Permutability in proof terms for intuitionistic sequent calculus with cuts

Detalhes bibliográficos
Autor(a) principal: Espírito Santo, José
Data de Publicação: 2018
Outros Autores: Frade, M. J., Pinto, Luís F.
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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spelling 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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