The syntactic side of autonomous categories enriched over generalised metric spaces
Autor(a) principal: | |
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Data de Publicação: | 2023 |
Outros Autores: | |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | https://hdl.handle.net/1822/89543 |
Resumo: | Programs with a continuous state space or that interact with physical processes often require notions of equivalence going beyond the standard binary setting in which equivalence either holds or does not hold. In this paper we explore the idea of equivalence taking values in a quantale V, which covers the cases of (in)equations and (ultra)metric equations among others. Our main result is the introduction of a V-equational deductive system for linear λ-calculus together with a proof that it is sound and complete. In fact we go further than this, by showing that linear λ-theories based on this V-equational system form a category equivalent to a category of autonomous categories enriched over ‘generalised metric spaces’. If we instantiate this result to inequations, we get an equivalence with autonomous categories enriched over partial orders. In the case of (ultra)metric equations, we get an equivalence with autonomous categories enriched over (ultra)metric spaces. Additionally, we show that this syntax-semantics correspondence extends to the affine setting. We use our results to develop examples of inequational and metric equational systems for higher-order programming in the setting of real-time, probabilistic, and quantum computing. |
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The syntactic side of autonomous categories enriched over generalised metric spacescategorical logicenriched category theoryquantaleλ-calculusCiências Naturais::Ciências da Computação e da InformaçãoEngenharia e Tecnologia::Engenharia Eletrotécnica, Eletrónica e InformáticaPrograms with a continuous state space or that interact with physical processes often require notions of equivalence going beyond the standard binary setting in which equivalence either holds or does not hold. In this paper we explore the idea of equivalence taking values in a quantale V, which covers the cases of (in)equations and (ultra)metric equations among others. Our main result is the introduction of a V-equational deductive system for linear λ-calculus together with a proof that it is sound and complete. In fact we go further than this, by showing that linear λ-theories based on this V-equational system form a category equivalent to a category of autonomous categories enriched over ‘generalised metric spaces’. If we instantiate this result to inequations, we get an equivalence with autonomous categories enriched over partial orders. In the case of (ultra)metric equations, we get an equivalence with autonomous categories enriched over (ultra)metric spaces. Additionally, we show that this syntax-semantics correspondence extends to the affine setting. We use our results to develop examples of inequational and metric equational systems for higher-order programming in the setting of real-time, probabilistic, and quantum computing.This work is financed by National Funds through FCT - Fundação para a Ciência e a Tecnologia, I.P. (Portuguese Foundation for Science and Technology) within project IBEX, with reference PTDC/CCI-COM/4280/2021. We are also thankful for the reviewers’ helpful feedback.Centre pour la Communication Scientifique Directe (CCSD)Universidade do MinhoDahlqvist, FredrikNeves, Renato Jorge Araújo20232023-01-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/1822/89543engDahlqvist, F., & Neves, R. (2023, December 18). The syntactic side of autonomous categories enriched over generalised metric spaces. Logical Methods in Computer Science. Centre pour la Communication Scientifique Directe (CCSD). http://doi.org/10.46298/lmcs-19(4:31)20231860-597410.46298/lmcs-19(4:31)2023https://lmcs.episciences.org/12719info:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2024-03-16T01:20:17Zoai:repositorium.sdum.uminho.pt:1822/89543Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-20T04:01:03.777513Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
The syntactic side of autonomous categories enriched over generalised metric spaces |
title |
The syntactic side of autonomous categories enriched over generalised metric spaces |
spellingShingle |
The syntactic side of autonomous categories enriched over generalised metric spaces Dahlqvist, Fredrik categorical logic enriched category theory quantale λ-calculus Ciências Naturais::Ciências da Computação e da Informação Engenharia e Tecnologia::Engenharia Eletrotécnica, Eletrónica e Informática |
title_short |
The syntactic side of autonomous categories enriched over generalised metric spaces |
title_full |
The syntactic side of autonomous categories enriched over generalised metric spaces |
title_fullStr |
The syntactic side of autonomous categories enriched over generalised metric spaces |
title_full_unstemmed |
The syntactic side of autonomous categories enriched over generalised metric spaces |
title_sort |
The syntactic side of autonomous categories enriched over generalised metric spaces |
author |
Dahlqvist, Fredrik |
author_facet |
Dahlqvist, Fredrik Neves, Renato Jorge Araújo |
author_role |
author |
author2 |
Neves, Renato Jorge Araújo |
author2_role |
author |
dc.contributor.none.fl_str_mv |
Universidade do Minho |
dc.contributor.author.fl_str_mv |
Dahlqvist, Fredrik Neves, Renato Jorge Araújo |
dc.subject.por.fl_str_mv |
categorical logic enriched category theory quantale λ-calculus Ciências Naturais::Ciências da Computação e da Informação Engenharia e Tecnologia::Engenharia Eletrotécnica, Eletrónica e Informática |
topic |
categorical logic enriched category theory quantale λ-calculus Ciências Naturais::Ciências da Computação e da Informação Engenharia e Tecnologia::Engenharia Eletrotécnica, Eletrónica e Informática |
description |
Programs with a continuous state space or that interact with physical processes often require notions of equivalence going beyond the standard binary setting in which equivalence either holds or does not hold. In this paper we explore the idea of equivalence taking values in a quantale V, which covers the cases of (in)equations and (ultra)metric equations among others. Our main result is the introduction of a V-equational deductive system for linear λ-calculus together with a proof that it is sound and complete. In fact we go further than this, by showing that linear λ-theories based on this V-equational system form a category equivalent to a category of autonomous categories enriched over ‘generalised metric spaces’. If we instantiate this result to inequations, we get an equivalence with autonomous categories enriched over partial orders. In the case of (ultra)metric equations, we get an equivalence with autonomous categories enriched over (ultra)metric spaces. Additionally, we show that this syntax-semantics correspondence extends to the affine setting. We use our results to develop examples of inequational and metric equational systems for higher-order programming in the setting of real-time, probabilistic, and quantum computing. |
publishDate |
2023 |
dc.date.none.fl_str_mv |
2023 2023-01-01T00:00:00Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
https://hdl.handle.net/1822/89543 |
url |
https://hdl.handle.net/1822/89543 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
Dahlqvist, F., & Neves, R. (2023, December 18). The syntactic side of autonomous categories enriched over generalised metric spaces. Logical Methods in Computer Science. Centre pour la Communication Scientifique Directe (CCSD). http://doi.org/10.46298/lmcs-19(4:31)2023 1860-5974 10.46298/lmcs-19(4:31)2023 https://lmcs.episciences.org/12719 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.publisher.none.fl_str_mv |
Centre pour la Communication Scientifique Directe (CCSD) |
publisher.none.fl_str_mv |
Centre pour la Communication Scientifique Directe (CCSD) |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
instname_str |
Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
collection |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
repository.name.fl_str_mv |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
repository.mail.fl_str_mv |
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1799138184899067904 |