Modal dependent type theory and dependent right adjoints

In recent years, we have seen several new models of dependent type theory extended with some form of modal necessity operator, including nominal type theory, guarded and clocked type theory and spatial and cohesive type theory. In this paper, we study modal dependent type theory : dependent type the...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Mathematical structures in computer science 2020-02, Vol.30 (2), p.118-138
Hauptverfasser: Birkedal, Lars, Clouston, Ranald, Mannaa, Bassel, Ejlers Møgelberg, Rasmus, Pitts, Andrew M., Spitters, Bas
Format: Artikel
Sprache:eng
Schlagworte:
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
container_end_page 138
container_issue 2
container_start_page 118
container_title Mathematical structures in computer science
container_volume 30
creator Birkedal, Lars
Clouston, Ranald
Mannaa, Bassel
Ejlers Møgelberg, Rasmus
Pitts, Andrew M.
Spitters, Bas
description In recent years, we have seen several new models of dependent type theory extended with some form of modal necessity operator, including nominal type theory, guarded and clocked type theory and spatial and cohesive type theory. In this paper, we study modal dependent type theory : dependent type theory with an operator satisfying (a dependent version of) the K axiom of modal logic. We investigate both semantics and syntax. For the semantics, we introduce categories with families with a dependent right adjoint (CwDRA) and show that the examples above can be presented as such. Indeed, we show that any category with finite limits and an adjunction of endofunctors give rise to a CwDRA via the local universe construction. For the syntax, we introduce a dependently typed extension of Fitch-style modal λ -calculus, show that it can be interpreted in any CwDRA, and build a term model. We extend the syntax and semantics with universes.
doi_str_mv 10.1017/S0960129519000197
format Article
fullrecord <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2374031248</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2374031248</sourcerecordid><originalsourceid>FETCH-LOGICAL-c273t-f82d65694090615e157f89f46e1ad55d5f2430249792f95c1ddcf1d7fc1d3e9d3</originalsourceid><addsrcrecordid>eNplkE1LxDAURYMoWEd_gLuC6-p7-Wj6ljKoI4y4UNehNInTMrY1ySz6720ZF4Kr--Ac3oXL2DXCLQLquzegEpCTQgIAJH3CMpQlFRVofsqyBRcLP2cXMXazIhAoY9XLYOt9bt3oeuv6lKdpdHnauSFMed3bPyS0n7uU17Yb2j7FS3bm6310V7-5Yh-PD-_rTbF9fXpe32-LhmuRCl9xW6qSJBCUqBwq7SvysnRYW6Ws8lwK4JI0cU-qQWsbj1b7-RKOrFixm-PfMQzfBxeT6YZD6OdKw4WWIJDLarbwaDVhiDE4b8bQftVhMghmGcj8G0j8AKRxVus</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2374031248</pqid></control><display><type>article</type><title>Modal dependent type theory and dependent right adjoints</title><source>Cambridge University Press Journals Complete</source><creator>Birkedal, Lars ; Clouston, Ranald ; Mannaa, Bassel ; Ejlers Møgelberg, Rasmus ; Pitts, Andrew M. ; Spitters, Bas</creator><creatorcontrib>Birkedal, Lars ; Clouston, Ranald ; Mannaa, Bassel ; Ejlers Møgelberg, Rasmus ; Pitts, Andrew M. ; Spitters, Bas</creatorcontrib><description>In recent years, we have seen several new models of dependent type theory extended with some form of modal necessity operator, including nominal type theory, guarded and clocked type theory and spatial and cohesive type theory. In this paper, we study modal dependent type theory : dependent type theory with an operator satisfying (a dependent version of) the K axiom of modal logic. We investigate both semantics and syntax. For the semantics, we introduce categories with families with a dependent right adjoint (CwDRA) and show that the examples above can be presented as such. Indeed, we show that any category with finite limits and an adjunction of endofunctors give rise to a CwDRA via the local universe construction. For the syntax, we introduce a dependently typed extension of Fitch-style modal λ -calculus, show that it can be interpreted in any CwDRA, and build a term model. We extend the syntax and semantics with universes.</description><identifier>ISSN: 0960-1295</identifier><identifier>EISSN: 1469-8072</identifier><identifier>DOI: 10.1017/S0960129519000197</identifier><language>eng</language><publisher>Cambridge: Cambridge University Press</publisher><subject>Adjoints ; Semantics ; Syntax</subject><ispartof>Mathematical structures in computer science, 2020-02, Vol.30 (2), p.118-138</ispartof><rights>Copyright Cambridge University Press Feb 2020</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c273t-f82d65694090615e157f89f46e1ad55d5f2430249792f95c1ddcf1d7fc1d3e9d3</citedby><cites>FETCH-LOGICAL-c273t-f82d65694090615e157f89f46e1ad55d5f2430249792f95c1ddcf1d7fc1d3e9d3</cites><orcidid>0000-0003-0097-6188 ; 0000-0001-7775-3471</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>315,781,785,27925,27926</link.rule.ids></links><search><creatorcontrib>Birkedal, Lars</creatorcontrib><creatorcontrib>Clouston, Ranald</creatorcontrib><creatorcontrib>Mannaa, Bassel</creatorcontrib><creatorcontrib>Ejlers Møgelberg, Rasmus</creatorcontrib><creatorcontrib>Pitts, Andrew M.</creatorcontrib><creatorcontrib>Spitters, Bas</creatorcontrib><title>Modal dependent type theory and dependent right adjoints</title><title>Mathematical structures in computer science</title><description>In recent years, we have seen several new models of dependent type theory extended with some form of modal necessity operator, including nominal type theory, guarded and clocked type theory and spatial and cohesive type theory. In this paper, we study modal dependent type theory : dependent type theory with an operator satisfying (a dependent version of) the K axiom of modal logic. We investigate both semantics and syntax. For the semantics, we introduce categories with families with a dependent right adjoint (CwDRA) and show that the examples above can be presented as such. Indeed, we show that any category with finite limits and an adjunction of endofunctors give rise to a CwDRA via the local universe construction. For the syntax, we introduce a dependently typed extension of Fitch-style modal λ -calculus, show that it can be interpreted in any CwDRA, and build a term model. We extend the syntax and semantics with universes.</description><subject>Adjoints</subject><subject>Semantics</subject><subject>Syntax</subject><issn>0960-1295</issn><issn>1469-8072</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><sourceid>ABUWG</sourceid><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><sourceid>GNUQQ</sourceid><recordid>eNplkE1LxDAURYMoWEd_gLuC6-p7-Wj6ljKoI4y4UNehNInTMrY1ySz6720ZF4Kr--Ac3oXL2DXCLQLquzegEpCTQgIAJH3CMpQlFRVofsqyBRcLP2cXMXazIhAoY9XLYOt9bt3oeuv6lKdpdHnauSFMed3bPyS0n7uU17Yb2j7FS3bm6310V7-5Yh-PD-_rTbF9fXpe32-LhmuRCl9xW6qSJBCUqBwq7SvysnRYW6Ws8lwK4JI0cU-qQWsbj1b7-RKOrFixm-PfMQzfBxeT6YZD6OdKw4WWIJDLarbwaDVhiDE4b8bQftVhMghmGcj8G0j8AKRxVus</recordid><startdate>202002</startdate><enddate>202002</enddate><creator>Birkedal, Lars</creator><creator>Clouston, Ranald</creator><creator>Mannaa, Bassel</creator><creator>Ejlers Møgelberg, Rasmus</creator><creator>Pitts, Andrew M.</creator><creator>Spitters, Bas</creator><general>Cambridge University Press</general><scope>AAYXX</scope><scope>CITATION</scope><scope>3V.</scope><scope>7SC</scope><scope>7XB</scope><scope>88I</scope><scope>8AL</scope><scope>8FD</scope><scope>8FE</scope><scope>8FG</scope><scope>8FK</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>ARAPS</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>GNUQQ</scope><scope>HCIFZ</scope><scope>JQ2</scope><scope>K7-</scope><scope>L6V</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>M0N</scope><scope>M2P</scope><scope>M7S</scope><scope>P5Z</scope><scope>P62</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope><scope>Q9U</scope><orcidid>https://orcid.org/0000-0003-0097-6188</orcidid><orcidid>https://orcid.org/0000-0001-7775-3471</orcidid></search><sort><creationdate>202002</creationdate><title>Modal dependent type theory and dependent right adjoints</title><author>Birkedal, Lars ; Clouston, Ranald ; Mannaa, Bassel ; Ejlers Møgelberg, Rasmus ; Pitts, Andrew M. ; Spitters, Bas</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c273t-f82d65694090615e157f89f46e1ad55d5f2430249792f95c1ddcf1d7fc1d3e9d3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Adjoints</topic><topic>Semantics</topic><topic>Syntax</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Birkedal, Lars</creatorcontrib><creatorcontrib>Clouston, Ranald</creatorcontrib><creatorcontrib>Mannaa, Bassel</creatorcontrib><creatorcontrib>Ejlers Møgelberg, Rasmus</creatorcontrib><creatorcontrib>Pitts, Andrew M.</creatorcontrib><creatorcontrib>Spitters, Bas</creatorcontrib><collection>CrossRef</collection><collection>ProQuest Central (Corporate)</collection><collection>Computer and Information Systems Abstracts</collection><collection>ProQuest Central (purchase pre-March 2016)</collection><collection>Science Database (Alumni Edition)</collection><collection>Computing Database (Alumni Edition)</collection><collection>Technology Research Database</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>ProQuest Central (Alumni) (purchase pre-March 2016)</collection><collection>Materials Science &amp; Engineering Collection</collection><collection>ProQuest Central (Alumni Edition)</collection><collection>ProQuest Central UK/Ireland</collection><collection>Advanced Technologies &amp; Aerospace Collection</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>ProQuest Central Student</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Computer Science Collection</collection><collection>Computer Science Database</collection><collection>ProQuest Engineering Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts – Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><collection>Computing Database</collection><collection>Science Database</collection><collection>Engineering Database</collection><collection>Advanced Technologies &amp; Aerospace Database</collection><collection>ProQuest Advanced Technologies &amp; Aerospace Collection</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering Collection</collection><collection>ProQuest Central Basic</collection><jtitle>Mathematical structures in computer science</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Birkedal, Lars</au><au>Clouston, Ranald</au><au>Mannaa, Bassel</au><au>Ejlers Møgelberg, Rasmus</au><au>Pitts, Andrew M.</au><au>Spitters, Bas</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Modal dependent type theory and dependent right adjoints</atitle><jtitle>Mathematical structures in computer science</jtitle><date>2020-02</date><risdate>2020</risdate><volume>30</volume><issue>2</issue><spage>118</spage><epage>138</epage><pages>118-138</pages><issn>0960-1295</issn><eissn>1469-8072</eissn><abstract>In recent years, we have seen several new models of dependent type theory extended with some form of modal necessity operator, including nominal type theory, guarded and clocked type theory and spatial and cohesive type theory. In this paper, we study modal dependent type theory : dependent type theory with an operator satisfying (a dependent version of) the K axiom of modal logic. We investigate both semantics and syntax. For the semantics, we introduce categories with families with a dependent right adjoint (CwDRA) and show that the examples above can be presented as such. Indeed, we show that any category with finite limits and an adjunction of endofunctors give rise to a CwDRA via the local universe construction. For the syntax, we introduce a dependently typed extension of Fitch-style modal λ -calculus, show that it can be interpreted in any CwDRA, and build a term model. We extend the syntax and semantics with universes.</abstract><cop>Cambridge</cop><pub>Cambridge University Press</pub><doi>10.1017/S0960129519000197</doi><tpages>21</tpages><orcidid>https://orcid.org/0000-0003-0097-6188</orcidid><orcidid>https://orcid.org/0000-0001-7775-3471</orcidid></addata></record>
fulltext fulltext
identifier ISSN: 0960-1295
ispartof Mathematical structures in computer science, 2020-02, Vol.30 (2), p.118-138
issn 0960-1295
1469-8072
language eng
recordid cdi_proquest_journals_2374031248
source Cambridge University Press Journals Complete
subjects Adjoints
Semantics
Syntax
title Modal dependent type theory and dependent right adjoints
url https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2024-12-18T14%3A13%3A43IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_cross&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Modal%20dependent%20type%20theory%20and%20dependent%20right%20adjoints&rft.jtitle=Mathematical%20structures%20in%20computer%20science&rft.au=Birkedal,%20Lars&rft.date=2020-02&rft.volume=30&rft.issue=2&rft.spage=118&rft.epage=138&rft.pages=118-138&rft.issn=0960-1295&rft.eissn=1469-8072&rft_id=info:doi/10.1017/S0960129519000197&rft_dat=%3Cproquest_cross%3E2374031248%3C/proquest_cross%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=2374031248&rft_id=info:pmid/&rfr_iscdi=true