{"id":"https://openalex.org/W2026399774","doi":"https://doi.org/10.1137/0206030","title":"A $2.5n$-Lower Bound on the Combinational Complexity of Boolean Functions","display_name":"A $2.5n$-Lower Bound on the Combinational Complexity of Boolean Functions","publication_year":1977,"publication_date":"1977-09-01","ids":{"openalex":"https://openalex.org/W2026399774","doi":"https://doi.org/10.1137/0206030","mag":"2026399774"},"language":"en","primary_location":{"id":"doi:10.1137/0206030","is_oa":false,"landing_page_url":"https://doi.org/10.1137/0206030","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/A5109888809","display_name":"Wolfgang J. Paul","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Wolfgang J. Paul","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5109888809"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.507,"has_fulltext":false,"cited_by_count":60,"citation_normalized_percentile":{"value":0.90789267,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"6","issue":"3","first_page":"427","last_page":"443"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","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"}},"topics":[{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9973000288009644,"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":0.9970999956130981,"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/boolean-function","display_name":"Boolean function","score":0.8219102621078491},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7601039409637451},{"id":"https://openalex.org/keywords/boolean-circuit","display_name":"Boolean circuit","score":0.5315629839897156},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.5249457359313965},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5207955241203308},{"id":"https://openalex.org/keywords/circuit-complexity","display_name":"Circuit complexity","score":0.5122310519218445},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.4870005249977112},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.48683589696884155},{"id":"https://openalex.org/keywords/computational-complexity-theory","display_name":"Computational complexity theory","score":0.46938058733940125},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.4528256058692932},{"id":"https://openalex.org/keywords/time-complexity","display_name":"Time complexity","score":0.44140517711639404},{"id":"https://openalex.org/keywords/complexity-index","display_name":"Complexity index","score":0.43703708052635193},{"id":"https://openalex.org/keywords/binary-logarithm","display_name":"Binary logarithm","score":0.4273070991039276},{"id":"https://openalex.org/keywords/boolean-expression","display_name":"Boolean expression","score":0.4239051342010498},{"id":"https://openalex.org/keywords/combinational-logic","display_name":"Combinational logic","score":0.41278141736984253},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.2898954153060913},{"id":"https://openalex.org/keywords/logic-gate","display_name":"Logic gate","score":0.12096279859542847},{"id":"https://openalex.org/keywords/electronic-circuit","display_name":"Electronic circuit","score":0.1063496470451355}],"concepts":[{"id":"https://openalex.org/C187455244","wikidata":"https://www.wikidata.org/wiki/Q942353","display_name":"Boolean function","level":2,"score":0.8219102621078491},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7601039409637451},{"id":"https://openalex.org/C141796577","wikidata":"https://www.wikidata.org/wiki/Q837479","display_name":"Boolean circuit","level":3,"score":0.5315629839897156},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.5249457359313965},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5207955241203308},{"id":"https://openalex.org/C90702460","wikidata":"https://www.wikidata.org/wiki/Q1055112","display_name":"Circuit complexity","level":3,"score":0.5122310519218445},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4870005249977112},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.48683589696884155},{"id":"https://openalex.org/C179799912","wikidata":"https://www.wikidata.org/wiki/Q205084","display_name":"Computational complexity theory","level":2,"score":0.46938058733940125},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.4528256058692932},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.44140517711639404},{"id":"https://openalex.org/C6435147","wikidata":"https://www.wikidata.org/wiki/Q5156627","display_name":"Complexity index","level":3,"score":0.43703708052635193},{"id":"https://openalex.org/C63553672","wikidata":"https://www.wikidata.org/wiki/Q581168","display_name":"Binary logarithm","level":2,"score":0.4273070991039276},{"id":"https://openalex.org/C158465420","wikidata":"https://www.wikidata.org/wiki/Q1979515","display_name":"Boolean expression","level":3,"score":0.4239051342010498},{"id":"https://openalex.org/C81409106","wikidata":"https://www.wikidata.org/wiki/Q76505","display_name":"Combinational logic","level":3,"score":0.41278141736984253},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2898954153060913},{"id":"https://openalex.org/C131017901","wikidata":"https://www.wikidata.org/wiki/Q170451","display_name":"Logic gate","level":2,"score":0.12096279859542847},{"id":"https://openalex.org/C134146338","wikidata":"https://www.wikidata.org/wiki/Q1815901","display_name":"Electronic circuit","level":2,"score":0.1063496470451355},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"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/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C119599485","wikidata":"https://www.wikidata.org/wiki/Q43035","display_name":"Electrical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/0206030","is_oa":false,"landing_page_url":"https://doi.org/10.1137/0206030","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:10.1.1.297.769","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.297.769","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.rutgers.edu/~zeilberg/EM11/WJpaul.pdf","raw_type":"text"},{"id":"pmh:oai:ecommons.cornell.edu:1813/6061","is_oa":false,"landing_page_url":"https://hdl.handle.net/1813/6061","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":5,"referenced_works":["https://openalex.org/W1508461556","https://openalex.org/W1982704699","https://openalex.org/W2119397055","https://openalex.org/W2401610261","https://openalex.org/W4250775169"],"related_works":["https://openalex.org/W2018822765","https://openalex.org/W3127266383","https://openalex.org/W2055594799","https://openalex.org/W3049543508","https://openalex.org/W3204916073","https://openalex.org/W2054417980","https://openalex.org/W1564702418","https://openalex.org/W2066362579","https://openalex.org/W3006210518","https://openalex.org/W2406619057"],"abstract_inverted_index":{"Consider":[0],"the":[1,9],"combinational":[2],"complexity":[3,79],"$L(f)$":[4],"of":[5,30,54],"Boolean":[6],"functions":[7,55],"over":[8],"basis":[10],"$\\Omega":[11],"=":[12],"\\{":[13,18,66],"f|f:\\{":[14],"0,1\\}":[15,19,58,67],"^2":[16],"\\to":[17,65],"\\}":[20],"$.":[21],"A":[22],"new":[23],"method":[24],"for":[25,50],"proving":[26],"linear":[27],"lower":[28],"bounds":[29],"size":[31,82],"$2n$":[32],"is":[33,83],"presented.":[34],"Combining":[35],"it":[36],"with":[37],"methods":[38],"presented":[39],"in":[40],"Savage":[41],"[13,":[42],"(1974)]":[43],"and":[44,80],"Schnorr":[45],"[18,":[46],"(1974)],":[47],"we":[48],"establish":[49],"a":[51,74],"special":[52],"sequence":[53],"$f_n":[56],":\\{":[57],"^{n":[59],"+":[60,63],"2\\log":[61],"(n)":[62],"1}":[64],":2.5n":[68],"\\leqq":[69,71],"L(f)":[70],"6n$.":[72],"Also":[73],"trade-off":[75],"result":[76],"between":[77],"circuit":[78],"formula":[81],"derived.":[84]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2012,"cited_by_count":6}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}