{"id":"https://openalex.org/W1966288006","doi":"https://doi.org/10.1007/978-3-540-87531-4_7","title":"Extensional Uniformity for Boolean Circuits","display_name":"Extensional Uniformity for Boolean Circuits","publication_year":2008,"publication_date":"2008-01-01","ids":{"openalex":"https://openalex.org/W1966288006","doi":"https://doi.org/10.1007/978-3-540-87531-4_7","mag":"1966288006"},"language":"en","primary_location":{"id":"doi:10.1007/978-3-540-87531-4_7","is_oa":false,"landing_page_url":"https://doi.org/10.1007/978-3-540-87531-4_7","pdf_url":null,"source":{"id":"https://openalex.org/S106296714","display_name":"Lecture notes in computer science","issn_l":"0302-9743","issn":["0302-9743","1611-3349"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"book series"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Lecture Notes in Computer Science","raw_type":"book-chapter"},"type":"conference-paper","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/A5039352699","display_name":"Pierre McKenzie","orcid":"https://orcid.org/0000-0002-2483-8305"},"institutions":[{"id":"https://openalex.org/I159129438","display_name":"Universit\u00e9 du Qu\u00e9bec \u00e0 Montr\u00e9al","ror":"https://ror.org/002rjbv21","country_code":"CA","type":"education","lineage":["https://openalex.org/I159129438","https://openalex.org/I49663120"]},{"id":"https://openalex.org/I4210111842","display_name":"Computer Research Institute of Montr\u00e9al","ror":"https://ror.org/0279d5115","country_code":"CA","type":"nonprofit","lineage":["https://openalex.org/I4210111842"]},{"id":"https://openalex.org/I4210161954","display_name":"D\u00e9partement d'Informatique","ror":"https://ror.org/05y6rqs46","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I1326498283","https://openalex.org/I2746051580","https://openalex.org/I29607241","https://openalex.org/I4210159245","https://openalex.org/I4210161954"]},{"id":"https://openalex.org/I70931966","display_name":"Universit\u00e9 de Montr\u00e9al","ror":"https://ror.org/0161xgx34","country_code":"CA","type":"education","lineage":["https://openalex.org/I70931966"]}],"countries":["CA","FR"],"is_corresponding":false,"raw_author_name":"Pierre McKenzie","raw_affiliation_strings":["D\u00e9p. d\u2019informatique et de recherche op\u00e9rationnelle, Universit\u00e9 de Montr\u00e9al, C.P. 6128, succ. Centre-Ville, Montr\u00e9al (Qu\u00e9bec), H3C 3J7, Canada","D\u00e9p. d'informatique et de recherche op\u00e9rationnelle, Universit\u00e9 de Montr\u00e9al, Montr\u00e9al (Qu\u00e9bec), Canada H3C 3J7#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"D\u00e9p. d\u2019informatique et de recherche op\u00e9rationnelle, Universit\u00e9 de Montr\u00e9al, C.P. 6128, succ. Centre-Ville, Montr\u00e9al (Qu\u00e9bec), H3C 3J7, Canada","institution_ids":["https://openalex.org/I159129438","https://openalex.org/I4210111842","https://openalex.org/I4210161954","https://openalex.org/I70931966"]},{"raw_affiliation_string":"D\u00e9p. d'informatique et de recherche op\u00e9rationnelle, Universit\u00e9 de Montr\u00e9al, Montr\u00e9al (Qu\u00e9bec), Canada H3C 3J7#TAB#","institution_ids":["https://openalex.org/I70931966"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5081736632","display_name":"Michael E. Thomas","orcid":"https://orcid.org/0009-0004-2121-6983"},"institutions":[{"id":"https://openalex.org/I114112103","display_name":"Leibniz University Hannover","ror":"https://ror.org/0304hq317","country_code":"DE","type":"education","lineage":["https://openalex.org/I114112103"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Michael Thomas","raw_affiliation_strings":["Institut f\u00fcr Theoretische Informatik, Leibniz Universit\u00e4t Hannover, Appelstr. 4, 30167, Hannover, Germany","Institut f\u00fcr Theoretische Informatik, Leibniz Universit\u00e4t Hannover, Hannover, Germany 30167#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Theoretische Informatik, Leibniz Universit\u00e4t Hannover, Appelstr. 4, 30167, Hannover, Germany","institution_ids":["https://openalex.org/I114112103"]},{"raw_affiliation_string":"Institut f\u00fcr Theoretische Informatik, Leibniz Universit\u00e4t Hannover, Hannover, Germany 30167#TAB#","institution_ids":["https://openalex.org/I114112103"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5008982209","display_name":"Heribert Vollmer","orcid":"https://orcid.org/0000-0002-9292-1960"},"institutions":[{"id":"https://openalex.org/I114112103","display_name":"Leibniz University Hannover","ror":"https://ror.org/0304hq317","country_code":"DE","type":"education","lineage":["https://openalex.org/I114112103"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Heribert Vollmer","raw_affiliation_strings":["Institut f\u00fcr Theoretische Informatik, Leibniz Universit\u00e4t Hannover, Appelstr. 4, 30167, Hannover, Germany","Institut f\u00fcr Theoretische Informatik, Leibniz Universit\u00e4t Hannover, Hannover, Germany 30167#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Theoretische Informatik, Leibniz Universit\u00e4t Hannover, Appelstr. 4, 30167, Hannover, Germany","institution_ids":["https://openalex.org/I114112103"]},{"raw_affiliation_string":"Institut f\u00fcr Theoretische Informatik, Leibniz Universit\u00e4t Hannover, Hannover, Germany 30167#TAB#","institution_ids":["https://openalex.org/I114112103"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":5,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":{"value":5000,"currency":"EUR","value_usd":5392},"apc_paid":null,"fwci":0.2856,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.45407098,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":"64","last_page":"78"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10142","display_name":"Formal Methods in Verification","score":0.9995999932289124,"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/T10142","display_name":"Formal Methods in Verification","score":0.9995999932289124,"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":0.9991000294685364,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/circuit-complexity","display_name":"Circuit complexity","score":0.6852753758430481},{"id":"https://openalex.org/keywords/duality","display_name":"Duality (order theory)","score":0.6847195625305176},{"id":"https://openalex.org/keywords/extensional-definition","display_name":"Extensional definition","score":0.6688230037689209},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.6252548694610596},{"id":"https://openalex.org/keywords/connection","display_name":"Connection (principal bundle)","score":0.5704692006111145},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.5581659078598022},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.5320682525634766},{"id":"https://openalex.org/keywords/property","display_name":"Property (philosophy)","score":0.5213248133659363},{"id":"https://openalex.org/keywords/coupling","display_name":"Coupling (piping)","score":0.478824645280838},{"id":"https://openalex.org/keywords/boolean-circuit","display_name":"Boolean circuit","score":0.4735114872455597},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.45367687940597534},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.44561058282852173},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.43149763345718384},{"id":"https://openalex.org/keywords/electronic-circuit","display_name":"Electronic circuit","score":0.4307723045349121},{"id":"https://openalex.org/keywords/complexity-class","display_name":"Complexity class","score":0.4136173129081726},{"id":"https://openalex.org/keywords/boolean-function","display_name":"Boolean function","score":0.3732464909553528},{"id":"https://openalex.org/keywords/topology","display_name":"Topology (electrical circuits)","score":0.3730412721633911},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3636612296104431},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.2569405436515808},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.17575249075889587},{"id":"https://openalex.org/keywords/time-complexity","display_name":"Time complexity","score":0.1454433798789978},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.13229501247406006},{"id":"https://openalex.org/keywords/materials-science","display_name":"Materials science","score":0.11654800176620483},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.11375734210014343},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.09146296977996826},{"id":"https://openalex.org/keywords/composite-material","display_name":"Composite material","score":0.06056302785873413}],"concepts":[{"id":"https://openalex.org/C90702460","wikidata":"https://www.wikidata.org/wiki/Q1055112","display_name":"Circuit complexity","level":3,"score":0.6852753758430481},{"id":"https://openalex.org/C2778023678","wikidata":"https://www.wikidata.org/wiki/Q554403","display_name":"Duality (order theory)","level":2,"score":0.6847195625305176},{"id":"https://openalex.org/C136137745","wikidata":"https://www.wikidata.org/wiki/Q19515659","display_name":"Extensional definition","level":3,"score":0.6688230037689209},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.6252548694610596},{"id":"https://openalex.org/C13355873","wikidata":"https://www.wikidata.org/wiki/Q2920850","display_name":"Connection (principal bundle)","level":2,"score":0.5704692006111145},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.5581659078598022},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5320682525634766},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.5213248133659363},{"id":"https://openalex.org/C131584629","wikidata":"https://www.wikidata.org/wiki/Q4308705","display_name":"Coupling (piping)","level":2,"score":0.478824645280838},{"id":"https://openalex.org/C141796577","wikidata":"https://www.wikidata.org/wiki/Q837479","display_name":"Boolean circuit","level":3,"score":0.4735114872455597},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.45367687940597534},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.44561058282852173},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.43149763345718384},{"id":"https://openalex.org/C134146338","wikidata":"https://www.wikidata.org/wiki/Q1815901","display_name":"Electronic circuit","level":2,"score":0.4307723045349121},{"id":"https://openalex.org/C39637292","wikidata":"https://www.wikidata.org/wiki/Q908207","display_name":"Complexity class","level":3,"score":0.4136173129081726},{"id":"https://openalex.org/C187455244","wikidata":"https://www.wikidata.org/wiki/Q942353","display_name":"Boolean function","level":2,"score":0.3732464909553528},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.3730412721633911},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3636612296104431},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.2569405436515808},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.17575249075889587},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.1454433798789978},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.13229501247406006},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.11654800176620483},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.11375734210014343},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.09146296977996826},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.06056302785873413},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","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},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C77928131","wikidata":"https://www.wikidata.org/wiki/Q193343","display_name":"Tectonics","level":2,"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":4,"locations":[{"id":"doi:10.1007/978-3-540-87531-4_7","is_oa":false,"landing_page_url":"https://doi.org/10.1007/978-3-540-87531-4_7","pdf_url":null,"source":{"id":"https://openalex.org/S106296714","display_name":"Lecture notes in computer science","issn_l":"0302-9743","issn":["0302-9743","1611-3349"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"book series"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Lecture Notes in Computer Science","raw_type":"book-chapter"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.312.6939","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.312.6939","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/0805.4072v2.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.313.1631","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.313.1631","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/0805.4072v1.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.683.2300","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.683.2300","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.thi.uni-hannover.de/fileadmin/forschung/publikationen/daten/mc-th-vo10.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":30,"referenced_works":["https://openalex.org/W41330591","https://openalex.org/W200063938","https://openalex.org/W299519334","https://openalex.org/W587223573","https://openalex.org/W1490189322","https://openalex.org/W1505482284","https://openalex.org/W1522715372","https://openalex.org/W1607580303","https://openalex.org/W1983845295","https://openalex.org/W1988899858","https://openalex.org/W2016206602","https://openalex.org/W2027501230","https://openalex.org/W2036308939","https://openalex.org/W2039393392","https://openalex.org/W2045583484","https://openalex.org/W2055874330","https://openalex.org/W2078123020","https://openalex.org/W2088141885","https://openalex.org/W2095274905","https://openalex.org/W2096648167","https://openalex.org/W2099197756","https://openalex.org/W2144337654","https://openalex.org/W2147340600","https://openalex.org/W2148224873","https://openalex.org/W2148302312","https://openalex.org/W2151033407","https://openalex.org/W2294009094","https://openalex.org/W2911599249","https://openalex.org/W2913344232","https://openalex.org/W4238852104"],"related_works":["https://openalex.org/W3204916073","https://openalex.org/W2009341260","https://openalex.org/W2055594799","https://openalex.org/W2403507963","https://openalex.org/W2611822924","https://openalex.org/W2486254595","https://openalex.org/W2799890067","https://openalex.org/W4286972813","https://openalex.org/W3202331273","https://openalex.org/W1492165320"],"abstract_inverted_index":null,"counts_by_year":[],"updated_date":"2026-08-26T07:47:46.906454","created_date":"2025-10-10T00:00:00"}