{"id":"https://openalex.org/W4252625889","doi":"https://doi.org/10.4086/toc.2014.v010a007","title":"","display_name":"","publication_year":2014,"publication_date":"2014-01-01","ids":{"openalex":"https://openalex.org/W4252625889","doi":"https://doi.org/10.4086/toc.2014.v010a007"},"language":"en","primary_location":{"id":"doi:10.4086/toc.2014.v010a007","is_oa":true,"landing_page_url":"https://doi.org/10.4086/toc.2014.v010a007","pdf_url":"https://theoryofcomputing.org/articles/v010a007/v010a007.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":"https://theoryofcomputing.org/articles/v010a007/v010a007.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5101599971","display_name":"Jin\u2010Yi Cai","orcid":"https://orcid.org/0009-0003-0675-6060"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Jin-Yi Cai","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5065286337","display_name":"Aaron Gorenstein","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Aaron Gorenstein","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5101599971"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":3.2901,"has_fulltext":true,"cited_by_count":18,"citation_normalized_percentile":{"value":0.92693325,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"10","issue":"1","first_page":"167","last_page":"197"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9871000051498413,"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":0.9871000051498413,"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/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.9832000136375427,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12029","display_name":"DNA and Biological Computing","score":0.9830999970436096,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/gadget","display_name":"Gadget","score":0.8214026689529419},{"id":"https://openalex.org/keywords/signature","display_name":"Signature (topology)","score":0.6658745408058167},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5815384387969971},{"id":"https://openalex.org/keywords/character","display_name":"Character (mathematics)","score":0.5699005722999573},{"id":"https://openalex.org/keywords/equivalence","display_name":"Equivalence (formal languages)","score":0.5657380819320679},{"id":"https://openalex.org/keywords/direct-proof","display_name":"Direct proof","score":0.5481744408607483},{"id":"https://openalex.org/keywords/crossover","display_name":"Crossover","score":0.47054487466812134},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4048141837120056},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3948381543159485},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.36850398778915405},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.35101228952407837},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.24809840321540833},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.09253177046775818}],"concepts":[{"id":"https://openalex.org/C119770614","wikidata":"https://www.wikidata.org/wiki/Q5516347","display_name":"Gadget","level":2,"score":0.8214026689529419},{"id":"https://openalex.org/C2779696439","wikidata":"https://www.wikidata.org/wiki/Q7512811","display_name":"Signature (topology)","level":2,"score":0.6658745408058167},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5815384387969971},{"id":"https://openalex.org/C2780861071","wikidata":"https://www.wikidata.org/wiki/Q1062934","display_name":"Character (mathematics)","level":2,"score":0.5699005722999573},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.5657380819320679},{"id":"https://openalex.org/C161505775","wikidata":"https://www.wikidata.org/wiki/Q3240280","display_name":"Direct proof","level":2,"score":0.5481744408607483},{"id":"https://openalex.org/C122507166","wikidata":"https://www.wikidata.org/wiki/Q628906","display_name":"Crossover","level":2,"score":0.47054487466812134},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4048141837120056},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3948381543159485},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.36850398778915405},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.35101228952407837},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.24809840321540833},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.09253177046775818},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.4086/toc.2014.v010a007","is_oa":true,"landing_page_url":"https://doi.org/10.4086/toc.2014.v010a007","pdf_url":"https://theoryofcomputing.org/articles/v010a007/v010a007.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.2014.v010a007","is_oa":true,"landing_page_url":"https://doi.org/10.4086/toc.2014.v010a007","pdf_url":"https://theoryofcomputing.org/articles/v010a007/v010a007.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/G2064676845","display_name":"AF: Small: Counting Problems, Holographic Algorithms and Dichotomy Theorems","funder_award_id":"1217549","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G431418324","display_name":"Counting Problems and Dichotomy Theorems","funder_award_id":"0914969","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8924900563","display_name":null,"funder_award_id":"CCF-1217549","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4252625889.pdf","grobid_xml":"https://content.openalex.org/works/W4252625889.grobid-xml"},"referenced_works_count":27,"referenced_works":["https://openalex.org/W105846551","https://openalex.org/W1503285044","https://openalex.org/W1533712308","https://openalex.org/W1749688215","https://openalex.org/W1971935176","https://openalex.org/W2000217912","https://openalex.org/W2032292489","https://openalex.org/W2036481581","https://openalex.org/W2054043146","https://openalex.org/W2065813152","https://openalex.org/W2071268262","https://openalex.org/W2074783224","https://openalex.org/W2076090007","https://openalex.org/W2083443606","https://openalex.org/W2084341218","https://openalex.org/W2149409888","https://openalex.org/W2288926289","https://openalex.org/W2293541501","https://openalex.org/W2294003223","https://openalex.org/W2358643362","https://openalex.org/W2913798276","https://openalex.org/W2950739138","https://openalex.org/W2962703159","https://openalex.org/W3004436327","https://openalex.org/W4206603976","https://openalex.org/W4226347946","https://openalex.org/W6910244139"],"related_works":["https://openalex.org/W4252293060","https://openalex.org/W4212943538","https://openalex.org/W2475971442","https://openalex.org/W4312756098","https://openalex.org/W3194202522","https://openalex.org/W4319586723","https://openalex.org/W2206440698","https://openalex.org/W3091274767","https://openalex.org/W4295163867","https://openalex.org/W3205495028"],"abstract_inverted_index":{"We":[0],"study":[1],"a":[2,39,98,121],"collection":[3],"of":[4,12,20,27,29,46,97,107,109,123,141],"concepts":[5],"and":[6,23,64,74,129],"theorems":[7],"that":[8,57,85],"laid":[9],"the":[10,17,24,58,79,86,90,105,110,126,130],"foundation":[11],"matchgate":[13,68,113],"computation.":[14],"This":[15,70,101,115],"includes":[16],"signature":[18,127],"theory":[19,26,128,132],"planar":[21,32],"matchgates,":[22],"parallel":[25],"characters":[28],"not":[30,76],"necessarily":[31],"matchgates.":[33],"Our":[34,50],"aim":[35],"is":[36,72,102,116],"to":[37,119],"present":[38],"unified":[40],"and,":[41],"whenever":[42],"possible,":[43],"simplified":[44,95],"account":[45],"this":[47],"challenging":[48],"theory.":[49,81],"results":[51],"include:":[52],"(1)":[53],"A":[54,83,94,138],"direct":[55,139],"proof":[56,71,84,106,122],"Matchgate":[59],"Identities":[60],"(MGI)":[61],"are":[62],"necessary":[63],"sufficient":[65],"conditions":[66],"for":[67,112],"signatures.":[69,114,147],"self-contained":[73],"does":[75],"go":[77],"through":[78],"character":[80,131],"(2)":[82],"MGI":[87,111],"already":[88],"imply":[89],"Parity":[91],"Condition.":[92],"(3)":[93],"construction":[96,140],"crossover":[99],"gadget.":[100],"used":[103,118],"in":[104],"sufficiency":[108],"also":[117],"give":[120],"equivalence":[124],"between":[125],"which":[133],"permits":[134],"omittable":[135],"nodes.":[136],"(4)":[137],"matchgates":[142],"realizing":[143],"all":[144],"matchgate-realizable":[145],"symmetric":[146]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":1}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2022-05-12T00:00:00"}