{"id":"https://openalex.org/W7164551468","doi":"https://doi.org/10.48550/arxiv.2606.12676","title":"A Calculus of Apartness over Separoids: Effective Convex Representation, Stratified Conservativity, and the Complexity of Entailment","display_name":"A Calculus of Apartness over Separoids: Effective Convex Representation, Stratified Conservativity, and the Complexity of Entailment","publication_year":2026,"publication_date":"2026-06-10","ids":{"openalex":"https://openalex.org/W7164551468","doi":"https://doi.org/10.48550/arxiv.2606.12676"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2606.12676","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.12676","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2606.12676","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5134351876","display_name":"Faruk Alpay","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Alpay, Faruk","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5138489540","display_name":"Baris Basaran","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Basaran, Baris","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.1437000036239624,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.1437000036239624,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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/T10249","display_name":"Distributed Control Multi-Agent Systems","score":0.14350000023841858,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/T11830","display_name":"Point processes and geometric inequalities","score":0.14100000262260437,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/polytope","display_name":"Polytope","score":0.5218999981880188},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.4300999939441681},{"id":"https://openalex.org/keywords/disjoint-sets","display_name":"Disjoint sets","score":0.42739999294281006},{"id":"https://openalex.org/keywords/decidability","display_name":"Decidability","score":0.41200000047683716},{"id":"https://openalex.org/keywords/lipschitz-continuity","display_name":"Lipschitz continuity","score":0.4056999981403351},{"id":"https://openalex.org/keywords/monotone-polygon","display_name":"Monotone polygon","score":0.4000000059604645},{"id":"https://openalex.org/keywords/monotonic-function","display_name":"Monotonic function","score":0.36629998683929443},{"id":"https://openalex.org/keywords/projection","display_name":"Projection (relational algebra)","score":0.34790000319480896},{"id":"https://openalex.org/keywords/polyhedron","display_name":"Polyhedron","score":0.3465000092983246}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8418999910354614},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5593000054359436},{"id":"https://openalex.org/C145691206","wikidata":"https://www.wikidata.org/wiki/Q747980","display_name":"Polytope","level":2,"score":0.5218999981880188},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.46630001068115234},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.4300999939441681},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.42739999294281006},{"id":"https://openalex.org/C153269930","wikidata":"https://www.wikidata.org/wiki/Q430001","display_name":"Decidability","level":2,"score":0.41200000047683716},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.4056999981403351},{"id":"https://openalex.org/C2834757","wikidata":"https://www.wikidata.org/wiki/Q4925424","display_name":"Monotone polygon","level":2,"score":0.4000000059604645},{"id":"https://openalex.org/C72169020","wikidata":"https://www.wikidata.org/wiki/Q194404","display_name":"Monotonic function","level":2,"score":0.36629998683929443},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.34790000319480896},{"id":"https://openalex.org/C54829058","wikidata":"https://www.wikidata.org/wiki/Q172937","display_name":"Polyhedron","level":2,"score":0.3465000092983246},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.3441999852657318},{"id":"https://openalex.org/C2781089630","wikidata":"https://www.wikidata.org/wiki/Q21856745","display_name":"Realization (probability)","level":2,"score":0.3312999904155731},{"id":"https://openalex.org/C28340159","wikidata":"https://www.wikidata.org/wiki/Q2256541","display_name":"Convex cone","level":5,"score":0.32749998569488525},{"id":"https://openalex.org/C13355873","wikidata":"https://www.wikidata.org/wiki/Q2920850","display_name":"Connection (principal bundle)","level":2,"score":0.32519999146461487},{"id":"https://openalex.org/C165064840","wikidata":"https://www.wikidata.org/wiki/Q1321061","display_name":"Matching (statistics)","level":2,"score":0.32179999351501465},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.3181000053882599},{"id":"https://openalex.org/C136979486","wikidata":"https://www.wikidata.org/wiki/Q773483","display_name":"Existential quantification","level":2,"score":0.29760000109672546},{"id":"https://openalex.org/C168773769","wikidata":"https://www.wikidata.org/wiki/Q1350299","display_name":"Satisfiability","level":2,"score":0.2906999886035919},{"id":"https://openalex.org/C206194317","wikidata":"https://www.wikidata.org/wiki/Q1138624","display_name":"Convex hull","level":3,"score":0.2888999879360199},{"id":"https://openalex.org/C31170391","wikidata":"https://www.wikidata.org/wiki/Q188619","display_name":"Hierarchy","level":2,"score":0.2879999876022339},{"id":"https://openalex.org/C186450821","wikidata":"https://www.wikidata.org/wiki/Q17295","display_name":"Euclidean space","level":2,"score":0.28349998593330383},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2752000093460083},{"id":"https://openalex.org/C2778023678","wikidata":"https://www.wikidata.org/wiki/Q554403","display_name":"Duality (order theory)","level":2,"score":0.2750000059604645},{"id":"https://openalex.org/C84496893","wikidata":"https://www.wikidata.org/wiki/Q1135782","display_name":"Urysohn and completely Hausdorff spaces","level":4,"score":0.2736000120639801},{"id":"https://openalex.org/C12108790","wikidata":"https://www.wikidata.org/wiki/Q2234833","display_name":"Convex analysis","level":4,"score":0.26930001378059387},{"id":"https://openalex.org/C2777573902","wikidata":"https://www.wikidata.org/wiki/Q1491747","display_name":"Galois connection","level":2,"score":0.26100000739097595},{"id":"https://openalex.org/C49870271","wikidata":"https://www.wikidata.org/wiki/Q193657","display_name":"Convex set","level":4,"score":0.25870001316070557},{"id":"https://openalex.org/C7655956","wikidata":"https://www.wikidata.org/wiki/Q5282259","display_name":"Disjoint union (topology)","level":2,"score":0.2538999915122986}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2606.12676","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.12676","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2606.12676","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.12676","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[{"score":0.6674127578735352,"id":"https://metadata.un.org/sdg/10","display_name":"Reduced inequalities"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Every":[0,81],"finite":[1,36,82],"family":[2],"of":[3,26,59,129],"compact":[4],"convex":[5,24,158],"bodies":[6,159],"in":[7,238],"Euclidean":[8,192],"space":[9],"induces":[10],"an":[11,66],"apartness":[12,41,83,224],"relation":[13],"between":[14],"disjoint":[15],"index":[16],"sets:":[17],"two":[18],"sets":[19],"are":[20,30,49,92],"apart":[21,137],"when":[22],"the":[23,27,35,43,56,75,115,180],"hulls":[25],"corresponding":[28],"unions":[29],"disjoint.":[31],"This":[32],"paper":[33],"studies":[34],"theory":[37,78],"obtained":[38],"by":[39,87,94,150],"taking":[40],"as":[42],"primitive":[44],"relation.":[45],"Its":[46],"basic":[47],"laws":[48],"symmetry,":[50],"bilateral":[51],"subsumption,":[52],"and":[53,74,99,122,155,175,197,205],"vacuity,":[54],"equivalently":[55],"separation-polarity":[57],"form":[58,231],"acyclic":[60],"separoids.":[61],"The":[62,133,162],"main":[63],"contribution":[64],"is":[65,85,185,194,200,203,208],"effective":[67],"rational":[68,88],"realization":[69,134],"theorem":[70,215],"with":[71,139],"uniform":[72],"margins":[73],"exact":[76],"consequence":[77,190,229],"it":[79],"supports.":[80],"separoid":[84,226],"realized":[86],"polytopes":[89],"whose":[90],"coordinates":[91],"indexed":[93],"maximal":[95,131],"separations.":[96],"Maximal":[97],"separations":[98],"minimal":[100],"Radon":[101],"partitions":[102],"can":[103],"be":[104],"enumerated":[105],"from":[106],"a":[107,112,126,232],"full":[108],"table,":[109],"generators,":[110],"or":[111],"membership":[113],"oracle;":[114],"coordinate":[116,124],"values":[117],"have":[118],"controlled":[119],"bit":[120],"height;":[121],"each":[123],"records":[125,165],"readable":[127],"certificate":[128],"one":[130],"separation.":[132],"separates":[135],"every":[136],"pair":[138],"clearance":[140],"at":[141],"least":[142],"2,":[143],"remains":[144],"correct":[145],"under":[146,173],"outer":[147,176],"parallel":[148,177],"enlargement":[149],"any":[151],"radius":[152],"below":[153],"1,":[154],"yields":[156],"full-dimensional":[157],"after":[160],"thickening.":[161],"distance-function":[163],"layer":[164],"standard":[166],"convex-analytic":[167],"stability":[168],"through":[169],"Lipschitz":[170],"comparison,":[171],"monotonicity":[172],"inclusion,":[174],"bodies.":[178],"On":[179],"logical":[181],"side,":[182],"positive":[183,206],"entailment":[184,207],"exactly":[186],"one-premise":[187],"subsumption.":[188],"Boolean":[189,218],"over":[191],"scenes":[193],"sound,":[195],"complete,":[196],"decidable;":[198],"satisfiability":[199],"NP-complete,":[201],"validity":[202],"coNP-complete,":[204],"linear":[209],"for":[210,243],"sorted":[211],"encodings.":[212],"A":[213],"stratification":[214],"shows":[216],"that":[217,236],"reasoning":[219],"introduces":[220],"no":[221],"new":[222],"atomic":[223],"beyond":[225],"closure.":[227],"Fixed-dimensional":[228],"relations":[230],"strictly":[233],"decreasing":[234],"hierarchy":[235],"stabilizes":[237],"dimension":[239],"n":[240,244],"minus":[241],"1":[242],"sites.":[245]},"counts_by_year":[],"updated_date":"2026-07-28T07:46:37.118299","created_date":"2026-06-13T00:00:00"}
