{"id":"https://openalex.org/W1971650774","doi":"https://doi.org/10.1137/s0097539702415950","title":"Counting Complexity Classes for Numeric Computations I: Semilinear Sets","display_name":"Counting Complexity Classes for Numeric Computations I: Semilinear Sets","publication_year":2003,"publication_date":"2003-01-01","ids":{"openalex":"https://openalex.org/W1971650774","doi":"https://doi.org/10.1137/s0097539702415950","mag":"1971650774"},"language":"en","primary_location":{"id":"doi:10.1137/s0097539702415950","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0097539702415950","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":true,"oa_status":"bronze","oa_url":"https://scholars.cityu.edu.hk/en/publications/counting-complexity-classes-for-numeric-computations-i-semilinear","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5029244957","display_name":"Peter B\u00fcrgisser","orcid":"https://orcid.org/0000-0001-8169-0514"},"institutions":[{"id":"https://openalex.org/I206945453","display_name":"Paderborn University","ror":"https://ror.org/058kzsd48","country_code":"DE","type":"education","lineage":["https://openalex.org/I206945453"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Peter B\u00fcrgisser","raw_affiliation_strings":["Universit\u00e4t Paderborn"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Universit\u00e4t Paderborn","institution_ids":["https://openalex.org/I206945453"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5079608792","display_name":"Felipe Cucker","orcid":"https://orcid.org/0000-0002-4569-3248"},"institutions":[{"id":"https://openalex.org/I530967","display_name":"Toronto Metropolitan University","ror":"https://ror.org/05g13zd79","country_code":"CA","type":"education","lineage":["https://openalex.org/I530967"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Felipe Cucker","raw_affiliation_strings":["Department of Mathematics"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics","institution_ids":["https://openalex.org/I530967"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.544,"has_fulltext":false,"cited_by_count":14,"citation_normalized_percentile":{"value":0.88566166,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"33","issue":"1","first_page":"227","last_page":"260"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9994000196456909,"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":0.9994000196456909,"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/T12002","display_name":"Computability, Logic, AI Algorithms","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/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.9988999962806702,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/betti-number","display_name":"Betti number","score":0.7777615785598755},{"id":"https://openalex.org/keywords/complexity-class","display_name":"Complexity class","score":0.7186298370361328},{"id":"https://openalex.org/keywords/completeness","display_name":"Completeness (order theory)","score":0.6573586463928223},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.6552774310112},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6153650879859924},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.5809834003448486},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.546213686466217},{"id":"https://openalex.org/keywords/characterization","display_name":"Characterization (materials science)","score":0.5248986482620239},{"id":"https://openalex.org/keywords/counting-problem","display_name":"Counting problem","score":0.48758643865585327},{"id":"https://openalex.org/keywords/eulers-formula","display_name":"Euler's formula","score":0.48314163088798523},{"id":"https://openalex.org/keywords/euler-characteristic","display_name":"Euler characteristic","score":0.4555659294128418},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.44314900040626526},{"id":"https://openalex.org/keywords/model-of-computation","display_name":"Model of computation","score":0.4388180673122406},{"id":"https://openalex.org/keywords/transfer","display_name":"Transfer (computing)","score":0.42056357860565186},{"id":"https://openalex.org/keywords/computational-complexity-theory","display_name":"Computational complexity theory","score":0.4188946783542633},{"id":"https://openalex.org/keywords/time-complexity","display_name":"Time complexity","score":0.32778263092041016},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.18317347764968872},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.17792049050331116},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.08559414744377136},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.07013782858848572}],"concepts":[{"id":"https://openalex.org/C129621563","wikidata":"https://www.wikidata.org/wiki/Q429593","display_name":"Betti number","level":2,"score":0.7777615785598755},{"id":"https://openalex.org/C39637292","wikidata":"https://www.wikidata.org/wiki/Q908207","display_name":"Complexity class","level":3,"score":0.7186298370361328},{"id":"https://openalex.org/C17231256","wikidata":"https://www.wikidata.org/wiki/Q5156540","display_name":"Completeness (order theory)","level":2,"score":0.6573586463928223},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.6552774310112},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6153650879859924},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.5809834003448486},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.546213686466217},{"id":"https://openalex.org/C2780841128","wikidata":"https://www.wikidata.org/wiki/Q5073781","display_name":"Characterization (materials science)","level":2,"score":0.5248986482620239},{"id":"https://openalex.org/C16592021","wikidata":"https://www.wikidata.org/wiki/Q5177154","display_name":"Counting problem","level":2,"score":0.48758643865585327},{"id":"https://openalex.org/C62884695","wikidata":"https://www.wikidata.org/wiki/Q184871","display_name":"Euler's formula","level":2,"score":0.48314163088798523},{"id":"https://openalex.org/C196433757","wikidata":"https://www.wikidata.org/wiki/Q852973","display_name":"Euler characteristic","level":2,"score":0.4555659294128418},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.44314900040626526},{"id":"https://openalex.org/C184596265","wikidata":"https://www.wikidata.org/wiki/Q2651576","display_name":"Model of computation","level":3,"score":0.4388180673122406},{"id":"https://openalex.org/C2776175482","wikidata":"https://www.wikidata.org/wiki/Q1195816","display_name":"Transfer (computing)","level":2,"score":0.42056357860565186},{"id":"https://openalex.org/C179799912","wikidata":"https://www.wikidata.org/wiki/Q205084","display_name":"Computational complexity theory","level":2,"score":0.4188946783542633},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.32778263092041016},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.18317347764968872},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.17792049050331116},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08559414744377136},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.07013782858848572},{"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/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.0},{"id":"https://openalex.org/C173608175","wikidata":"https://www.wikidata.org/wiki/Q232661","display_name":"Parallel computing","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/s0097539702415950","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0097539702415950","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"},{"is_oa":true,"landing_page_url":"https://scholars.cityu.edu.hk/en/publications/counting-complexity-classes-for-numeric-computations-i-semilinear","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:10.1.1.409.4033","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.409.4033","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://math-www.upb.de/agpb/work/2rev_add_count.pdf","raw_type":"text"},{"id":"pmh:oai:pure.atira.dk:publications/8191fa73-c85f-4964-9d8b-c1af8990e078","is_oa":true,"landing_page_url":"https://hdl.handle.net/2031/8191fa73-c85f-4964-9d8b-c1af8990e078","pdf_url":null,"source":{"id":"https://openalex.org/S7407055387","display_name":"CityU Scholars","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"B\u00fcrgisser, P & Cucker, F 2003, 'Counting complexity classes for numeric computations I : Semilinear sets', SIAM Journal on Computing, vol. 33, no. 1, pp. 227-260. https://doi.org/10.1137/S0097539702415950","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"is_oa":true,"landing_page_url":"https://scholars.cityu.edu.hk/en/publications/counting-complexity-classes-for-numeric-computations-i-semilinear","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"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":44,"referenced_works":["https://openalex.org/W1496383623","https://openalex.org/W1509580513","https://openalex.org/W1545819195","https://openalex.org/W1550331971","https://openalex.org/W1561337879","https://openalex.org/W1576854535","https://openalex.org/W1968206655","https://openalex.org/W1979572578","https://openalex.org/W1982538209","https://openalex.org/W1986662193","https://openalex.org/W1997744504","https://openalex.org/W2004419188","https://openalex.org/W2006912660","https://openalex.org/W2007014039","https://openalex.org/W2033040247","https://openalex.org/W2039302045","https://openalex.org/W2042106485","https://openalex.org/W2048353152","https://openalex.org/W2050190588","https://openalex.org/W2059985428","https://openalex.org/W2064463780","https://openalex.org/W2068645950","https://openalex.org/W2071846104","https://openalex.org/W2080928066","https://openalex.org/W2088141885","https://openalex.org/W2103985627","https://openalex.org/W2107326100","https://openalex.org/W2114861296","https://openalex.org/W2115204359","https://openalex.org/W2115826669","https://openalex.org/W2134424832","https://openalex.org/W2137407595","https://openalex.org/W2150121468","https://openalex.org/W2170831059","https://openalex.org/W2170975953","https://openalex.org/W2240977055","https://openalex.org/W2486762503","https://openalex.org/W2610884651","https://openalex.org/W2611147814","https://openalex.org/W3040586665","https://openalex.org/W4213009134","https://openalex.org/W4243643814","https://openalex.org/W4248249872","https://openalex.org/W4301168154"],"related_works":["https://openalex.org/W3153979038","https://openalex.org/W4319591071","https://openalex.org/W2070001083","https://openalex.org/W4213087886","https://openalex.org/W2963511174","https://openalex.org/W4309871294","https://openalex.org/W4298446668","https://openalex.org/W2623021320","https://openalex.org/W3011095105","https://openalex.org/W1992921769"],"abstract_inverted_index":{"We":[0,39,119],"define":[1],"a":[2,25],"counting":[3,32],"class":[4,21,33],"${\\rm":[5,88,105],"#P}_\\add$":[6],"in":[7,27,58,131],"the":[8,15,30,48,52,63,81,84,97,100,109,123,132],"Blum--Shub--Smale":[9],"setting":[10],"of":[11,19,29,60,65,70,83,99,122],"additive":[12,50,75],"computations":[13],"over":[14],"reals.":[16],"Structural":[17],"properties":[18],"this":[20],"are":[22],"studied,":[23],"including":[24],"characterization":[26],"terms":[28,59],"classical":[31,133],"$#{\\sf":[34],"P}$":[35],"introduced":[36],"by":[37,74],"Valiant.":[38],"also":[40],"establish":[41],"transfer":[42],"theorems":[43],"for":[44,94],"both":[45],"directions":[46],"between":[47],"real":[49],"and":[51],"discrete":[53],"setting.":[54,134],"Then":[55],"we":[56],"characterize":[57],"completeness":[61,129],"results":[62,130],"complexity":[64,116],"computing":[66],"basic":[67],"topological":[68],"invariants":[69],"semilinear":[71],"sets":[72],"given":[73],"circuits.":[76],"It":[77],"turns":[78],"out":[79],"that":[80],"computation":[82,98],"Euler":[85],"characteristic":[86],"is":[87,104,111],"FP}_{\\rm":[89],"add}^{{\\rm":[90],"#P}_{\\rm":[91],"add}}$-complete,":[92],"while":[93],"fixed":[95],"k":[96],"kth":[101],"Betti":[102],"number":[103],"FPAR}_{\\rm":[106],"add}$-complete.":[107],"Thus":[108],"latter":[110],"more":[112],"difficult":[113],"under":[114],"standard":[115],"theoretic":[117],"assumptions.":[118],"use":[120],"all":[121],"above":[124],"to":[125],"prove":[126],"some":[127],"analogous":[128]},"counts_by_year":[{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-07-25T09:21:30.201066","created_date":"2025-10-10T00:00:00"}
