Permutability in proof terms for intuitionistic sequent calculus with cuts
Autor(a) principal: | |
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Data de Publicação: | 2018 |
Outros Autores: | , |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/1822/58009 |
Resumo: | This paper gives a comprehensive and coherent view on permutability in the intuitionistic sequent calculus with cuts. Specifically we show that, once permutability is packaged into appropriate global reduction procedures, it organizes the internal structure of the system and determines fragments with computational interest, both for the computation-as-proof-normalization and the computation-as-proof-search paradigms. The vehicle of the study is a lambda-calculus of multiary proof terms with generalized application, previously developed by the authors (the paper argues this system represents the simplest fragment of ordinary sequent calculus that does not fall into mere natural deduction). We start by adapting to our setting the concept of normal proof, developed by Mints, Dyckhoff, and Pinto, and by defining natural proofs, so that a proof is normal iff it is natural and cut-free. Natural proofs form a subsystem with a transparent Curry-Howard interpretation (a kind of formal vector notation for lambda-terms with vectors consisting of lists of lists of arguments), while searching for normal proofs corresponds to a slight relaxation of focusing (in the sense of LJT). Next, we define a process of permutative conversion to natural form, and show that its combination with cut elimination gives a concept of normalization for the sequent calculus. We derive a systematic picture of the full system comprehending a rich set of reduction procedures (cut elimination, flattening, permutative conversion, normalization, focalization), organizing the relevant subsystems and the important subclasses of cut-free, normal, and focused proofs. |
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Permutability in proof terms for intuitionistic sequent calculus with cutsSequent calculusPermutative conversionCurry-Howard isomorphismVector of argumentsGeneralized applicationNormal proofNatural proofCut eliminationFlatenningNormalizationFocalizationCiências Naturais::MatemáticasCiências Naturais::Ciências da Computação e da InformaçãoThis paper gives a comprehensive and coherent view on permutability in the intuitionistic sequent calculus with cuts. Specifically we show that, once permutability is packaged into appropriate global reduction procedures, it organizes the internal structure of the system and determines fragments with computational interest, both for the computation-as-proof-normalization and the computation-as-proof-search paradigms. The vehicle of the study is a lambda-calculus of multiary proof terms with generalized application, previously developed by the authors (the paper argues this system represents the simplest fragment of ordinary sequent calculus that does not fall into mere natural deduction). We start by adapting to our setting the concept of normal proof, developed by Mints, Dyckhoff, and Pinto, and by defining natural proofs, so that a proof is normal iff it is natural and cut-free. Natural proofs form a subsystem with a transparent Curry-Howard interpretation (a kind of formal vector notation for lambda-terms with vectors consisting of lists of lists of arguments), while searching for normal proofs corresponds to a slight relaxation of focusing (in the sense of LJT). Next, we define a process of permutative conversion to natural form, and show that its combination with cut elimination gives a concept of normalization for the sequent calculus. We derive a systematic picture of the full system comprehending a rich set of reduction procedures (cut elimination, flattening, permutative conversion, normalization, focalization), organizing the relevant subsystems and the important subclasses of cut-free, normal, and focused proofs.Partially financed by FCT through project UID/MAT/00013/2013, and by COST action CA15123 EUTYPES. The first and the last authors were partially financed by Fundação para a Ciência e a Tecnologia (FCT) through project UID/MAT/00013/2013. The first author got financial support by the COST action CA15123 EUTYPES.info:eu-repo/semantics/publishedVersionSchloss Dagstuhl – Leibniz-Zentrum für Informatik GmbHUniversidade do MinhoEspírito Santo, JoséFrade, M. J.Pinto, Luís F.20182018-01-01T00:00:00Zconference paperinfo:eu-repo/semantics/publishedVersionapplication/pdfhttp://hdl.handle.net/1822/58009eng97839597706511868-896910.4230/LIPIcs.TYPES.2016.10info: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:RCAAP2024-05-11T07:05:00Zoai:repositorium.sdum.uminho.pt:1822/58009Portal AgregadorONGhttps://www.rcaap.pt/oai/openairemluisa.alvim@gmail.comopendoar:71602024-05-11T07:05Repositório 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 |
Permutability in proof terms for intuitionistic sequent calculus with cuts |
title |
Permutability in proof terms for intuitionistic sequent calculus with cuts |
spellingShingle |
Permutability in proof terms for intuitionistic sequent calculus with cuts Espírito Santo, José Sequent calculus Permutative conversion Curry-Howard isomorphism Vector of arguments Generalized application Normal proof Natural proof Cut elimination Flatenning Normalization Focalization Ciências Naturais::Matemáticas Ciências Naturais::Ciências da Computação e da Informação |
title_short |
Permutability in proof terms for intuitionistic sequent calculus with cuts |
title_full |
Permutability in proof terms for intuitionistic sequent calculus with cuts |
title_fullStr |
Permutability in proof terms for intuitionistic sequent calculus with cuts |
title_full_unstemmed |
Permutability in proof terms for intuitionistic sequent calculus with cuts |
title_sort |
Permutability in proof terms for intuitionistic sequent calculus with cuts |
author |
Espírito Santo, José |
author_facet |
Espírito Santo, José Frade, M. J. Pinto, Luís F. |
author_role |
author |
author2 |
Frade, M. J. Pinto, Luís F. |
author2_role |
author author |
dc.contributor.none.fl_str_mv |
Universidade do Minho |
dc.contributor.author.fl_str_mv |
Espírito Santo, José Frade, M. J. Pinto, Luís F. |
dc.subject.por.fl_str_mv |
Sequent calculus Permutative conversion Curry-Howard isomorphism Vector of arguments Generalized application Normal proof Natural proof Cut elimination Flatenning Normalization Focalization Ciências Naturais::Matemáticas Ciências Naturais::Ciências da Computação e da Informação |
topic |
Sequent calculus Permutative conversion Curry-Howard isomorphism Vector of arguments Generalized application Normal proof Natural proof Cut elimination Flatenning Normalization Focalization Ciências Naturais::Matemáticas Ciências Naturais::Ciências da Computação e da Informação |
description |
This paper gives a comprehensive and coherent view on permutability in the intuitionistic sequent calculus with cuts. Specifically we show that, once permutability is packaged into appropriate global reduction procedures, it organizes the internal structure of the system and determines fragments with computational interest, both for the computation-as-proof-normalization and the computation-as-proof-search paradigms. The vehicle of the study is a lambda-calculus of multiary proof terms with generalized application, previously developed by the authors (the paper argues this system represents the simplest fragment of ordinary sequent calculus that does not fall into mere natural deduction). We start by adapting to our setting the concept of normal proof, developed by Mints, Dyckhoff, and Pinto, and by defining natural proofs, so that a proof is normal iff it is natural and cut-free. Natural proofs form a subsystem with a transparent Curry-Howard interpretation (a kind of formal vector notation for lambda-terms with vectors consisting of lists of lists of arguments), while searching for normal proofs corresponds to a slight relaxation of focusing (in the sense of LJT). Next, we define a process of permutative conversion to natural form, and show that its combination with cut elimination gives a concept of normalization for the sequent calculus. We derive a systematic picture of the full system comprehending a rich set of reduction procedures (cut elimination, flattening, permutative conversion, normalization, focalization), organizing the relevant subsystems and the important subclasses of cut-free, normal, and focused proofs. |
publishDate |
2018 |
dc.date.none.fl_str_mv |
2018 2018-01-01T00:00:00Z |
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/1822/58009 |
url |
http://hdl.handle.net/1822/58009 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
9783959770651 1868-8969 10.4230/LIPIcs.TYPES.2016.10 |
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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1817545197688455168 |