{"id":"https://openalex.org/W2077721565","doi":"https://doi.org/10.2307/2275099","title":"Working below a high recursively enumerable degree","display_name":"Working below a high recursively enumerable degree","publication_year":1993,"publication_date":"1993-09-01","ids":{"openalex":"https://openalex.org/W2077721565","doi":"https://doi.org/10.2307/2275099","mag":"2077721565"},"language":"en","primary_location":{"id":"doi:10.2307/2275099","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2275099","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/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, New York 14853, E-mail:","Department Of Mathematics, Cornell University, Ithaca, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Cornell University, Ithaca, New York 14853, E-mail:","institution_ids":["https://openalex.org/I205783295"]},{"raw_affiliation_string":"Department Of Mathematics, Cornell University, Ithaca, USA","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/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":"Theodore A. Slaman","raw_affiliation_strings":["Department of Mathematics, The University of Chicago, Chicago, Illinois 60637, E-mail:","Dept. of Mathematics, Univ. of Chicago, Chicago, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, The University of Chicago, Chicago, Illinois 60637, E-mail:","institution_ids":["https://openalex.org/I40347166"]},{"raw_affiliation_string":"Dept. of Mathematics, Univ. of Chicago, Chicago, USA","institution_ids":["https://openalex.org/I40347166"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":4.173,"has_fulltext":false,"cited_by_count":41,"citation_normalized_percentile":{"value":0.94439452,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"58","issue":"3","first_page":"824","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":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/T11151","display_name":"Advanced Topology and Set Theory","score":0.9951000213623047,"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"}},{"id":"https://openalex.org/T11567","display_name":"semigroups and automata theory","score":0.9932000041007996,"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/recursively-enumerable-language","display_name":"Recursively enumerable language","score":0.9554022550582886},{"id":"https://openalex.org/keywords/recursively-enumerable-set","display_name":"Recursively enumerable set","score":0.6510164737701416},{"id":"https://openalex.org/keywords/jump","display_name":"Jump","score":0.6347476243972778},{"id":"https://openalex.org/keywords/equivalence-relation","display_name":"Equivalence relation","score":0.6016305685043335},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5213904976844788},{"id":"https://openalex.org/keywords/turing","display_name":"Turing","score":0.5123394727706909},{"id":"https://openalex.org/keywords/equivalence","display_name":"Equivalence (formal languages)","score":0.48824480175971985},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4459974765777588},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.363675594329834},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2505567967891693},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.12410661578178406}],"concepts":[{"id":"https://openalex.org/C203702658","wikidata":"https://www.wikidata.org/wiki/Q1073063","display_name":"Recursively enumerable language","level":2,"score":0.9554022550582886},{"id":"https://openalex.org/C52703633","wikidata":"https://www.wikidata.org/wiki/Q676835","display_name":"Recursively enumerable set","level":3,"score":0.6510164737701416},{"id":"https://openalex.org/C2780695682","wikidata":"https://www.wikidata.org/wiki/Q4005959","display_name":"Jump","level":2,"score":0.6347476243972778},{"id":"https://openalex.org/C156103551","wikidata":"https://www.wikidata.org/wiki/Q130998","display_name":"Equivalence relation","level":2,"score":0.6016305685043335},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5213904976844788},{"id":"https://openalex.org/C9870796","wikidata":"https://www.wikidata.org/wiki/Q490481","display_name":"Turing","level":2,"score":0.5123394727706909},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.48824480175971985},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4459974765777588},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.363675594329834},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2505567967891693},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.12410661578178406},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.2307/2275099","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2275099","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/1183744300","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1183744300","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":[{"score":0.8299999833106995,"id":"https://metadata.un.org/sdg/14","display_name":"Life below water"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":21,"referenced_works":["https://openalex.org/W130517038","https://openalex.org/W243218583","https://openalex.org/W1592162066","https://openalex.org/W1964249676","https://openalex.org/W1981962996","https://openalex.org/W2029985505","https://openalex.org/W2033957959","https://openalex.org/W2038443311","https://openalex.org/W2042742137","https://openalex.org/W2045263624","https://openalex.org/W2056414301","https://openalex.org/W2082637799","https://openalex.org/W2083623332","https://openalex.org/W2083737626","https://openalex.org/W2101746971","https://openalex.org/W2167819410","https://openalex.org/W2167930025","https://openalex.org/W2500837594","https://openalex.org/W4232277851","https://openalex.org/W4240721023","https://openalex.org/W6658100611"],"related_works":["https://openalex.org/W2037833610","https://openalex.org/W1972481159","https://openalex.org/W1973052363","https://openalex.org/W2094468224","https://openalex.org/W1980641386","https://openalex.org/W2132512348","https://openalex.org/W798745779","https://openalex.org/W1555094413","https://openalex.org/W2181593437","https://openalex.org/W2117594216"],"abstract_inverted_index":{"In":[0,37,278],"recent":[1],"work,":[2],"Cooper":[3,46],"[3,":[4],"1990]":[5,218],"has":[6],"extended":[7],"results":[8,392],"of":[9,34,40,54,78,88,119,145,149,180,231,239,252,255,301,320,323,338,346,418],"Jockusch":[10],"and":[11,31,63,137,174,215,219,245,295,326,412],"Shore":[12,214,262],"[6,":[13],"1984]":[14],"to":[15,114,130,168,177,199,204,221,227,264,286,312,388,403],"show":[16,169],"that":[17,57,70,162,170,268,333],"the":[18,24,28,32,55,74,80,86,94,109,117,146,150,171,178,190,201,205,224,232,235,253,269,299,321,329,336,347,394,419],"Turing":[19,29,35,151,191,324],"jump":[20,87,110,120,172,181,247,257,339,348],"is":[21,73,90,112,127,153,159,276,410,423],"definable":[22,133,184,317,411,424],"in":[23,61,185,259,290,318,425],"structure":[25],"given":[26],"by":[27,52,92],"degrees":[30,56,81,97,100,152,296,332],"ordering":[33,322],"reducibility.":[36],"his":[38,163,194],"definition":[39],"x":[41,44,62,65,71,89,122,136,271],"\u2032":[42,72,123,126],"from":[43],",":[45,200,260],"identifies":[47],"an":[48,229,250],"order-theoretic":[49],"property":[50,417],"shared":[51],"all":[53,79],"are":[58,102,183],"recursively":[59,95,104,206,241,306,330,395,407,420],"enumerable":[60,96,207,242,331,396,408,421],"above":[64],".":[66,139,426],"He":[67],"then":[68,382],"shows":[69,267],"least":[75],"upper":[76],"bound":[77],"with":[82,98],"this":[83,279,302],"property.":[84],"Thus,":[85],"identified":[91],"comparing":[93],"other":[99,389],"which":[101],"not":[103,197],"enumerable.":[105],"Of":[106],"course,":[107],"once":[108],"operator":[111,173],"known":[113,129],"be":[115,131,166,305,383,400],"definable,":[116],"relation":[118,134,179,270,337],"equivalence":[121,182,258,340],"=":[124,273,352,358,366,372],"y":[125,138,274],"also":[128],"a":[132,240,377,386],"on":[135],"If":[140],"we":[141,281],"consider":[142],"how":[143],"much":[144],"global":[147],"theory":[148],"sufficient":[154],"for":[155,298,393],"Cooper's":[156],"methods,":[157],"it":[158],"immediately":[160],"clear":[161],"methods":[164,195],"can":[165],"implemented":[167],"its":[175,246],"weakening":[176],"any":[186],"ideal":[187],"closed":[188],"under":[189],"jump.":[192],"However,":[193],"do":[196,213],"localize":[198],"degrees,":[202],"or":[203,341,344,363],"degrees.":[208,397],"This":[209],"paper":[210,303],"fits,":[211],"as":[212,385],"Slaman":[216],"[16,":[217],"[17,":[220],"appear],":[222],"within":[223],"general":[225,390],"project":[226],"develop":[228],"understanding":[230],"relationship":[233],"between":[234],"local":[236],"degree-theoretic":[237,315],"properties":[238,316],"set":[243],"A":[244],"class.":[248],"For":[249],"analysis":[251],"possibility":[254],"defining":[256],"consult":[261],"[15,":[263],"appear]":[265],"who":[266],"(3)":[272,275],"definable.":[277],"paper,":[280],"will":[282,304],"restrict":[283],"our":[284],"attention":[285],"definitions":[287],"expressed":[288],"completely":[289],"\u211b":[291],"(Note:":[292],"All":[293],"sets":[294,422],"discussed":[297],"remainder":[300],"enumerable.)":[307],"Ultimately,":[308],"one":[309,343],"would":[310,334,399],"like":[311],"find":[313],"some":[314],"terms":[319],"reducibility":[325],"quantifiers":[327],"over":[328],"define":[335,342],"more":[345],"classes":[349],"H":[350],"n":[351,357,360,365,371,374],"{":[353,367],"w":[354,356,368,370],"\u2223":[355,369],"0":[359,373],"+1":[361],"}":[362],"L":[364],"}.":[375],"Such":[376],"result":[378],"could":[379],"very":[380],"likely":[381],"used":[384],"springboard":[387],"definability":[391],"It":[398],"especially":[401],"interesting":[402],"know":[404],"whether":[405,413],"every":[406,414],"degree":[409],"arithmetical":[415],"degree-invariant":[416]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2012,"cited_by_count":3}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
