{"id":"https://openalex.org/W1964013156","doi":"https://doi.org/10.1137/s0097539701417723","title":"Dimension in Complexity Classes","display_name":"Dimension in Complexity Classes","publication_year":2003,"publication_date":"2003-01-01","ids":{"openalex":"https://openalex.org/W1964013156","doi":"https://doi.org/10.1137/s0097539701417723","mag":"1964013156"},"language":"en","primary_location":{"id":"doi:10.1137/s0097539701417723","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0097539701417723","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Computing","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/A5064796448","display_name":"Jack H. Lutz","orcid":"https://orcid.org/0000-0003-1004-3891"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Jack H. Lutz","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5064796448"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":16.7184,"has_fulltext":false,"cited_by_count":139,"citation_normalized_percentile":{"value":0.99407937,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"32","issue":"5","first_page":"1236","last_page":"1259"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","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/T12002","display_name":"Computability, Logic, AI Algorithms","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/T11567","display_name":"semigroups and automata theory","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/T11269","display_name":"Algorithms and Data Compression","score":0.9962999820709229,"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/hausdorff-dimension","display_name":"Hausdorff dimension","score":0.7729716300964355},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6820769309997559},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6785973906517029},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.5900688767433167},{"id":"https://openalex.org/keywords/decidability","display_name":"Decidability","score":0.5619114637374878},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5466086268424988},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.4834086000919342},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.08667948842048645}],"concepts":[{"id":"https://openalex.org/C194198291","wikidata":"https://www.wikidata.org/wiki/Q565186","display_name":"Hausdorff dimension","level":2,"score":0.7729716300964355},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6820769309997559},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6785973906517029},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5900688767433167},{"id":"https://openalex.org/C153269930","wikidata":"https://www.wikidata.org/wiki/Q430001","display_name":"Decidability","level":2,"score":0.5619114637374878},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5466086268424988},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.4834086000919342},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.08667948842048645}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/s0097539701417723","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0097539701417723","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Computing","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.23.388","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.23.388","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.iastate.edu/~lutz/=PAPERS/dcc.ps","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.342.8666","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.342.8666","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/cs/0203016v1.pdf","raw_type":"text"}],"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/W1970972184","https://openalex.org/W1995279130","https://openalex.org/W2020728339","https://openalex.org/W2029071561","https://openalex.org/W2057621739","https://openalex.org/W2068422506","https://openalex.org/W2086968855","https://openalex.org/W2176419554","https://openalex.org/W2499081645"],"related_works":["https://openalex.org/W2157707209","https://openalex.org/W2354545140","https://openalex.org/W2053762111","https://openalex.org/W2440791454","https://openalex.org/W4297670065","https://openalex.org/W1618267488","https://openalex.org/W1543051890","https://openalex.org/W2611810425","https://openalex.org/W4205250428","https://openalex.org/W1563786920"],"abstract_inverted_index":{"A":[0],"theory":[1],"of":[2,13,22,42,62,78,101,126,135,143,166,173],"resource-bounded":[3],"dimension":[4,29,35,48,82,139,182],"is":[5,25,30,70,86],"developed":[6],"using":[7],"gales,":[8],"which":[9,85],"are":[10,107],"natural":[11],"generalizations":[12],"martingales.":[14],"When":[15],"the":[16,23,27,32,43,60,99,120,158],"resource":[17],"bound":[18],"$\\Delta$":[19,45],"(a":[20],"parameter":[21,44],"theory)":[24],"unrestricted,":[26],"resulting":[28],"precisely":[31],"classical":[33],"Hausdorff":[34],"(sometimes":[36],"called":[37],"\"fractal":[38],"dimension\").":[39],"Other":[40],"choices":[41],"yield":[46],"internal":[47],"theories":[49],"in":[50,59,83,147,184],"E,":[51],"E2,":[52],"ESPACE,":[53],"and":[54,58,146],"other":[55],"complexity":[56],"classes,":[57],"class":[61],"all":[63,127,136,167],"decidable":[64,169],"problems.":[65],"In":[66],"general,":[67],"if":[68],"$\\mathcal{C}$":[69],"such":[71],"a":[72,81,87],"class,":[73],"then":[74],"every":[75,110,150],"set":[76,121,159],"X":[77],"languages":[79,128,168],"has":[80,138,181],"$\\mathcal{C}$,":[84],"real":[88,111,151],"number":[89,112,152],"$\\dim":[90],"(X":[91],"\\mid":[92],"\\mathcal{C})":[93],"\\in":[94],"[0,":[95],"1]$.":[96],"Along":[97],"with":[98],"elements":[100],"this":[102],"theory,":[103],"two":[104],"preliminary":[105],"applications":[106],"presented:":[108],"For":[109,149],"$0":[113,153],"\\le":[114,116,154,156],"\\alpha":[115,155],"\\frac":[117,162,177],"1":[118],"2$,":[119],"${\\rm":[122,160],"FREQ}(\\le":[123],"\\alpha)$,":[124],"consisting":[125,165],"that":[129],"asymptotically":[130],"contain":[131],"at":[132,174],"most":[133,175],"$\\alpha$":[134,183],"strings,":[137],"$\\mathcal{H}(\\alpha)$---the":[140],"binary":[141],"entropy":[142],"$\\alpha$---in":[144],"E":[145],"E2.":[148],"1$,":[157],"SIZE}(\\alpha":[161],"{2^n}":[163,178],"n)$,":[164],"by":[170],"Boolean":[171],"circuits":[172],"$\\alpha":[176],"n$":[179],"gates,":[180],"ESPACE.":[185]},"counts_by_year":[{"year":2025,"cited_by_count":5},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":7},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":5},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":4},{"year":2012,"cited_by_count":5}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
