{"id":"https://openalex.org/W2137407595","doi":"https://doi.org/10.1137/0220053","title":"PP is as Hard as the Polynomial-Time Hierarchy","display_name":"PP is as Hard as the Polynomial-Time Hierarchy","publication_year":1991,"publication_date":"1991-10-01","ids":{"openalex":"https://openalex.org/W2137407595","doi":"https://doi.org/10.1137/0220053","mag":"2137407595"},"language":"en","primary_location":{"id":"doi:10.1137/0220053","is_oa":false,"landing_page_url":"https://doi.org/10.1137/0220053","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Computing","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/A5053366001","display_name":"Seinosuke Toda","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Seinosuke Toda","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5053366001"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":15.9126,"has_fulltext":false,"cited_by_count":673,"citation_normalized_percentile":{"value":0.99380218,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":99,"max":100},"biblio":{"volume":"20","issue":"5","first_page":"865","last_page":"877"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.998199999332428,"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":0.998199999332428,"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":0.9976999759674072,"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/T12162","display_name":"Cellular Automata and Applications","score":0.9976999759674072,"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/combinatorics","display_name":"Combinatorics","score":0.6723719835281372},{"id":"https://openalex.org/keywords/time-complexity","display_name":"Time complexity","score":0.6263489723205566},{"id":"https://openalex.org/keywords/hierarchy","display_name":"Hierarchy","score":0.6074094176292419},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.5648515224456787},{"id":"https://openalex.org/keywords/turing","display_name":"Turing","score":0.5593302249908447},{"id":"https://openalex.org/keywords/polynomial-hierarchy","display_name":"Polynomial hierarchy","score":0.5435073971748352},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5311768054962158},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.503375232219696},{"id":"https://openalex.org/keywords/turing-machine","display_name":"Turing machine","score":0.5007340908050537},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.321489542722702},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1376480758190155},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.12972503900527954}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6723719835281372},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.6263489723205566},{"id":"https://openalex.org/C31170391","wikidata":"https://www.wikidata.org/wiki/Q188619","display_name":"Hierarchy","level":2,"score":0.6074094176292419},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5648515224456787},{"id":"https://openalex.org/C9870796","wikidata":"https://www.wikidata.org/wiki/Q490481","display_name":"Turing","level":2,"score":0.5593302249908447},{"id":"https://openalex.org/C2776013622","wikidata":"https://www.wikidata.org/wiki/Q2103021","display_name":"Polynomial hierarchy","level":3,"score":0.5435073971748352},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5311768054962158},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.503375232219696},{"id":"https://openalex.org/C29248071","wikidata":"https://www.wikidata.org/wiki/Q163310","display_name":"Turing machine","level":3,"score":0.5007340908050537},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.321489542722702},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1376480758190155},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.12972503900527954},{"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/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","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},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/0220053","is_oa":false,"landing_page_url":"https://doi.org/10.1137/0220053","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Computing","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":22,"referenced_works":["https://openalex.org/W1552288906","https://openalex.org/W1638461282","https://openalex.org/W1967159037","https://openalex.org/W1996184580","https://openalex.org/W1999384241","https://openalex.org/W2002385864","https://openalex.org/W2003513300","https://openalex.org/W2005440350","https://openalex.org/W2006912660","https://openalex.org/W2011358434","https://openalex.org/W2014603110","https://openalex.org/W2020831804","https://openalex.org/W2024309621","https://openalex.org/W2033323875","https://openalex.org/W2036095420","https://openalex.org/W2044504659","https://openalex.org/W2057512592","https://openalex.org/W2059442002","https://openalex.org/W2065251436","https://openalex.org/W2078096580","https://openalex.org/W2115826669","https://openalex.org/W4383163603"],"related_works":["https://openalex.org/W2119244842","https://openalex.org/W4285436350","https://openalex.org/W2090113742","https://openalex.org/W4324016496","https://openalex.org/W2215138167","https://openalex.org/W4254404873","https://openalex.org/W4231495498","https://openalex.org/W4312201671","https://openalex.org/W2306527987","https://openalex.org/W3160192459"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"two":[3],"interesting":[4],"complexity":[5],"classes,":[6],"PP":[7],"and":[8,36],"$":[9],"\\oplus":[10,43],"{\\text{P}}$,":[11],"are":[12],"compared":[13],"with":[14],"PH,":[15],"the":[16,49],"polynomial-time":[17,28,78],"hierarchy.":[18],"It":[19],"is":[20,27,38,70,77],"shown":[21],"that":[22,53],"every":[23,73],"set":[24,33,74,83],"in":[25,34,40,75,84],"PH":[26,37],"Turing":[29,79],"reducible":[30,80],"to":[31,81],"a":[32,46,63,82],"PP,":[35],"included":[39],"${\\text{BP}}":[41],"\\cdot":[42],"{\\text{P}}$.":[44],"As":[45],"consequence":[47],"of":[48,65],"results,":[50],"it":[51],"follows":[52],"${\\text{PP}}":[54],"\\subseteq":[55,60],"{\\text{PH}}$":[56],"(or":[57],"$\\oplus":[58],"{\\text{P}}":[59],"{\\text{PH}}$)":[61],"implies":[62],"collapse":[64],"PH.":[66],"A":[67],"stronger":[68],"result":[69],"also":[71],"shown:":[72],"PP(PH)":[76],"PP.":[85]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":10},{"year":2024,"cited_by_count":11},{"year":2023,"cited_by_count":29},{"year":2022,"cited_by_count":18},{"year":2021,"cited_by_count":29},{"year":2020,"cited_by_count":28},{"year":2019,"cited_by_count":15},{"year":2018,"cited_by_count":26},{"year":2017,"cited_by_count":36},{"year":2016,"cited_by_count":26},{"year":2015,"cited_by_count":28},{"year":2014,"cited_by_count":33},{"year":2013,"cited_by_count":24},{"year":2012,"cited_by_count":20}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}
