Curry-Howard for sequent calculus at last!

Detalhes bibliográficos
Autor(a) principal: Espírito Santo, José
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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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
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