Curry-Howard for sequent calculus at last!
Autor(a) principal: | |
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Data de Publicação: | 2015 |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/1822/39302 |
Resumo: | This paper tries to remove what seems to be the remaining stumbling blocks in the way to a full understanding of the Curry-Howard isomorphism for sequent calculus, namely the questions: What do variables in proof terms stand for? What is co-control and a co-continuation? How to define the dual of Parigot's mu-operator so that it is a co-control operator? Answering these questions leads to the interpretation that sequent calculus is a formal vector notation with first-class co-control. But this is just the "internal" interpretation, which has to be developed simultaneously with, and is justified by, an "external" one, offered by natural deduction: the sequent calculus corresponds to a bi-directional, agnostic (w.r.t. the call strategy), computational lambda-calculus. Next, the duality between control and co-control is studied and proved in the context of classical logic, where one discovers that the classical sequent calculus has a distortion towards control, and that sequent calculus is the de Morgan dual of natural deduction. |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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7160 |
spelling |
Curry-Howard for sequent calculus at last!Co-controlCo-continuationVector notationLet-expressionFormal substitutionContext substitutionComputational lambda-calculusClassical logicde Morgan dualityCiências Naturais::MatemáticasThis paper tries to remove what seems to be the remaining stumbling blocks in the way to a full understanding of the Curry-Howard isomorphism for sequent calculus, namely the questions: What do variables in proof terms stand for? What is co-control and a co-continuation? How to define the dual of Parigot's mu-operator so that it is a co-control operator? Answering these questions leads to the interpretation that sequent calculus is a formal vector notation with first-class co-control. But this is just the "internal" interpretation, which has to be developed simultaneously with, and is justified by, an "external" one, offered by natural deduction: the sequent calculus corresponds to a bi-directional, agnostic (w.r.t. the call strategy), computational lambda-calculus. Next, the duality between control and co-control is studied and proved in the context of classical logic, where one discovers that the classical sequent calculus has a distortion towards control, and that sequent calculus is the de Morgan dual of natural deduction.(undefined)Schloss Dagstuhl – Leibniz-Zentrum für Informatik GmbHUniversidade do MinhoEspírito Santo, José2015-06-122015-06-12T00:00:00Zconference paperinfo:eu-repo/semantics/publishedVersionapplication/pdfhttp://hdl.handle.net/1822/39302eng97839398978731868-896910.4230/LIPIcs.TLCA.2015.165info: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:RCAAP2024-05-11T07:06:02Zoai:repositorium.sdum.uminho.pt:1822/39302Portal AgregadorONGhttps://www.rcaap.pt/oai/openairemluisa.alvim@gmail.comopendoar:71602024-05-11T07:06:02Repositório 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 |
Curry-Howard for sequent calculus at last! |
title |
Curry-Howard for sequent calculus at last! |
spellingShingle |
Curry-Howard for sequent calculus at last! Espírito Santo, José Co-control Co-continuation Vector notation Let-expression Formal substitution Context substitution Computational lambda-calculus Classical logic de Morgan duality Ciências Naturais::Matemáticas |
title_short |
Curry-Howard for sequent calculus at last! |
title_full |
Curry-Howard for sequent calculus at last! |
title_fullStr |
Curry-Howard for sequent calculus at last! |
title_full_unstemmed |
Curry-Howard for sequent calculus at last! |
title_sort |
Curry-Howard for sequent calculus at last! |
author |
Espírito Santo, José |
author_facet |
Espírito Santo, José |
author_role |
author |
dc.contributor.none.fl_str_mv |
Universidade do Minho |
dc.contributor.author.fl_str_mv |
Espírito Santo, José |
dc.subject.por.fl_str_mv |
Co-control Co-continuation Vector notation Let-expression Formal substitution Context substitution Computational lambda-calculus Classical logic de Morgan duality Ciências Naturais::Matemáticas |
topic |
Co-control Co-continuation Vector notation Let-expression Formal substitution Context substitution Computational lambda-calculus Classical logic de Morgan duality Ciências Naturais::Matemáticas |
description |
This paper tries to remove what seems to be the remaining stumbling blocks in the way to a full understanding of the Curry-Howard isomorphism for sequent calculus, namely the questions: What do variables in proof terms stand for? What is co-control and a co-continuation? How to define the dual of Parigot's mu-operator so that it is a co-control operator? Answering these questions leads to the interpretation that sequent calculus is a formal vector notation with first-class co-control. But this is just the "internal" interpretation, which has to be developed simultaneously with, and is justified by, an "external" one, offered by natural deduction: the sequent calculus corresponds to a bi-directional, agnostic (w.r.t. the call strategy), computational lambda-calculus. Next, the duality between control and co-control is studied and proved in the context of classical logic, where one discovers that the classical sequent calculus has a distortion towards control, and that sequent calculus is the de Morgan dual of natural deduction. |
publishDate |
2015 |
dc.date.none.fl_str_mv |
2015-06-12 2015-06-12T00:00:00Z |
dc.type.driver.fl_str_mv |
conference paper |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/1822/39302 |
url |
http://hdl.handle.net/1822/39302 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
9783939897873 1868-8969 10.4230/LIPIcs.TLCA.2015.165 |
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 |
Schloss Dagstuhl – Leibniz-Zentrum für Informatik GmbH |
publisher.none.fl_str_mv |
Schloss Dagstuhl – Leibniz-Zentrum für Informatik GmbH |
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 |
mluisa.alvim@gmail.com |
_version_ |
1817545204187529216 |