{"id":"https://openalex.org/W2770784480","doi":"https://doi.org/10.23638/lmcs-13(4:19)2017","title":"Statman's Hierarchy Theorem","display_name":"Statman's Hierarchy Theorem","publication_year":2017,"publication_date":"2017-11-27","ids":{"openalex":"https://openalex.org/W2770784480","doi":"https://doi.org/10.23638/lmcs-13(4:19)2017","mag":"2770784480"},"language":"en","primary_location":{"id":"doi:10.23638/lmcs-13(4:19)2017","is_oa":true,"landing_page_url":"https://doi.org/10.23638/lmcs-13(4:19)2017","pdf_url":"https://lmcs.episciences.org/4089/pdf","source":{"id":"https://openalex.org/S114379355","display_name":"Logical Methods in Computer Science","issn_l":"1860-5974","issn":["1860-5974"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310313916","host_organization_name":"Logical Methods in Computer Science e.V.","host_organization_lineage":["https://openalex.org/P4310313916"],"host_organization_lineage_names":["Logical Methods in Computer Science e.V."],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Logical Methods in Computer Science","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://lmcs.episciences.org/4089/pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5073018635","display_name":"Bram Westerbaan","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Bram Westerbaan","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5031739211","display_name":"Bas Westerbaan","orcid":"https://orcid.org/0000-0002-3195-6238"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Bas Westerbaan","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5057158738","display_name":"Rutger Kuyper","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Rutger Kuyper","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5069604653","display_name":"Carst Tankink","orcid":"https://orcid.org/0000-0003-4748-5773"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Carst Tankink","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5044874394","display_name":"Remy Viehoff","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Remy Viehoff","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5016150987","display_name":"Henk Barendregt","orcid":"https://orcid.org/0000-0002-3735-4078"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Henk Barendregt","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5073018635"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.17771318,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"Volume 13, Issue 4","issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.9984999895095825,"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"}},"topics":[{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.9984999895095825,"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/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9850999712944031,"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/T11010","display_name":"Logic, Reasoning, and Knowledge","score":0.983299970626831,"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/type","display_name":"Type (biology)","score":0.7089504599571228},{"id":"https://openalex.org/keywords/injective-function","display_name":"Injective function","score":0.6332399845123291},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.5902186036109924},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.5857512950897217},{"id":"https://openalex.org/keywords/order-type","display_name":"Order type","score":0.5576721429824829},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.553886353969574},{"id":"https://openalex.org/keywords/term","display_name":"Term (time)","score":0.4928014576435089},{"id":"https://openalex.org/keywords/lambda","display_name":"Lambda","score":0.4522857367992401},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4218255579471588},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.4159408211708069},{"id":"https://openalex.org/keywords/equivalence-relation","display_name":"Equivalence relation","score":0.4147685170173645},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4071805477142334},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.10425800085067749}],"concepts":[{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.7089504599571228},{"id":"https://openalex.org/C128107574","wikidata":"https://www.wikidata.org/wiki/Q182003","display_name":"Injective function","level":2,"score":0.6332399845123291},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.5902186036109924},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.5857512950897217},{"id":"https://openalex.org/C138890893","wikidata":"https://www.wikidata.org/wiki/Q620006","display_name":"Order type","level":2,"score":0.5576721429824829},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.553886353969574},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.4928014576435089},{"id":"https://openalex.org/C2778113609","wikidata":"https://www.wikidata.org/wiki/Q10897","display_name":"Lambda","level":2,"score":0.4522857367992401},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4218255579471588},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4159408211708069},{"id":"https://openalex.org/C156103551","wikidata":"https://www.wikidata.org/wiki/Q130998","display_name":"Equivalence relation","level":2,"score":0.4147685170173645},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4071805477142334},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.10425800085067749},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.23638/lmcs-13(4:19)2017","is_oa":true,"landing_page_url":"https://doi.org/10.23638/lmcs-13(4:19)2017","pdf_url":"https://lmcs.episciences.org/4089/pdf","source":{"id":"https://openalex.org/S114379355","display_name":"Logical Methods in Computer Science","issn_l":"1860-5974","issn":["1860-5974"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310313916","host_organization_name":"Logical Methods in Computer Science e.V.","host_organization_lineage":["https://openalex.org/P4310313916"],"host_organization_lineage_names":["Logical Methods in Computer Science e.V."],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Logical Methods in Computer Science","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1711.05497","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1711.05497","pdf_url":"https://arxiv.org/pdf/1711.05497","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.715.2443","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.715.2443","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://bram.westerbaan.name/hierarchy.pdf","raw_type":"text"},{"id":"doi:10.48550/arxiv.1711.05497","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1711.05497","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"doi:10.23638/lmcs-13(4:19)2017","is_oa":true,"landing_page_url":"https://doi.org/10.23638/lmcs-13(4:19)2017","pdf_url":"https://lmcs.episciences.org/4089/pdf","source":{"id":"https://openalex.org/S114379355","display_name":"Logical Methods in Computer Science","issn_l":"1860-5974","issn":["1860-5974"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310313916","host_organization_name":"Logical Methods in Computer Science e.V.","host_organization_lineage":["https://openalex.org/P4310313916"],"host_organization_lineage_names":["Logical Methods in Computer Science e.V."],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Logical Methods in Computer Science","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W2770784480.pdf"},"referenced_works_count":6,"referenced_works":["https://openalex.org/W1494407580","https://openalex.org/W2074282135","https://openalex.org/W2080452006","https://openalex.org/W2154273644","https://openalex.org/W2267324769","https://openalex.org/W2398664406"],"related_works":["https://openalex.org/W238609530","https://openalex.org/W2734192830","https://openalex.org/W3016395638","https://openalex.org/W1636163329","https://openalex.org/W3045121788","https://openalex.org/W3201655533","https://openalex.org/W3134442314","https://openalex.org/W2951695430","https://openalex.org/W4235371823","https://openalex.org/W4285242600"],"abstract_inverted_index":{"In":[0,65],"the":[1,7,39,51,56,66,75,135,149,157,169],"Simply":[2],"Typed":[3],"$\\lambda$-calculus":[4],"Statman":[5],"investigates":[6],"reducibility":[8],"relation":[9,70,90],"$\\leq_{\\beta\\eta}$":[10],"between":[11],"types:":[12],"for":[13,119],"$A,B":[14],"\\in":[15],"\\mathbb{T}^0$,":[16],"types":[17],"freely":[18],"generated":[19],"using":[20],"$\\rightarrow$":[21],"and":[22,85,112,126,160],"a":[23,35,68,82,93,108,166],"single":[24],"ground":[25],"type":[26,43,48,122,129],"$0$,":[27],"define":[28],"$A":[29],"\\leq_{\\beta\\eta}":[30],"B$":[31],"if":[32],"there":[33],"exists":[34],"$\\lambda$-definable":[36],"injection":[37,77],"from":[38],"closed":[40,172],"terms":[41,173],"of":[42,47,96,134,138,145,156,171,174],"$A$":[44],"into":[45],"those":[46],"$B$.":[49],"Unexpectedly,":[50],"induced":[52],"partial":[53],"order":[54,60],"is":[55,72,78,100],"(linear)":[57],"well-ordering":[58],"(of":[59],"type)":[61],"$\\omega":[62,123],"+":[63,124],"4$.":[64],"proof":[67],"finer":[69],"$\\leq_{h}$":[71],"used,":[73],"where":[74],"above":[76],"required":[79],"to":[80,141,148],"be":[81],"B\\"ohm":[83,97],"transformation,":[84],"an":[86],"(a":[87],"posteriori)":[88],"coarser":[89],"$\\leq_{h^+}$,":[91],"requiring":[92],"finite":[94],"family":[95],"transformations":[98],"that":[99],"jointly":[101],"injective.":[102],"We":[103],"present":[104],"this":[105],"result":[106],"in":[107],"self-contained,":[109],"syntactic,":[110],"constructive":[111],"simplified":[113],"manner.":[114],"En":[115],"route":[116],"similar":[117],"results":[118],"$\\leq_h$":[120],"(order":[121,128],"5$)":[125],"$\\leq_{h^+}$":[127,139],"$8$)":[130],"are":[131],"obtained.":[132],"Five":[133],"equivalence":[136],"classes":[137],"correspond":[140],"canonical":[142],"term":[143,151],"models":[144],"Statman,":[146],"one":[147,161],"trivial":[150],"model":[152,167],"collapsing":[153],"all":[154],"elements":[155],"same":[158],"type,":[159],"does":[162],"not":[163],"even":[164],"form":[165],"by":[168],"lack":[170],"many":[175],"types.":[176]},"counts_by_year":[],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}