;":[200],"[1958],":[202],"Sacks":[203],"[1963]).":[204],"Every":[205],"countable":[206],"ordering":[208],"even":[210],"embedded":[214],"into":[215],"1.2":[219],"(Sacks":[220,249],"Splitting":[221],"Theorem":[222,251],"[1963b]).":[223],"For":[224,253],"every":[225,254],"nonrecursive":[226,257],"degrees":[233,259],"b,":[234],"c":[235,242,268,273],"<":[236,261,272,274],"such":[238,269],"b":[240,262,275],"\u2228":[241],"=":[243],"1.3":[248],"Density":[250],"[1964]).":[252],"pair":[255]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2017,"cited_by_count":6},{"year":2016,"cited_by_count":1},{"year":2012,"cited_by_count":2}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}