{"id":"https://openalex.org/W2026143994","doi":"https://doi.org/10.1007/s000370050030","title":"Lower bounds for modular counting by circuits with modular gates","display_name":"Lower bounds for modular counting by circuits with modular gates","publication_year":1999,"publication_date":"1999-12-01","ids":{"openalex":"https://openalex.org/W2026143994","doi":"https://doi.org/10.1007/s000370050030","mag":"2026143994"},"language":"en","primary_location":{"id":"doi:10.1007/s000370050030","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s000370050030","pdf_url":null,"source":{"id":"https://openalex.org/S193995496","display_name":"Computational Complexity","issn_l":"1016-3328","issn":["1016-3328","1420-8954"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320186","host_organization_name":"Birkh\u00e4user","host_organization_lineage":["https://openalex.org/P4310320186","https://openalex.org/P4310319900"],"host_organization_lineage_names":["Birkh\u00e4user","Springer Science+Business Media"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Complexity","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/A5082512839","display_name":"David A. Mix Barrington","orcid":"https://orcid.org/0000-0002-5958-2792"},"institutions":[{"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":true,"raw_author_name":"D. A. Mix Barrington","raw_affiliation_strings":["Computer Science Department, University of Massachusetts, Amherst, MA 01003, USA, e-mail: barrington@cs.umass.edu, , , , , , US","COINS Department, University of Massachusetts, Amherst, USA"],"affiliations":[{"raw_affiliation_string":"Computer Science Department, University of Massachusetts, Amherst, MA 01003, USA, e-mail: barrington@cs.umass.edu, , , , , , US","institution_ids":["https://openalex.org/I24603500"]},{"raw_affiliation_string":"COINS Department, University of Massachusetts, Amherst, USA","institution_ids":["https://openalex.org/I24603500"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5084561937","display_name":"Howard Straubing","orcid":null},"institutions":[{"id":"https://openalex.org/I103531236","display_name":"Boston College","ror":"https://ror.org/02n2fzt79","country_code":"US","type":"education","lineage":["https://openalex.org/I103531236"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"H. Straubing","raw_affiliation_strings":["Computer Science Department, Boston College, Chestnut Hill, MA 02167, USA, e-mail: straubing@bc.edu, , , , , , US","Computer Science Dept., Boston College, Chestnut Hill, USA"],"affiliations":[{"raw_affiliation_string":"Computer Science Department, Boston College, Chestnut Hill, MA 02167, USA, e-mail: straubing@bc.edu, , , , , , US","institution_ids":["https://openalex.org/I103531236"]},{"raw_affiliation_string":"Computer Science Dept., Boston College, Chestnut Hill, USA","institution_ids":["https://openalex.org/I103531236"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5082512839"],"corresponding_institution_ids":["https://openalex.org/I24603500"],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.16203743,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"8","issue":"3","first_page":"258","last_page":"272"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9998000264167786,"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":0.9998000264167786,"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":0.9976999759674072,"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":0.9975000023841858,"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/constant","display_name":"Constant (computer programming)","score":0.6449334025382996},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6288353204727173},{"id":"https://openalex.org/keywords/modular-design","display_name":"Modular design","score":0.6123449802398682},{"id":"https://openalex.org/keywords/electronic-circuit","display_name":"Electronic circuit","score":0.6024898886680603},{"id":"https://openalex.org/keywords/prime","display_name":"Prime (order theory)","score":0.5845926403999329},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.5246003866195679},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.522961437702179},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5058788657188416},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.47684016823768616},{"id":"https://openalex.org/keywords/topology","display_name":"Topology (electrical circuits)","score":0.3862679898738861},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.34011760354042053},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2782977819442749},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.17171263694763184},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.10483798384666443},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.08413678407669067}],"concepts":[{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.6449334025382996},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6288353204727173},{"id":"https://openalex.org/C101468663","wikidata":"https://www.wikidata.org/wiki/Q1620158","display_name":"Modular design","level":2,"score":0.6123449802398682},{"id":"https://openalex.org/C134146338","wikidata":"https://www.wikidata.org/wiki/Q1815901","display_name":"Electronic circuit","level":2,"score":0.6024898886680603},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.5845926403999329},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.5246003866195679},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.522961437702179},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5058788657188416},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.47684016823768616},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.3862679898738861},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.34011760354042053},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2782977819442749},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.17171263694763184},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.10483798384666443},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.08413678407669067},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","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/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s000370050030","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s000370050030","pdf_url":null,"source":{"id":"https://openalex.org/S193995496","display_name":"Computational Complexity","issn_l":"1016-3328","issn":["1016-3328","1420-8954"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320186","host_organization_name":"Birkh\u00e4user","host_organization_lineage":["https://openalex.org/P4310320186","https://openalex.org/P4310319900"],"host_organization_lineage_names":["Birkh\u00e4user","Springer Science+Business Media"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Complexity","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":9,"referenced_works":["https://openalex.org/W200063938","https://openalex.org/W1500453128","https://openalex.org/W1982818703","https://openalex.org/W2036656440","https://openalex.org/W2060270693","https://openalex.org/W2062223744","https://openalex.org/W2070678975","https://openalex.org/W2097575812","https://openalex.org/W2913344232"],"related_works":["https://openalex.org/W1978042415","https://openalex.org/W2185671459","https://openalex.org/W2404664685","https://openalex.org/W2010720345","https://openalex.org/W4234591301","https://openalex.org/W2976111252","https://openalex.org/W2893190530","https://openalex.org/W1614158749","https://openalex.org/W2128510017","https://openalex.org/W1516895011"],"abstract_inverted_index":null,"counts_by_year":[{"year":2016,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}