{"id":"https://openalex.org/W2158841525","doi":"https://doi.org/10.1002/malq.200710005","title":"Primitive recursive real numbers","display_name":"Primitive recursive real numbers","publication_year":2007,"publication_date":"2007-07-26","ids":{"openalex":"https://openalex.org/W2158841525","doi":"https://doi.org/10.1002/malq.200710005","mag":"2158841525"},"language":"en","primary_location":{"id":"doi:10.1002/malq.200710005","is_oa":false,"landing_page_url":"https://doi.org/10.1002/malq.200710005","pdf_url":null,"source":{"id":"https://openalex.org/S172351290","display_name":"Mathematical logic quarterly","issn_l":"0044-3050","issn":["0044-3050","0942-5616","1521-3870"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Logic Quarterly","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/A5016292021","display_name":"Qingliang Chen","orcid":"https://orcid.org/0000-0001-5849-8268"},"institutions":[{"id":"https://openalex.org/I159948400","display_name":"Jinan University","ror":"https://ror.org/02xe5ns62","country_code":"CN","type":"education","lineage":["https://openalex.org/I159948400"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Qingliang Chen","raw_affiliation_strings":["Department of Computer Science, Jinan University, Guangzhou 510632, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science, Jinan University, Guangzhou 510632, China","institution_ids":["https://openalex.org/I159948400"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5037325282","display_name":"Kaile Su","orcid":"https://orcid.org/0000-0001-6741-9699"},"institutions":[{"id":"https://openalex.org/I11701301","display_name":"Griffith University","ror":"https://ror.org/02sc3r913","country_code":"AU","type":"education","lineage":["https://openalex.org/I11701301"]},{"id":"https://openalex.org/I20231570","display_name":"Peking University","ror":"https://ror.org/02v51f717","country_code":"CN","type":"education","lineage":["https://openalex.org/I20231570"]}],"countries":["AU","CN"],"is_corresponding":true,"raw_author_name":"Kaile Su","raw_affiliation_strings":["Institute for Integrated and Intelligent Systems, Griffith University, Brisbane, Qld 4111, Australia","Key Laboratory of High Confidence Software Technologies of Ministry of Education, School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute for Integrated and Intelligent Systems, Griffith University, Brisbane, Qld 4111, Australia","institution_ids":["https://openalex.org/I11701301"]},{"raw_affiliation_string":"Key Laboratory of High Confidence Software Technologies of Ministry of Education, School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, China","institution_ids":["https://openalex.org/I20231570"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5102089636","display_name":"Xizhong Zheng","orcid":null},"institutions":[{"id":"https://openalex.org/I115592961","display_name":"Jiangsu University","ror":"https://ror.org/03jc41j30","country_code":"CN","type":"education","lineage":["https://openalex.org/I115592961"]},{"id":"https://openalex.org/I51783024","display_name":"Brandenburg University of Technology Cottbus-Senftenberg","ror":"https://ror.org/02wxx3e24","country_code":"DE","type":"education","lineage":["https://openalex.org/I51783024"]}],"countries":["CN","DE"],"is_corresponding":true,"raw_author_name":"Xizhong Zheng","raw_affiliation_strings":["Department of Computer Science, Jiangsu University, Zhenjiang 212013, China","Theoretische Informatik, BTU Cottbus, Cottbus 03044, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science, Jiangsu University, Zhenjiang 212013, China","institution_ids":["https://openalex.org/I115592961"]},{"raw_affiliation_string":"Theoretische Informatik, BTU Cottbus, Cottbus 03044, Germany","institution_ids":["https://openalex.org/I51783024"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5016292021","https://openalex.org/A5037325282","https://openalex.org/A5102089636"],"corresponding_institution_ids":["https://openalex.org/I115592961","https://openalex.org/I11701301","https://openalex.org/I159948400","https://openalex.org/I20231570","https://openalex.org/I51783024"],"apc_list":{"value":2630,"currency":"USD","value_usd":2630},"apc_paid":null,"fwci":0.7364,"has_fulltext":false,"cited_by_count":8,"citation_normalized_percentile":{"value":0.74808723,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"53","issue":"4-5","first_page":"365","last_page":"380"},"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/T11269","display_name":"Algorithms and Data Compression","score":0.9624000191688538,"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/T13720","display_name":"Benford\u2019s Law and Fraud Detection","score":0.9559000134468079,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/real-number","display_name":"Real number","score":0.8214839696884155},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7935423254966736},{"id":"https://openalex.org/keywords/representation","display_name":"Representation (politics)","score":0.7204136252403259},{"id":"https://openalex.org/keywords/dedekind-cut","display_name":"Dedekind cut","score":0.642996609210968},{"id":"https://openalex.org/keywords/computable-number","display_name":"Computable number","score":0.5129101276397705},{"id":"https://openalex.org/keywords/hierarchy","display_name":"Hierarchy","score":0.4819202721118927},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.39586174488067627},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3901170492172241},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.2954978942871094},{"id":"https://openalex.org/keywords/computable-function","display_name":"Computable function","score":0.26752421259880066},{"id":"https://openalex.org/keywords/computable-analysis","display_name":"Computable analysis","score":0.1796397566795349}],"concepts":[{"id":"https://openalex.org/C94020503","wikidata":"https://www.wikidata.org/wiki/Q12916","display_name":"Real number","level":2,"score":0.8214839696884155},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7935423254966736},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.7204136252403259},{"id":"https://openalex.org/C76449584","wikidata":"https://www.wikidata.org/wiki/Q851333","display_name":"Dedekind cut","level":2,"score":0.642996609210968},{"id":"https://openalex.org/C156382255","wikidata":"https://www.wikidata.org/wiki/Q818895","display_name":"Computable number","level":4,"score":0.5129101276397705},{"id":"https://openalex.org/C31170391","wikidata":"https://www.wikidata.org/wiki/Q188619","display_name":"Hierarchy","level":2,"score":0.4819202721118927},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.39586174488067627},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3901170492172241},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2954978942871094},{"id":"https://openalex.org/C54271186","wikidata":"https://www.wikidata.org/wiki/Q1148456","display_name":"Computable function","level":2,"score":0.26752421259880066},{"id":"https://openalex.org/C27318111","wikidata":"https://www.wikidata.org/wiki/Q5157264","display_name":"Computable analysis","level":3,"score":0.1796397566795349},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","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/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","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/C94625758","wikidata":"https://www.wikidata.org/wiki/Q7163","display_name":"Politics","level":2,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1002/malq.200710005","is_oa":false,"landing_page_url":"https://doi.org/10.1002/malq.200710005","pdf_url":null,"source":{"id":"https://openalex.org/S172351290","display_name":"Mathematical logic quarterly","issn_l":"0044-3050","issn":["0044-3050","0942-5616","1521-3870"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Logic Quarterly","raw_type":"journal-article"},{"id":"pmh:oai:research-repository.griffith.edu.au:10072/19008","is_oa":false,"landing_page_url":"http://hdl.handle.net/10072/19008","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Journal article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":19,"referenced_works":["https://openalex.org/W293447284","https://openalex.org/W1483568252","https://openalex.org/W1497475168","https://openalex.org/W1518163798","https://openalex.org/W1964728647","https://openalex.org/W1975452830","https://openalex.org/W1985920135","https://openalex.org/W2016407984","https://openalex.org/W2025476736","https://openalex.org/W2059410566","https://openalex.org/W2092853606","https://openalex.org/W2126160338","https://openalex.org/W2170080480","https://openalex.org/W3175367423","https://openalex.org/W4231629838","https://openalex.org/W4243563333","https://openalex.org/W4245102239","https://openalex.org/W4247585886","https://openalex.org/W4254793172"],"related_works":["https://openalex.org/W1638251065","https://openalex.org/W2962816983","https://openalex.org/W2158841525","https://openalex.org/W2019323726","https://openalex.org/W2143028857","https://openalex.org/W2034440051","https://openalex.org/W1593659144","https://openalex.org/W4298840716","https://openalex.org/W2796974007","https://openalex.org/W2002700168"],"abstract_inverted_index":{"Abstract":[0],"In":[1,98],"mathematics,":[2],"various":[3,60],"representations":[4,13,35,144],"of":[5,33,47,80,145],"real":[6,23,49,54,111,147],"numbers":[7,55,112],"have":[8],"been":[9],"investigated.":[10],"All":[11],"these":[12,34,74],"are":[14,36],"mathematically":[15],"equivalent":[16,37,61],"because":[17],"they":[18,42],"lead":[19,76],"to":[20,77,134],"the":[21,30,39,44,52,91,138,143,146],"same":[22,45],"structure":[24],"\u2013":[25],"Dedekind\u2010complete":[26],"ordered":[27],"field.":[28],"Even":[29],"effective":[31],"versions":[32],"in":[38,59,86],"sense":[40],"that":[41,102],"define":[43],"notion":[46],"computable":[48,53],"numbers.":[50,148],"Although":[51],"can":[56],"be":[57],"defined":[58],"ways,":[62],"if":[63],"\u201ccomputable\u201d":[64],"is":[65,104,133],"replaced":[66],"by":[67,113],"\u201cprimitive":[68],"recursive\u201d":[69],"(p.":[70],"r.,":[71],"for":[72],"short),":[73],"definitions":[75],"a":[78,105],"number":[79],"different":[81],"concepts,":[82],"which":[83],"we":[84,100],"compare":[85],"this":[87],"article.":[88],"We":[89],"summarize":[90],"known":[92],"results":[93],"and":[94,126],"add":[95],"new":[96],"ones.":[97],"particular":[99],"show":[101],"there":[103],"proper":[106],"hierarchy":[107],"among":[108],"p.":[109],"r.":[110],"nested":[114],"interval":[115],"representation,":[116,118,122,125],"Cauchy":[117],"b":[119],"\u2010adic":[120],"expansion":[121,129],"Dedekind":[123],"cut":[124],"continued":[127],"fraction":[128],"representation.":[130],"Our":[131],"goal":[132],"clarify":[135],"systematically":[136],"how":[137],"primitive":[139],"recursiveness":[140],"depends":[141],"on":[142],"(\u00a9":[149],"2007":[150],"WILEY\u2010VCH":[151],"Verlag":[152],"GmbH":[153],"&amp;":[154],"Co.":[155],"KGaA,":[156],"Weinheim)":[157]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
