{"id":"https://openalex.org/W1576431088","doi":"https://doi.org/10.1007/3-540-48523-6_12","title":"Bounded Depth Arithmetic Circuits: Counting and Closure","display_name":"Bounded Depth Arithmetic Circuits: Counting and Closure","publication_year":1999,"publication_date":"1999-01-01","ids":{"openalex":"https://openalex.org/W1576431088","doi":"https://doi.org/10.1007/3-540-48523-6_12","mag":"1576431088"},"language":"en","primary_location":{"id":"doi:10.1007/3-540-48523-6_12","is_oa":false,"landing_page_url":"https://doi.org/10.1007/3-540-48523-6_12","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":"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/A5078734824","display_name":"Eric Allender","orcid":"https://orcid.org/0000-0002-0650-028X"},"institutions":[{"id":"https://openalex.org/I102322142","display_name":"Rutgers, The State University of New Jersey","ror":"https://ror.org/05vt9qd57","country_code":"US","type":"education","lineage":["https://openalex.org/I102322142"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Eric Allender","raw_affiliation_strings":["Department of Computer Science, Rutgers University, Piscataway, USA","Rutgers, University"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science, Rutgers University, Piscataway, USA","institution_ids":["https://openalex.org/I102322142"]},{"raw_affiliation_string":"Rutgers, University","institution_ids":["https://openalex.org/I102322142"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5056318386","display_name":"Samir Datta","orcid":"https://orcid.org/0000-0003-2196-2308"},"institutions":[{"id":"https://openalex.org/I102322142","display_name":"Rutgers, The State University of New Jersey","ror":"https://ror.org/05vt9qd57","country_code":"US","type":"education","lineage":["https://openalex.org/I102322142"]},{"id":"https://openalex.org/I2803209242","display_name":"University of California System","ror":"https://ror.org/00pjdza24","country_code":"US","type":"education","lineage":["https://openalex.org/I2803209242"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Samir Datta","raw_affiliation_strings":["Department of Computer Science, Rutgers University, Piscataway, USA","University of California"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science, Rutgers University, Piscataway, USA","institution_ids":["https://openalex.org/I102322142"]},{"raw_affiliation_string":"University of California","institution_ids":["https://openalex.org/I2803209242"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5045669197","display_name":"Andris Ambainis","orcid":"https://orcid.org/0000-0002-8716-001X"},"institutions":[{"id":"https://openalex.org/I95457486","display_name":"University of California, Berkeley","ror":"https://ror.org/01an7q238","country_code":"US","type":"education","lineage":["https://openalex.org/I95457486"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Andris Ambainis","raw_affiliation_strings":["Computer Science Division, University of California, Berkeley, USA","University of Massachusetts"],"affiliations":[{"raw_affiliation_string":"Computer Science Division, University of California, Berkeley, USA","institution_ids":["https://openalex.org/I95457486"]},{"raw_affiliation_string":"University of Massachusetts","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5082512839","display_name":"David A. Mix Barrington","orcid":"https://orcid.org/0000-0002-5958-2792"},"institutions":[{"id":"https://openalex.org/I102322142","display_name":"Rutgers, The State University of New Jersey","ror":"https://ror.org/05vt9qd57","country_code":"US","type":"education","lineage":["https://openalex.org/I102322142"]},{"id":"https://openalex.org/I24603500","display_name":"University of Massachusetts Amherst","ror":"https://ror.org/0072zz521","country_code":"US","type":"education","lineage":["https://openalex.org/I24603500"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"David A. Mix Barrington","raw_affiliation_strings":["Department of Computer Science, University of Massachusetts, Amherst, USA","Rutgers, University"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science, University of Massachusetts, Amherst, USA","institution_ids":["https://openalex.org/I24603500"]},{"raw_affiliation_string":"Rutgers, University","institution_ids":["https://openalex.org/I102322142"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5062023561","display_name":"Huong L\u00eaThanh","orcid":null},"institutions":[{"id":"https://openalex.org/I102197404","display_name":"Universit\u00e9 Paris-Sud","ror":"https://ror.org/028rypz17","country_code":"FR","type":"education","lineage":["https://openalex.org/I102197404"]},{"id":"https://openalex.org/I204730241","display_name":"Universit\u00e9 Paris Cit\u00e9","ror":"https://ror.org/05f82e368","country_code":"FR","type":"education","lineage":["https://openalex.org/I204730241"]},{"id":"https://openalex.org/I4210144804","display_name":"Laboratoire de Recherche en Informatique","ror":"https://ror.org/04e3ktk27","country_code":"FR","type":"facility","lineage":["https://openalex.org/I102197404","https://openalex.org/I1294671590","https://openalex.org/I1326498283","https://openalex.org/I4210144804","https://openalex.org/I4210159245"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Huong L\u00eaThanh","raw_affiliation_strings":["Laboratoire de Recherche en Informatique, Universit\u00e9 de Paris-Sud, Paris, France","Universit\u00e9 de Paris-Sud"],"affiliations":[{"raw_affiliation_string":"Laboratoire de Recherche en Informatique, Universit\u00e9 de Paris-Sud, Paris, France","institution_ids":["https://openalex.org/I4210144804","https://openalex.org/I204730241"]},{"raw_affiliation_string":"Universit\u00e9 de Paris-Sud","institution_ids":["https://openalex.org/I102197404","https://openalex.org/I204730241"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5078734824"],"corresponding_institution_ids":["https://openalex.org/I102322142"],"apc_list":{"value":5000,"currency":"EUR","value_usd":5392},"apc_paid":null,"fwci":0.4371,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.64555954,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"6","issue":null,"first_page":"149","last_page":"158"},"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/T10374","display_name":"Advanced Graph Theory Research","score":0.9991999864578247,"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":0.9986000061035156,"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/bounded-function","display_name":"Bounded function","score":0.7666763067245483},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6660555005073547},{"id":"https://openalex.org/keywords/closure","display_name":"Closure (psychology)","score":0.6610537171363831},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5690873861312866},{"id":"https://openalex.org/keywords/electronic-circuit","display_name":"Electronic circuit","score":0.5393815040588379},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4931454062461853},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.4689846634864807},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.41851842403411865},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3762992024421692},{"id":"https://openalex.org/keywords/topology","display_name":"Topology (electrical circuits)","score":0.33734405040740967},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2500229477882385},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.08010560274124146},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.06491008400917053}],"concepts":[{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.7666763067245483},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6660555005073547},{"id":"https://openalex.org/C146834321","wikidata":"https://www.wikidata.org/wiki/Q2979672","display_name":"Closure (psychology)","level":2,"score":0.6610537171363831},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5690873861312866},{"id":"https://openalex.org/C134146338","wikidata":"https://www.wikidata.org/wiki/Q1815901","display_name":"Electronic circuit","level":2,"score":0.5393815040588379},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4931454062461853},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.4689846634864807},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.41851842403411865},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3762992024421692},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.33734405040740967},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2500229477882385},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.08010560274124146},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.06491008400917053},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"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/C34447519","wikidata":"https://www.wikidata.org/wiki/Q179522","display_name":"Market economy","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":8,"locations":[{"id":"doi:10.1007/3-540-48523-6_12","is_oa":false,"landing_page_url":"https://doi.org/10.1007/3-540-48523-6_12","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:scholarworks.umass.edu:cs_faculty_pubs-1087","is_oa":false,"landing_page_url":"https://scholarworks.umass.edu/context/cs_faculty_pubs/article/1087/viewcontent/Eric_Allender.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306402057","display_name":"Scholarworks (University of Massachusetts Amherst)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I24603500","host_organization_name":"University of Massachusetts Amherst","host_organization_lineage":["https://openalex.org/I24603500"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Computer Science Department Faculty Publication Series","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.3.373","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.3.373","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://ftp.cs.rutgers.edu/pub/allender/icalp99.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.35.6303","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.35.6303","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.lri.fr/~huong/Publi/icalp067.ps","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.45.5525","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.45.5525","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ftp://ftp.cs.rutgers.edu/pub/allender/icalp99.ps.gz","raw_type":"text"},{"id":"pmh:oai:alma.01RUT_INST:11663476840004646","is_oa":false,"landing_page_url":"https://scholarship.libraries.rutgers.edu/esploro/outputs/bookChapter/Bounded-Depth-Arithmetic-Circuits-Counting-and/991031654514504646","pdf_url":null,"source":{"id":"https://openalex.org/S4210197018","display_name":"View","issn_l":"2688-268X","issn":["2688-268X","2688-3988"],"is_oa":false,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Book Section"},{"id":"pmh:oai:scholarworks.umass.edu:20.500.14394/10471","is_oa":false,"landing_page_url":"https://hdl.handle.net/20.500.14394/10471","pdf_url":null,"source":{"id":"https://openalex.org/S4306402240","display_name":"ScholarWorks@UMassAmherst (University of Massachusetts Amherst)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I24603500","host_organization_name":"University of Massachusetts Amherst","host_organization_lineage":["https://openalex.org/I24603500"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"published","raw_type":"Article"},{"id":"mag:1576431088","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":null}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":27,"referenced_works":["https://openalex.org/W1486976598","https://openalex.org/W1495935433","https://openalex.org/W1592302951","https://openalex.org/W1592405134","https://openalex.org/W1623271446","https://openalex.org/W1878491781","https://openalex.org/W1984509356","https://openalex.org/W1984534637","https://openalex.org/W1990538552","https://openalex.org/W2006912660","https://openalex.org/W2007535638","https://openalex.org/W2024994589","https://openalex.org/W2044997341","https://openalex.org/W2063584607","https://openalex.org/W2070678975","https://openalex.org/W2080370600","https://openalex.org/W2083553352","https://openalex.org/W2085012646","https://openalex.org/W2104952158","https://openalex.org/W2105859078","https://openalex.org/W2123466414","https://openalex.org/W2152708687","https://openalex.org/W2165293955","https://openalex.org/W2171197316","https://openalex.org/W2580001362","https://openalex.org/W2580193495","https://openalex.org/W4240781210"],"related_works":["https://openalex.org/W2738243513","https://openalex.org/W2178086598","https://openalex.org/W1496129971","https://openalex.org/W2785960849","https://openalex.org/W1982984553","https://openalex.org/W1992071680","https://openalex.org/W2058474793","https://openalex.org/W2167660903","https://openalex.org/W1566003175","https://openalex.org/W302036035","https://openalex.org/W2765993145","https://openalex.org/W2150144243","https://openalex.org/W2460503887","https://openalex.org/W1601605622","https://openalex.org/W250538733","https://openalex.org/W2394746612","https://openalex.org/W2537001886","https://openalex.org/W1880954240","https://openalex.org/W2952025700","https://openalex.org/W2571401227"],"abstract_inverted_index":null,"counts_by_year":[],"updated_date":"2026-03-25T23:56:10.502304","created_date":"2025-10-10T00:00:00"}
