{"id":"https://openalex.org/W1984477611","doi":"https://doi.org/10.1145/1490270.1490272","title":"Algebrization","display_name":"Algebrization","publication_year":2009,"publication_date":"2009-02-01","ids":{"openalex":"https://openalex.org/W1984477611","doi":"https://doi.org/10.1145/1490270.1490272","mag":"1984477611"},"language":"en","primary_location":{"id":"doi:10.1145/1490270.1490272","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1490270.1490272","pdf_url":null,"source":{"id":"https://openalex.org/S205019203","display_name":"ACM Transactions on Computation Theory","issn_l":"1942-3454","issn":["1942-3454","1942-3462"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319798","host_organization_name":"Association for Computing Machinery","host_organization_lineage":["https://openalex.org/P4310319798"],"host_organization_lineage_names":["Association for Computing Machinery"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"ACM Transactions on Computation Theory","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/A5020769134","display_name":"Scott Aaronson","orcid":null},"institutions":[{"id":"https://openalex.org/I4210109586","display_name":"Moscow Institute of Thermal Technology","ror":"https://ror.org/021es5e59","country_code":"RU","type":"facility","lineage":["https://openalex.org/I4210109586"]}],"countries":["RU"],"is_corresponding":true,"raw_author_name":"Scott Aaronson","raw_affiliation_strings":["MIT"],"affiliations":[{"raw_affiliation_string":"MIT","institution_ids":["https://openalex.org/I4210109586"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5039043184","display_name":"Avi Wigderson","orcid":"https://orcid.org/0000-0002-1539-1417"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Avi Wigderson","raw_affiliation_strings":["Institute for Advanced Study"],"affiliations":[{"raw_affiliation_string":"Institute for Advanced Study","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5020769134"],"corresponding_institution_ids":["https://openalex.org/I4210109586"],"apc_list":null,"apc_paid":null,"fwci":7.1474,"has_fulltext":false,"cited_by_count":156,"citation_normalized_percentile":{"value":0.97305735,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":98,"max":100},"biblio":{"volume":"1","issue":"1","first_page":"1","last_page":"54"},"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/T10237","display_name":"Cryptography and Data Security","score":0.9997000098228455,"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/T11797","display_name":"graph theory and CDMA systems","score":0.9966999888420105,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/complexity-class","display_name":"Complexity class","score":0.6929078698158264},{"id":"https://openalex.org/keywords/mathematical-proof","display_name":"Mathematical proof","score":0.5924282670021057},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5691025853157043},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.47889676690101624},{"id":"https://openalex.org/keywords/oracle","display_name":"Oracle","score":0.47809550166130066},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.46458250284194946},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.45680540800094604},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.44562190771102905},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.43623408675193787},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.4336010813713074},{"id":"https://openalex.org/keywords/descriptive-complexity-theory","display_name":"Descriptive complexity theory","score":0.4267251491546631},{"id":"https://openalex.org/keywords/pspace","display_name":"PSPACE","score":0.4170491099357605},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.41217195987701416},{"id":"https://openalex.org/keywords/computational-complexity-theory","display_name":"Computational complexity theory","score":0.36841967701911926},{"id":"https://openalex.org/keywords/time-complexity","display_name":"Time complexity","score":0.3299635052680969},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3193141520023346},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.20817089080810547},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.1306762993335724}],"concepts":[{"id":"https://openalex.org/C39637292","wikidata":"https://www.wikidata.org/wiki/Q908207","display_name":"Complexity class","level":3,"score":0.6929078698158264},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.5924282670021057},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5691025853157043},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.47889676690101624},{"id":"https://openalex.org/C55166926","wikidata":"https://www.wikidata.org/wiki/Q2892946","display_name":"Oracle","level":2,"score":0.47809550166130066},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.46458250284194946},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.45680540800094604},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.44562190771102905},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.43623408675193787},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4336010813713074},{"id":"https://openalex.org/C133142742","wikidata":"https://www.wikidata.org/wiki/Q1200755","display_name":"Descriptive complexity theory","level":3,"score":0.4267251491546631},{"id":"https://openalex.org/C197685441","wikidata":"https://www.wikidata.org/wiki/Q500716","display_name":"PSPACE","level":3,"score":0.4170491099357605},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.41217195987701416},{"id":"https://openalex.org/C179799912","wikidata":"https://www.wikidata.org/wiki/Q205084","display_name":"Computational complexity theory","level":2,"score":0.36841967701911926},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.3299635052680969},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3193141520023346},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.20817089080810547},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.1306762993335724},{"id":"https://openalex.org/C115903868","wikidata":"https://www.wikidata.org/wiki/Q80993","display_name":"Software engineering","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},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"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/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1145/1490270.1490272","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1490270.1490272","pdf_url":null,"source":{"id":"https://openalex.org/S205019203","display_name":"ACM Transactions on Computation Theory","issn_l":"1942-3454","issn":["1942-3454","1942-3462"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319798","host_organization_name":"Association for Computing Machinery","host_organization_lineage":["https://openalex.org/P4310319798"],"host_organization_lineage_names":["Association for Computing Machinery"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"ACM Transactions on Computation Theory","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"display_name":"Reduced inequalities","id":"https://metadata.un.org/sdg/10","score":0.7300000190734863}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":45,"referenced_works":["https://openalex.org/W50958263","https://openalex.org/W188525652","https://openalex.org/W1205694693","https://openalex.org/W1484001631","https://openalex.org/W1979479253","https://openalex.org/W1980223785","https://openalex.org/W1991730713","https://openalex.org/W1993138363","https://openalex.org/W2011112377","https://openalex.org/W2012496360","https://openalex.org/W2025279645","https://openalex.org/W2045876200","https://openalex.org/W2045924252","https://openalex.org/W2049016755","https://openalex.org/W2053590396","https://openalex.org/W2060270693","https://openalex.org/W2062767125","https://openalex.org/W2062934582","https://openalex.org/W2070486524","https://openalex.org/W2070678975","https://openalex.org/W2082647621","https://openalex.org/W2085025327","https://openalex.org/W2088492763","https://openalex.org/W2092018479","https://openalex.org/W2092429268","https://openalex.org/W2097206148","https://openalex.org/W2114522915","https://openalex.org/W2119407990","https://openalex.org/W2125495920","https://openalex.org/W2130515620","https://openalex.org/W2137407595","https://openalex.org/W2140270919","https://openalex.org/W2152813177","https://openalex.org/W2156001253","https://openalex.org/W2158747250","https://openalex.org/W2163605617","https://openalex.org/W2169453948","https://openalex.org/W2293988196","https://openalex.org/W2296506534","https://openalex.org/W2487893106","https://openalex.org/W2612879297","https://openalex.org/W2912080987","https://openalex.org/W4256317935","https://openalex.org/W4285719527","https://openalex.org/W6628771293"],"related_works":["https://openalex.org/W2003513300","https://openalex.org/W2327776271","https://openalex.org/W1504200807","https://openalex.org/W4388327399","https://openalex.org/W2082636129","https://openalex.org/W3003471035","https://openalex.org/W1583470866","https://openalex.org/W2970311880","https://openalex.org/W2075449032","https://openalex.org/W1501278571"],"abstract_inverted_index":{"Any":[0],"proof":[1],"of":[2,45,108,132,174,225,235,255,306],"P":[3,179,182],"\u2260":[4,320],"NP":[5],"will":[6],"have":[7,23,33,211],"to":[8,52,97,104,128,198,285],"overcome":[9,37],"two":[10],"barriers:":[11],"relativization":[12,73],"and":[13,122,154,158,185,246,262,275],"natural":[14],"proofs":[15],".":[16,76,192],"Yet":[17],"over":[18,110],"the":[19,42,55,91,130,133,175,290],"last":[20],"decade,":[21],"we":[22,64,70,82,88,138,169,210,241,281],"seen":[24],"circuit":[25,213],"lower":[26,214,229,257,304],"bounds":[27,215,230,258],"(e.g.,":[28],"that":[29,36,140,171],"PP":[30],"does":[31],"not":[32,95,219],"linear-size":[34],"circuits)":[35],"both":[38],"barriers":[39],"simultaneously.":[40],"So":[41],"question":[43],"arises":[44],"whether":[46],"there":[47],"is":[48,79,248],"a":[49,67,105,111,232,269,303,311],"third":[50],"barrier":[51],"progress":[53,202],"on":[54,146],"central":[56],"questions":[57],"in":[58,125,231,243,250],"complexity":[59,85,126,238,313],"theory.":[60],"In":[61,193],"this":[62,244,279],"article,":[63],"present":[65],"such":[66,149,160],"barrier,":[68],"which":[69,240,247],"call":[71],"algebraic":[72,236,263],"or":[74,114],"algebrization":[75,135,196],"The":[77],"idea":[78],"that,":[80],"when":[81],"relativize":[83],"some":[84,194],"class":[86],"inclusion,":[87],"should":[89],"give":[90,286],"simulating":[92],"machine":[93],"access":[94],"only":[96],"an":[98,287],"oracle":[99],"A":[100,109],",":[101,239],"but":[102,218],"also":[103,283],"low-degree":[106],"extension":[107],"finite":[112],"field":[113],"ring.":[115],"We":[116],"systematically":[117],"go":[118],"through":[119],"basic":[120],"results":[121,144,226],"open":[123,177],"problems":[124],"theory":[127],"delineate":[129],"power":[131],"new":[134,233],"barrier.":[136],"First,":[137],"show":[139,170],"all":[141,173],"known":[142],"nonrelativizing":[143],"based":[145],"arithmetization---both":[147],"inclusions":[148],"as":[150,161,308,310],"IP":[151],"=":[152,156],"PSPACE":[153],"MIP":[155],"NEXP,":[157],"separations":[159],"MA":[162],"EXP":[163],"\u2284":[164],"P/poly---do":[165],"indeed":[166],"algebrize.":[167],"Second,":[168],"almost":[172],"major":[176],"problems---including":[178],"versus":[180,183,187],"NP,":[181],"RP,":[184],"NEXP":[186],"P/poly---will":[188],"require":[189],"non-algebrizing":[190],"techniques":[191],"cases,":[195],"seems":[197],"explain":[199],"exactly":[200],"why":[201,209],"stopped":[203],"where":[204],"it":[205],"did:":[206],"for":[207,216,220,289],"example,":[208],"superlinear":[212],"PromiseMA":[217],"NP.":[221,321],"Our":[222],"second":[223],"set":[224],"follows":[227],"from":[228,268],"model":[234,274],"query":[237],"introduce":[242],"article":[245],"interesting":[249],"its":[251],"own":[252],"right.":[253],"Some":[254],"our":[256,273],"use":[259],"direct":[260],"combinatorial":[261],"arguments,":[264],"while":[265],"others":[266],"stem":[267],"surprising":[270],"connection":[271],"between":[272],"communication":[276,300,312],"complexity.":[277],"Using":[278],"connection,":[280],"are":[282],"able":[284],"MA-protocol":[288],"Inner":[291],"Product":[292],"function":[293],"with":[294],"O":[295],"(\u221an":[296],"log":[297],"n":[298],")":[299],"(essentially":[301],"matching":[302],"bound":[305],"Klauck),":[307],"well":[309],"conjecture":[314],"whose":[315],"truth":[316],"would":[317],"imply":[318],"NL":[319]},"counts_by_year":[{"year":2025,"cited_by_count":8},{"year":2024,"cited_by_count":7},{"year":2023,"cited_by_count":5},{"year":2022,"cited_by_count":8},{"year":2021,"cited_by_count":6},{"year":2020,"cited_by_count":13},{"year":2019,"cited_by_count":10},{"year":2018,"cited_by_count":13},{"year":2017,"cited_by_count":11},{"year":2016,"cited_by_count":13},{"year":2015,"cited_by_count":11},{"year":2014,"cited_by_count":13},{"year":2013,"cited_by_count":17},{"year":2012,"cited_by_count":5}],"updated_date":"2026-04-04T08:04:53.788161","created_date":"2016-06-24T00:00:00"}
