{"id":"https://openalex.org/W2278710187","doi":"https://doi.org/10.4086/toc.2011.v007a004","title":"","display_name":"","publication_year":2011,"publication_date":"2011-01-01","ids":{"openalex":"https://openalex.org/W2278710187","doi":"https://doi.org/10.4086/toc.2011.v007a004","mag":"2278710187"},"language":"en","primary_location":{"id":"doi:10.4086/toc.2011.v007a004","is_oa":true,"landing_page_url":"https://doi.org/10.4086/toc.2011.v007a004","pdf_url":"http://www.theoryofcomputing.org/articles/v007a004/v007a004.pdf","source":{"id":"https://openalex.org/S142665794","display_name":"Theory of Computing","issn_l":"1557-2862","issn":["1557-2862"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310315690","host_organization_name":"University of Chicago","host_organization_lineage":["https://openalex.org/P4310315690"],"host_organization_lineage_names":["University of Chicago"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Theory of Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"http://www.theoryofcomputing.org/articles/v007a004/v007a004.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5050504059","display_name":"Kazuyuki Amano","orcid":"https://orcid.org/0000-0003-2322-6072"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Kazuyuki Amano","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5050504059"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":4.5238,"has_fulltext":true,"cited_by_count":20,"citation_normalized_percentile":{"value":0.95200995,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"7","issue":"1","first_page":"45","last_page":"48"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9998999834060669,"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/T10720","display_name":"Complexity and Algorithms in Graphs","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/T12072","display_name":"Machine Learning and Algorithms","score":0.9987999796867371,"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/T11130","display_name":"Coding theory and cryptography","score":0.9979000091552734,"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/mathematics","display_name":"Mathematics","score":0.7211619019508362},{"id":"https://openalex.org/keywords/boolean-function","display_name":"Boolean function","score":0.6989673376083374},{"id":"https://openalex.org/keywords/parity-function","display_name":"Parity function","score":0.5920365452766418},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5902905464172363},{"id":"https://openalex.org/keywords/parity","display_name":"Parity (physics)","score":0.552635908126831},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.5477827787399292},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.535152018070221},{"id":"https://openalex.org/keywords/sensitivity","display_name":"Sensitivity (control systems)","score":0.5130702257156372},{"id":"https://openalex.org/keywords/random-variable","display_name":"Random variable","score":0.4682043194770813},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.4354900121688843},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.16956359148025513},{"id":"https://openalex.org/keywords/boolean-expression","display_name":"Boolean expression","score":0.16521751880645752},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.10604360699653625},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.07688412070274353},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.07226178050041199}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7211619019508362},{"id":"https://openalex.org/C187455244","wikidata":"https://www.wikidata.org/wiki/Q942353","display_name":"Boolean function","level":2,"score":0.6989673376083374},{"id":"https://openalex.org/C188183281","wikidata":"https://www.wikidata.org/wiki/Q2898309","display_name":"Parity function","level":4,"score":0.5920365452766418},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5902905464172363},{"id":"https://openalex.org/C2777151079","wikidata":"https://www.wikidata.org/wiki/Q141160","display_name":"Parity (physics)","level":2,"score":0.552635908126831},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.5477827787399292},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.535152018070221},{"id":"https://openalex.org/C21200559","wikidata":"https://www.wikidata.org/wiki/Q7451068","display_name":"Sensitivity (control systems)","level":2,"score":0.5130702257156372},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.4682043194770813},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4354900121688843},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.16956359148025513},{"id":"https://openalex.org/C158465420","wikidata":"https://www.wikidata.org/wiki/Q1979515","display_name":"Boolean expression","level":3,"score":0.16521751880645752},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.10604360699653625},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.07688412070274353},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.07226178050041199},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C24326235","wikidata":"https://www.wikidata.org/wiki/Q126095","display_name":"Electronic engineering","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.4086/toc.2011.v007a004","is_oa":true,"landing_page_url":"https://doi.org/10.4086/toc.2011.v007a004","pdf_url":"http://www.theoryofcomputing.org/articles/v007a004/v007a004.pdf","source":{"id":"https://openalex.org/S142665794","display_name":"Theory of Computing","issn_l":"1557-2862","issn":["1557-2862"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310315690","host_organization_name":"University of Chicago","host_organization_lineage":["https://openalex.org/P4310315690"],"host_organization_lineage_names":["University of Chicago"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Theory of Computing","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.4086/toc.2011.v007a004","is_oa":true,"landing_page_url":"https://doi.org/10.4086/toc.2011.v007a004","pdf_url":"http://www.theoryofcomputing.org/articles/v007a004/v007a004.pdf","source":{"id":"https://openalex.org/S142665794","display_name":"Theory of Computing","issn_l":"1557-2862","issn":["1557-2862"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310315690","host_organization_name":"University of Chicago","host_organization_lineage":["https://openalex.org/P4310315690"],"host_organization_lineage_names":["University of Chicago"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Theory of Computing","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1318710975","display_name":"Establishing Computer Assisted Proof Methods for Computational Intractability","funder_award_id":"21500005","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"}],"funders":[{"id":"https://openalex.org/F4320334764","display_name":"Japan Society for the Promotion of Science","ror":"https://ror.org/00hhkn466"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2278710187.pdf","grobid_xml":"https://content.openalex.org/works/W2278710187.grobid-xml"},"referenced_works_count":9,"referenced_works":["https://openalex.org/W1577016621","https://openalex.org/W1594550269","https://openalex.org/W1986296546","https://openalex.org/W2023691706","https://openalex.org/W2042194938","https://openalex.org/W2088559598","https://openalex.org/W2092928619","https://openalex.org/W2885803187","https://openalex.org/W4241017158"],"related_works":["https://openalex.org/W2003722472","https://openalex.org/W2379318067","https://openalex.org/W2106622045","https://openalex.org/W2136871609","https://openalex.org/W2374311587","https://openalex.org/W2999312112","https://openalex.org/W4301915461","https://openalex.org/W2340850974","https://openalex.org/W2405110775","https://openalex.org/W2952185983"],"abstract_inverted_index":{"The":[0],"average":[1,49,66],"sensitivity":[2,50,67],"of":[3,15,18,27],"a":[4,11,31,43,46],"Boolean":[5,39],"function":[6,40,61],"is":[7,56],"the":[8,16,25,28,59],"expectation,":[9],"given":[10],"uniformly":[12],"random":[13],"input,":[14],"number":[17],"input":[19],"bits":[20],"which":[21],"when":[22],"flipped":[23],"change":[24],"output":[26],"function.":[29],"Answering":[30],"question":[32],"by":[33,42],"O'Donnell,":[34],"we":[35],"show":[36],"that":[37],"every":[38],"represented":[41],"k-CNF":[44],"(or":[45],"k-DNF)":[47],"has":[48,65],"at":[51],"most":[52],"k.":[53,68],"This":[54],"bound":[55],"tight":[57],"since":[58],"parity":[60],"on":[62],"k":[63],"variables":[64]},"counts_by_year":[{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":5},{"year":2012,"cited_by_count":3}],"updated_date":"2026-04-21T08:09:41.155169","created_date":"2016-06-24T00:00:00"}