{"id":"https://openalex.org/W2050186596","doi":"https://doi.org/10.1137/s0097539790178069","title":"The Boolean Hierarchy and the Polynomial Hierarchy: A Closer Connection","display_name":"The Boolean Hierarchy and the Polynomial Hierarchy: A Closer Connection","publication_year":1996,"publication_date":"1996-04-01","ids":{"openalex":"https://openalex.org/W2050186596","doi":"https://doi.org/10.1137/s0097539790178069","mag":"2050186596"},"language":"en","primary_location":{"id":"doi:10.1137/s0097539790178069","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0097539790178069","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/A5082398104","display_name":"Richard Chang","orcid":"https://orcid.org/0000-0001-5278-7958"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Richard Chang","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5113697744","display_name":"Jim Kadin","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Jim Kadin","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5082398104"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":8.76,"has_fulltext":false,"cited_by_count":66,"citation_normalized_percentile":{"value":0.981828,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"25","issue":"2","first_page":"340","last_page":"354"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9994000196456909,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9994000196456909,"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/T12072","display_name":"Machine Learning and Algorithms","score":0.9970999956130981,"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/T11269","display_name":"Algorithms and Data Compression","score":0.9965000152587891,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/hierarchy","display_name":"Hierarchy","score":0.7500799894332886},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6906224489212036},{"id":"https://openalex.org/keywords/connection","display_name":"Connection (principal bundle)","score":0.4972880184650421},{"id":"https://openalex.org/keywords/polynomial-hierarchy","display_name":"Polynomial hierarchy","score":0.4514327943325043},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.4329807460308075},{"id":"https://openalex.org/keywords/time-complexity","display_name":"Time complexity","score":0.42892611026763916},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4251856207847595},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.38667818903923035}],"concepts":[{"id":"https://openalex.org/C31170391","wikidata":"https://www.wikidata.org/wiki/Q188619","display_name":"Hierarchy","level":2,"score":0.7500799894332886},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6906224489212036},{"id":"https://openalex.org/C13355873","wikidata":"https://www.wikidata.org/wiki/Q2920850","display_name":"Connection (principal bundle)","level":2,"score":0.4972880184650421},{"id":"https://openalex.org/C2776013622","wikidata":"https://www.wikidata.org/wiki/Q2103021","display_name":"Polynomial hierarchy","level":3,"score":0.4514327943325043},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.4329807460308075},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.42892611026763916},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4251856207847595},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.38667818903923035},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"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/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/s0097539790178069","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0097539790178069","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"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.31.5459","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.31.5459","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.umbc.edu/~chang/papers/closer/bhe2.ps","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.77.4186","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.77.4186","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.umbc.edu/~chang/papers/closer/bhe2.pdf","raw_type":"text"},{"id":"pmh:oai:ecommons.cornell.edu:1813/6808","is_oa":false,"landing_page_url":"https://hdl.handle.net/1813/6808","pdf_url":null,"source":{"id":"https://openalex.org/S4306400757","display_name":"eCommons (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":null,"raw_type":"technical report"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":6,"referenced_works":["https://openalex.org/W1752076080","https://openalex.org/W1986602165","https://openalex.org/W2053094807","https://openalex.org/W2090039116","https://openalex.org/W2168714844","https://openalex.org/W4255515059"],"related_works":["https://openalex.org/W1895405313","https://openalex.org/W4388962799","https://openalex.org/W2041517264","https://openalex.org/W1522180487","https://openalex.org/W4294789866","https://openalex.org/W1969386550","https://openalex.org/W2982557863","https://openalex.org/W2590361123","https://openalex.org/W1575303034","https://openalex.org/W2057722348"],"abstract_inverted_index":{"We":[0],"show":[1,43],"that":[2,44,66],"if":[3],"the":[4,12,23,27,39,45,63,71,79,92],"Boolean":[5,28,72,93],"hierarchy":[6,14,29,47,73],"collapses":[7,15],"to":[8,16,50,86],"level":[9,25],"k,":[10],"then":[11],"polynomial":[13,46,80],"${\\text{BH}}_3":[17,20],"(k)$,":[18],"where":[19],"(k)$":[21],"is":[22,35,57],"kth":[24],"of":[26,70,78,97],"over":[30,38],"$\\Sigma":[31],"_2^{\\text{P}}":[32,99],"$.":[33,104],"This":[34,55],"an":[36],"improvement":[37],"known":[40],"results,":[41],"which":[42],"would":[48],"collapse":[49,69,77],"${\\text{P}}^{{\\text{NP}}^{{\\text{NP}}}":[51],"}":[52,130,136],"[O(\\log":[53],"n)]$.":[54],"result":[56,84],"significant":[58],"in":[59],"two":[60],"ways.":[61],"First,":[62],"theorem":[64],"says":[65],"a":[67,75],"deeper":[68,76],"implies":[74],"hierarchy.":[81],"Also,":[82],"this":[83],"points":[85],"some":[87],"previously":[88],"unexplored":[89],"connections":[90],"between":[91],"and":[94,101],"query":[95],"hierarchies":[96],"$\\Delta":[98,102],"$":[100],"_3^{\\text{P}}":[103],"Namely,\\[":[105],"\\begin{gathered}":[106],"{\\text{BH}}(k)":[107],"=":[108,115,123,134],"{\\text{co":[109,116],"-":[110,117],"BH}}(k)":[111],"\\Rightarrow":[112,128],"{\\text{BH}}_3":[113],"(k)":[114],"BH}}_3":[118],"(k),":[119],"\\\\":[120,140],"{\\text{P}}^{{\\text{NP}}}":[121,124],"\\|[k]":[122],"\\|[k":[125,131,137],"+":[126,132,138],"1]":[127,133],"{\\text{P}}^{{\\text{NP}}^{{\\text{NP}}}":[129,135],"2].":[139],"\\end{gathered}":[141],"\\]":[142]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2019,"cited_by_count":2},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":2},{"year":2012,"cited_by_count":1}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}