{"id":"https://openalex.org/W2093225431","doi":"https://doi.org/10.1007/s00165-005-0061-1","title":"Not necessarily closed convex polyhedra and the double description method","display_name":"Not necessarily closed convex polyhedra and the double description method","publication_year":2005,"publication_date":"2005-06-18","ids":{"openalex":"https://openalex.org/W2093225431","doi":"https://doi.org/10.1007/s00165-005-0061-1","mag":"2093225431"},"language":"en","primary_location":{"id":"doi:10.1007/s00165-005-0061-1","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s00165-005-0061-1","pdf_url":null,"source":{"id":"https://openalex.org/S128250343","display_name":"Formal Aspects of Computing","issn_l":"0934-5043","issn":["0934-5043","1433-299X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formal Aspects of 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/A5057370941","display_name":"Roberto Bagnara","orcid":"https://orcid.org/0000-0002-6163-6278"},"institutions":[{"id":"https://openalex.org/I124601658","display_name":"University of Parma","ror":"https://ror.org/02k7wn190","country_code":"IT","type":"education","lineage":["https://openalex.org/I124601658"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"Roberto Bagnara","raw_affiliation_strings":["Department of Mathematics, University of Parma, Italy","Department of Mathematics, University of Parma, Italy#TAB#"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Parma, Italy","institution_ids":["https://openalex.org/I124601658"]},{"raw_affiliation_string":"Department of Mathematics, University of Parma, Italy#TAB#","institution_ids":["https://openalex.org/I124601658"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5111600664","display_name":"Patricia M. Hill","orcid":null},"institutions":[{"id":"https://openalex.org/I130828816","display_name":"University of Leeds","ror":"https://ror.org/024mrxd33","country_code":"GB","type":"education","lineage":["https://openalex.org/I130828816"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Patricia M. Hill","raw_affiliation_strings":["School of Computing, University of Leeds, UK","School of Computing, University of Leeds. UK"],"affiliations":[{"raw_affiliation_string":"School of Computing, University of Leeds, UK","institution_ids":["https://openalex.org/I130828816"]},{"raw_affiliation_string":"School of Computing, University of Leeds. UK","institution_ids":["https://openalex.org/I130828816"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5065794938","display_name":"Enea Zaffanella","orcid":"https://orcid.org/0000-0001-6388-2053"},"institutions":[{"id":"https://openalex.org/I124601658","display_name":"University of Parma","ror":"https://ror.org/02k7wn190","country_code":"IT","type":"education","lineage":["https://openalex.org/I124601658"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Enea Zaffanella","raw_affiliation_strings":["Department of Mathematics, University of Parma, Italy","Department of Mathematics, University of Parma, Italy#TAB#"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Parma, Italy","institution_ids":["https://openalex.org/I124601658"]},{"raw_affiliation_string":"Department of Mathematics, University of Parma, Italy#TAB#","institution_ids":["https://openalex.org/I124601658"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5057370941"],"corresponding_institution_ids":["https://openalex.org/I124601658"],"apc_list":null,"apc_paid":null,"fwci":2.6933,"has_fulltext":false,"cited_by_count":40,"citation_normalized_percentile":{"value":0.90428187,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"17","issue":"2","first_page":"222","last_page":"257"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10142","display_name":"Formal Methods in Verification","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/T10142","display_name":"Formal Methods in Verification","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/T10904","display_name":"Embedded Systems Design Techniques","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/1708","display_name":"Hardware and Architecture"},"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/T11032","display_name":"VLSI and Analog Circuit Testing","score":0.9990000128746033,"subfield":{"id":"https://openalex.org/subfields/1708","display_name":"Hardware and Architecture"},"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/polyhedron","display_name":"Polyhedron","score":0.9797371625900269},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.6269941926002502},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.5907715559005737},{"id":"https://openalex.org/keywords/theory-of-computation","display_name":"Theory of computation","score":0.5453464984893799},{"id":"https://openalex.org/keywords/convex-polytope","display_name":"Convex polytope","score":0.5262400507926941},{"id":"https://openalex.org/keywords/implementation","display_name":"Implementation","score":0.5218604803085327},{"id":"https://openalex.org/keywords/dual-polyhedron","display_name":"Dual polyhedron","score":0.5084189772605896},{"id":"https://openalex.org/keywords/embedding","display_name":"Embedding","score":0.49842143058776855},{"id":"https://openalex.org/keywords/domain","display_name":"Domain (mathematical analysis)","score":0.44363078474998474},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.37544727325439453},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.33028408885002136},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.299439013004303},{"id":"https://openalex.org/keywords/convex-set","display_name":"Convex set","score":0.27302420139312744},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.1944323480129242},{"id":"https://openalex.org/keywords/convex-optimization","display_name":"Convex optimization","score":0.17036816477775574},{"id":"https://openalex.org/keywords/programming-language","display_name":"Programming language","score":0.14622551202774048},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.11269897222518921},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.09814098477363586}],"concepts":[{"id":"https://openalex.org/C54829058","wikidata":"https://www.wikidata.org/wiki/Q172937","display_name":"Polyhedron","level":2,"score":0.9797371625900269},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6269941926002502},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.5907715559005737},{"id":"https://openalex.org/C24858836","wikidata":"https://www.wikidata.org/wiki/Q844718","display_name":"Theory of computation","level":2,"score":0.5453464984893799},{"id":"https://openalex.org/C110943637","wikidata":"https://www.wikidata.org/wiki/Q4129097","display_name":"Convex polytope","level":5,"score":0.5262400507926941},{"id":"https://openalex.org/C26713055","wikidata":"https://www.wikidata.org/wiki/Q245962","display_name":"Implementation","level":2,"score":0.5218604803085327},{"id":"https://openalex.org/C105313907","wikidata":"https://www.wikidata.org/wiki/Q1053857","display_name":"Dual polyhedron","level":2,"score":0.5084189772605896},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.49842143058776855},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.44363078474998474},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.37544727325439453},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.33028408885002136},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.299439013004303},{"id":"https://openalex.org/C49870271","wikidata":"https://www.wikidata.org/wiki/Q193657","display_name":"Convex set","level":4,"score":0.27302420139312744},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.1944323480129242},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.17036816477775574},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.14622551202774048},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.11269897222518921},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.09814098477363586},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1007/s00165-005-0061-1","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s00165-005-0061-1","pdf_url":null,"source":{"id":"https://openalex.org/S128250343","display_name":"Formal Aspects of Computing","issn_l":"0934-5043","issn":["0934-5043","1433-299X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formal Aspects of Computing","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.10.8646","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.10.8646","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.unipr.it/~bagnara/Papers/PostScript/600dpi/NNC-600DPI.ps.gz","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.9.8455","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.9.8455","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.unipr.it/~bagnara/Papers/PDF/NNC.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.5799999833106995,"id":"https://metadata.un.org/sdg/9","display_name":"Industry, innovation and infrastructure"}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":28,"referenced_works":["https://openalex.org/W1471753124","https://openalex.org/W1487657751","https://openalex.org/W1489150842","https://openalex.org/W1494436834","https://openalex.org/W1502028089","https://openalex.org/W1505135855","https://openalex.org/W1535550067","https://openalex.org/W1551571370","https://openalex.org/W1558952152","https://openalex.org/W1559274767","https://openalex.org/W1637976039","https://openalex.org/W1963547452","https://openalex.org/W2032391783","https://openalex.org/W2043100293","https://openalex.org/W2060095702","https://openalex.org/W2081041113","https://openalex.org/W2092017668","https://openalex.org/W2132331020","https://openalex.org/W2132661148","https://openalex.org/W2144494996","https://openalex.org/W2173534792","https://openalex.org/W2272316422","https://openalex.org/W2492353954","https://openalex.org/W2602645338","https://openalex.org/W3022177319","https://openalex.org/W3029115295","https://openalex.org/W4213255380","https://openalex.org/W4239558088"],"related_works":["https://openalex.org/W2008231869","https://openalex.org/W2090263210","https://openalex.org/W2949692952","https://openalex.org/W2197678707","https://openalex.org/W1995855104","https://openalex.org/W2900515541","https://openalex.org/W2254121517","https://openalex.org/W2090864852","https://openalex.org/W2067082491","https://openalex.org/W1979597944"],"abstract_inverted_index":{"Abstract":[0],"Since":[1],"the":[2,9,21,34,39,46,93,109,122,151,161,171],"seminal":[3],"work":[4,104],"of":[5,11,25,33,62,135,142,153,163,173],"Cousot":[6],"and":[7,23,27,54,91,95,107,145,160],"Halbwachs,":[8],"domain":[10,61],"convex":[12,63],"polyhedra":[13,40,64,77,84],"has":[14],"been":[15],"employed":[16],"in":[17,85],"several":[18,52],"systems":[19],"for":[20,99,115,167],"analysis":[22],"verification":[24],"hardware":[26],"software":[28],"components.":[29],"Although":[30],"most":[31],"implementations":[32,117],"polyhedral":[35],"operations":[36],"assume":[37],"that":[38,65,118],"are":[41,50,66,119,150],"topologically":[42],"closed":[43,69,83,100],"(i.e.,":[44],"all":[45],"constraints":[47,137],"defining":[48],"them":[49,81],"non-strict),":[51],"analyzers":[53],"verifiers":[55],"need":[56],"to":[57,74,79],"compute":[58],"on":[59,121],"a":[60,86,127,133,140,154],"not":[67],"necessarily":[68],"(NNC).":[70],"The":[71],"usual":[72],"approach":[73],"implementing":[75],"NNC":[76,174],"is":[78],"embed":[80],"into":[82],"higher":[87],"dimensional":[88],"vector":[89],"space":[90],"reuse":[92],"tools":[94],"techniques":[96],"already":[97],"available":[98],"polyhedra.":[101,175],"In":[102],"this":[103],"we":[105],"highlight":[106],"discuss":[108],"issues":[110],"underlying":[111],"such":[112],"an":[113,164],"embedding":[114],"those":[116],"based":[120],"double":[123],"description":[124],"method,":[125],"where":[126],"polyhedron":[128],"may":[129],"be":[130],"described":[131],"by":[132,139],"system":[134,141],"linear":[136],"or":[138],"generating":[143],"rays":[144],"points.":[146],"Two":[147],"major":[148],"achievements":[149],"definition":[152],"theoretically":[155],"clean,":[156],"high-level":[157],"user":[158],"interface":[159],"specification":[162],"efficient":[165],"procedure":[166],"removing":[168],"redundancies":[169],"from":[170],"descriptions":[172]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":5},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":3},{"year":2012,"cited_by_count":4}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}
