{"id":"https://openalex.org/W2082558313","doi":"https://doi.org/10.1016/s1571-0661(04)80911-3","title":"Modal Rules are Co-Implications","display_name":"Modal Rules are Co-Implications","publication_year":2001,"publication_date":"2001-05-01","ids":{"openalex":"https://openalex.org/W2082558313","doi":"https://doi.org/10.1016/s1571-0661(04)80911-3","mag":"2082558313"},"language":"en","primary_location":{"id":"doi:10.1016/s1571-0661(04)80911-3","is_oa":true,"landing_page_url":"https://doi.org/10.1016/s1571-0661(04)80911-3","pdf_url":null,"source":{"id":"https://openalex.org/S50368787","display_name":"Electronic Notes in Theoretical Computer Science","issn_l":"1571-0661","issn":["1571-0661"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Electronic Notes in Theoretical Computer Science","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://doi.org/10.1016/s1571-0661(04)80911-3","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5057868877","display_name":"Alexander Kurz","orcid":"https://orcid.org/0000-0002-8685-5207"},"institutions":[{"id":"https://openalex.org/I1341640284","display_name":"Centrum Wiskunde & Informatica","ror":"https://ror.org/00x7ekv49","country_code":"NL","type":"facility","lineage":["https://openalex.org/I1341640284","https://openalex.org/I2800991832"]}],"countries":["NL"],"is_corresponding":true,"raw_author_name":"Alexander Kurz","raw_affiliation_strings":["CWI, P.O. Box 94079, NL-1090 GB Amsterdam, The Netherlands"],"affiliations":[{"raw_affiliation_string":"CWI, P.O. Box 94079, NL-1090 GB Amsterdam, The Netherlands","institution_ids":["https://openalex.org/I1341640284"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5057868877"],"corresponding_institution_ids":["https://openalex.org/I1341640284"],"apc_list":null,"apc_paid":null,"fwci":2.1805,"has_fulltext":false,"cited_by_count":8,"citation_normalized_percentile":{"value":0.89661876,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"44","issue":"1","first_page":"241","last_page":"253"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.9997000098228455,"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":0.9997000098228455,"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/T11010","display_name":"Logic, Reasoning, and Knowledge","score":0.9975000023841858,"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/T10142","display_name":"Formal Methods in Verification","score":0.9857000112533569,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/modal-logic","display_name":"Modal logic","score":0.8624313473701477},{"id":"https://openalex.org/keywords/normal-modal-logic","display_name":"Normal modal logic","score":0.7950418591499329},{"id":"https://openalex.org/keywords/disjoint-sets","display_name":"Disjoint sets","score":0.7745468616485596},{"id":"https://openalex.org/keywords/modal","display_name":"Modal","score":0.7689031362533569},{"id":"https://openalex.org/keywords/modal-\u03bc-calculus","display_name":"Modal \u03bc-calculus","score":0.7017406225204468},{"id":"https://openalex.org/keywords/multimodal-logic","display_name":"Multimodal logic","score":0.622079610824585},{"id":"https://openalex.org/keywords/expressive-power","display_name":"Expressive power","score":0.5906964540481567},{"id":"https://openalex.org/keywords/s5","display_name":"S5","score":0.5785518884658813},{"id":"https://openalex.org/keywords/accessibility-relation","display_name":"Accessibility relation","score":0.5568554401397705},{"id":"https://openalex.org/keywords/dynamic-logic","display_name":"Dynamic logic (digital electronics)","score":0.5497820377349854},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.5478070974349976},{"id":"https://openalex.org/keywords/corollary","display_name":"Corollary","score":0.5388054847717285},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5249300599098206},{"id":"https://openalex.org/keywords/modal-operator","display_name":"Modal operator","score":0.5065947771072388},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.47570520639419556},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3575246036052704},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.35046565532684326},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3411110043525696},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.21008500456809998},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.16537263989448547},{"id":"https://openalex.org/keywords/description-logic","display_name":"Description logic","score":0.16082894802093506},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.08550915122032166}],"concepts":[{"id":"https://openalex.org/C27508121","wikidata":"https://www.wikidata.org/wiki/Q210841","display_name":"Modal logic","level":3,"score":0.8624313473701477},{"id":"https://openalex.org/C114092440","wikidata":"https://www.wikidata.org/wiki/Q840226","display_name":"Normal modal logic","level":4,"score":0.7950418591499329},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.7745468616485596},{"id":"https://openalex.org/C71139939","wikidata":"https://www.wikidata.org/wiki/Q910194","display_name":"Modal","level":2,"score":0.7689031362533569},{"id":"https://openalex.org/C173125200","wikidata":"https://www.wikidata.org/wiki/Q5275729","display_name":"Modal \u03bc-calculus","level":5,"score":0.7017406225204468},{"id":"https://openalex.org/C203659156","wikidata":"https://www.wikidata.org/wiki/Q6934805","display_name":"Multimodal logic","level":3,"score":0.622079610824585},{"id":"https://openalex.org/C195818886","wikidata":"https://www.wikidata.org/wiki/Q5421724","display_name":"Expressive power","level":2,"score":0.5906964540481567},{"id":"https://openalex.org/C192609573","wikidata":"https://www.wikidata.org/wiki/Q7388341","display_name":"S5","level":5,"score":0.5785518884658813},{"id":"https://openalex.org/C11191006","wikidata":"https://www.wikidata.org/wiki/Q2663311","display_name":"Accessibility relation","level":4,"score":0.5568554401397705},{"id":"https://openalex.org/C2777796570","wikidata":"https://www.wikidata.org/wiki/Q2351326","display_name":"Dynamic logic (digital electronics)","level":4,"score":0.5497820377349854},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.5478070974349976},{"id":"https://openalex.org/C2780012671","wikidata":"https://www.wikidata.org/wiki/Q1343870","display_name":"Corollary","level":2,"score":0.5388054847717285},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5249300599098206},{"id":"https://openalex.org/C78344899","wikidata":"https://www.wikidata.org/wiki/Q638442","display_name":"Modal operator","level":4,"score":0.5065947771072388},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.47570520639419556},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3575246036052704},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.35046565532684326},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3411110043525696},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.21008500456809998},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.16537263989448547},{"id":"https://openalex.org/C102993220","wikidata":"https://www.wikidata.org/wiki/Q387196","display_name":"Description logic","level":2,"score":0.16082894802093506},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08550915122032166},{"id":"https://openalex.org/C165801399","wikidata":"https://www.wikidata.org/wiki/Q25428","display_name":"Voltage","level":2,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C188027245","wikidata":"https://www.wikidata.org/wiki/Q750446","display_name":"Polymer chemistry","level":1,"score":0.0},{"id":"https://openalex.org/C172385210","wikidata":"https://www.wikidata.org/wiki/Q5339","display_name":"Transistor","level":3,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1016/s1571-0661(04)80911-3","is_oa":true,"landing_page_url":"https://doi.org/10.1016/s1571-0661(04)80911-3","pdf_url":null,"source":{"id":"https://openalex.org/S50368787","display_name":"Electronic Notes in Theoretical Computer Science","issn_l":"1571-0661","issn":["1571-0661"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Electronic Notes in Theoretical Computer Science","raw_type":"journal-article"},{"id":"pmh:oai:digitalcommons.chapman.edu:engineering_articles-1034","is_oa":true,"landing_page_url":"https://digitalcommons.chapman.edu/engineering_articles/35","pdf_url":null,"source":{"id":"https://openalex.org/S4377196324","display_name":"Chapman University Digital Commons (Chapman University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I185071736","host_organization_name":"Chapman University","host_organization_lineage":["https://openalex.org/I185071736"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Engineering Faculty Articles and Research","raw_type":null}],"best_oa_location":{"id":"doi:10.1016/s1571-0661(04)80911-3","is_oa":true,"landing_page_url":"https://doi.org/10.1016/s1571-0661(04)80911-3","pdf_url":null,"source":{"id":"https://openalex.org/S50368787","display_name":"Electronic Notes in Theoretical Computer Science","issn_l":"1571-0661","issn":["1571-0661"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Electronic Notes in Theoretical Computer Science","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":24,"referenced_works":["https://openalex.org/W45032386","https://openalex.org/W132152209","https://openalex.org/W1541645275","https://openalex.org/W1543688617","https://openalex.org/W1561011284","https://openalex.org/W1578654632","https://openalex.org/W1973536814","https://openalex.org/W2049261036","https://openalex.org/W2056138173","https://openalex.org/W2068003272","https://openalex.org/W2077760000","https://openalex.org/W2088189323","https://openalex.org/W2107725559","https://openalex.org/W2154547008","https://openalex.org/W2158921667","https://openalex.org/W2493333586","https://openalex.org/W2587256088","https://openalex.org/W4213277014","https://openalex.org/W4235545750","https://openalex.org/W6632558099","https://openalex.org/W6633548484","https://openalex.org/W6661793311","https://openalex.org/W6667126014","https://openalex.org/W6670310965"],"related_works":["https://openalex.org/W2534562289","https://openalex.org/W4233254255","https://openalex.org/W2188462059","https://openalex.org/W2795425183","https://openalex.org/W4318188140","https://openalex.org/W2123071520","https://openalex.org/W1566488688","https://openalex.org/W2077061463","https://openalex.org/W2554822349","https://openalex.org/W2566122784"],"abstract_inverted_index":{"In":[0],"[13],":[1],"it":[2,38],"was":[3],"shown":[4,26],"that":[5,27],"modal":[6,20,35,58],"logic":[7,12,59],"for":[8,13],"coalgebras":[9,31],"dualises\u2014concerning":[10],"definability\u2014equational":[11],"algebras.":[14],"This":[15],"paper":[16],"establishes":[17],"that,":[18],"similarly,":[19],"rules":[21,36,55],"dualise":[22],"implications:":[23],"It":[24],"is":[25,32,39,63],"a":[28,49],"class":[29],"of":[30,54,56],"definable":[33],"by":[34],"iff":[37],"closed":[40],"under":[41],"H":[42],"(images)":[43],"and":[44],"\u03a3":[45],"(disjoint":[46],"unions).":[47],"As":[48],"corollary":[50],"the":[51],"expressive":[52],"power":[53],"infinitary":[57],"on":[60],"Kripke":[61],"frames":[62],"characterised.":[64]},"counts_by_year":[{"year":2021,"cited_by_count":1},{"year":2014,"cited_by_count":1}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
