{"id":"https://openalex.org/W2000860116","doi":"https://doi.org/10.1017/s0022481200051136","title":"On the complexity of the theories of weak direct powers","display_name":"On the complexity of the theories of weak direct powers","publication_year":1976,"publication_date":"1976-09-01","ids":{"openalex":"https://openalex.org/W2000860116","doi":"https://doi.org/10.1017/s0022481200051136","mag":"2000860116"},"language":"en","primary_location":{"id":"doi:10.1017/s0022481200051136","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0022481200051136","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/A5071766529","display_name":"Charles Rackoff","orcid":null},"institutions":[{"id":"https://openalex.org/I185261750","display_name":"University of Toronto","ror":"https://ror.org/03dbr7087","country_code":"CA","type":"education","lineage":["https://openalex.org/I185261750"]}],"countries":["CA"],"is_corresponding":true,"raw_author_name":"Charles Rackoff","raw_affiliation_strings":["University of Toronto, Toronto, Canada M55 1A7","University of Toronto"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Toronto, Toronto, Canada M55 1A7","institution_ids":["https://openalex.org/I185261750"]},{"raw_affiliation_string":"University of Toronto","institution_ids":["https://openalex.org/I185261750"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5071766529"],"corresponding_institution_ids":["https://openalex.org/I185261750"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.13341094,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"41","issue":"3","first_page":"561","last_page":"573"},"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":0.9987999796867371,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9975000023841858,"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/decidability","display_name":"Decidability","score":0.7564880847930908},{"id":"https://openalex.org/keywords/nondeterministic-algorithm","display_name":"Nondeterministic algorithm","score":0.7249953746795654},{"id":"https://openalex.org/keywords/time-hierarchy-theorem","display_name":"Time hierarchy theorem","score":0.7085061073303223},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6942087411880493},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5092226266860962},{"id":"https://openalex.org/keywords/elementary-theory","display_name":"Elementary theory","score":0.5085476636886597},{"id":"https://openalex.org/keywords/abelian-group","display_name":"Abelian group","score":0.4713142514228821},{"id":"https://openalex.org/keywords/turing","display_name":"Turing","score":0.4706909954547882},{"id":"https://openalex.org/keywords/turing-machine","display_name":"Turing machine","score":0.4641140103340149},{"id":"https://openalex.org/keywords/multiplication","display_name":"Multiplication (music)","score":0.4590825140476227},{"id":"https://openalex.org/keywords/complexity-class","display_name":"Complexity class","score":0.4451879560947418},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.31053000688552856},{"id":"https://openalex.org/keywords/time-complexity","display_name":"Time complexity","score":0.30594736337661743},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.17809224128723145},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.15199488401412964},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.09302020072937012}],"concepts":[{"id":"https://openalex.org/C153269930","wikidata":"https://www.wikidata.org/wiki/Q430001","display_name":"Decidability","level":2,"score":0.7564880847930908},{"id":"https://openalex.org/C176181172","wikidata":"https://www.wikidata.org/wiki/Q3490301","display_name":"Nondeterministic algorithm","level":2,"score":0.7249953746795654},{"id":"https://openalex.org/C48415503","wikidata":"https://www.wikidata.org/wiki/Q5251122","display_name":"Time hierarchy theorem","level":5,"score":0.7085061073303223},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6942087411880493},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5092226266860962},{"id":"https://openalex.org/C2779851814","wikidata":"https://www.wikidata.org/wiki/Q5358916","display_name":"Elementary theory","level":2,"score":0.5085476636886597},{"id":"https://openalex.org/C136170076","wikidata":"https://www.wikidata.org/wiki/Q181296","display_name":"Abelian group","level":2,"score":0.4713142514228821},{"id":"https://openalex.org/C9870796","wikidata":"https://www.wikidata.org/wiki/Q490481","display_name":"Turing","level":2,"score":0.4706909954547882},{"id":"https://openalex.org/C29248071","wikidata":"https://www.wikidata.org/wiki/Q163310","display_name":"Turing machine","level":3,"score":0.4641140103340149},{"id":"https://openalex.org/C2780595030","wikidata":"https://www.wikidata.org/wiki/Q3860309","display_name":"Multiplication (music)","level":2,"score":0.4590825140476227},{"id":"https://openalex.org/C39637292","wikidata":"https://www.wikidata.org/wiki/Q908207","display_name":"Complexity class","level":3,"score":0.4451879560947418},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.31053000688552856},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.30594736337661743},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.17809224128723145},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.15199488401412964},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.09302020072937012},{"id":"https://openalex.org/C131671149","wikidata":"https://www.wikidata.org/wiki/Q2703890","display_name":"Universal Turing machine","level":4,"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:10.1017/s0022481200051136","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0022481200051136","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/1183739822","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1183739822","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":[{"id":"https://metadata.un.org/sdg/16","display_name":"Peace, Justice and strong institutions","score":0.8299999833106995}],"awards":[],"funders":[{"id":"https://openalex.org/F4320310153","display_name":"Ryerson University","ror":"https://ror.org/05g13zd79"},{"id":"https://openalex.org/F4320322015","display_name":"University of Toronto","ror":"https://ror.org/03dbr7087"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":13,"referenced_works":["https://openalex.org/W27784635","https://openalex.org/W296038034","https://openalex.org/W1508780728","https://openalex.org/W1511526676","https://openalex.org/W1555138513","https://openalex.org/W1556844982","https://openalex.org/W1972603893","https://openalex.org/W1981324778","https://openalex.org/W1999590237","https://openalex.org/W2007464118","https://openalex.org/W2021592352","https://openalex.org/W4206797071","https://openalex.org/W4213257073"],"related_works":["https://openalex.org/W2075449032","https://openalex.org/W2169625092","https://openalex.org/W2144557478","https://openalex.org/W1483438220","https://openalex.org/W2000805093","https://openalex.org/W2951313060","https://openalex.org/W4300950794","https://openalex.org/W1593050105","https://openalex.org/W2000860116","https://openalex.org/W2465024128"],"abstract_inverted_index":{"Abstract":[0],"Mostowski":[1],"[11]":[2],"shows":[3],"that":[4,77,139],"if":[5],"a":[6,9,48],"structure":[7,114],"has":[8,17],"decidable":[10],"theory,":[11],"then":[12],"its":[13],"weak":[14,44,61,83],"direct":[15,45,62,84],"power":[16,46,85],"one":[18],"as":[19],"well;":[20],"his":[21],"proof":[22],"however":[23],"never":[24],"produces":[25],"decision":[26,58],"procedures":[27,59],"which":[28,50,66],"are":[29,36],"elementary":[30,56,69],"recursive.":[31],"Some":[32],"very":[33],"general":[34],"results":[35],"obtained":[37,108],"here":[38],"about":[39],"the":[40,43,82,103,110,113,127,140],"nature":[41],"of":[42,47,64,86,112,120,129,142],"structure,":[49],"in":[51,94],"most":[52],"cases":[53],"lead":[54],"to":[55],"recursive":[57,70],"for":[60,96,109,126],"powers":[63],"structures":[65],"themselves":[67],"have":[68,137],"procedures.":[71],"In":[72],"particular,":[73],"it":[74],"is":[75,107],"shown":[76,138],"\u3008":[78,87,115,143],"N":[79,88,116,144],"*,":[80],"+\u3009,":[81,90],",":[89,118],"can":[91],"be":[92],"decided":[93],"space":[95],"some":[97],"constant":[98],"c":[99],".":[100],"As":[101],"corollaries,":[102],"same":[104],"upper":[105],"bound":[106],"theory":[111,128,141],"+":[117],"\u00b7\u3009":[119],"positive":[121],"integers":[122],"under":[123],"multiplication,":[124],"and":[125,134],"finite":[130],"abelian":[131],"groups.":[132],"Fischer":[133],"Rabin":[135],"[7]":[136],",*":[145],"+\u3009":[146],"requires":[147],"time":[148],"even":[149],"on":[150],"nondeterministic":[151],"Turing":[152],"machines.":[153]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}