Variations and interpretations of naturality in call-by-name lambda-calculi with generalized applications
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/1822/87622 |
Resumo: | In the context of intuitionistic sequent calculus, “naturality” means permutation-freeness (the terminology is essentially due to Mints). We study naturality in the context of the lambda-calculus with generalized applications and its multiary extension, to cover, under the Curry-Howard correspondence, proof systems ranging from natural deduction (with and without general elimination rules) to a fragment of sequent calculus with an iterable left-introduction rule, and which can still be recognized as a call-by-name lambda-calculus. In this context, naturality consists of a certain restricted use of generalized applications. We consider the further restriction obtained by the combination of naturality with normality w.r.t. the commutative conversion engendered by generalized applications. This combination sheds light on the interpretation of naturality as a vectorization mechanism, allowing a multitude of different ways of structuring lambda-terms, and the structuring of a multitude of interesting fragments of the systems under study. We also consider a relaxation of naturality, called weak naturality: this not only brings similar structural benefits, but also suggests a new “weak” system of natural deduction with generalized applications which is exempt from commutative conversions. In the end, we use all of this evidence as a stepping stone to propose a computational interpretation of generalized application (whether multiary or not, and without any restriction): it includes, alongside the argument(s) for the function, a general list – a new, very general, vectorization mechanism, that structures the continuation of the computation. |
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Variations and interpretations of naturality in call-by-name lambda-calculi with generalized applicationsCommutative conversionNatural deductionNormal and natural proofsPermutative conversionSequent calculusVector notationScience & TechnologyIn the context of intuitionistic sequent calculus, “naturality” means permutation-freeness (the terminology is essentially due to Mints). We study naturality in the context of the lambda-calculus with generalized applications and its multiary extension, to cover, under the Curry-Howard correspondence, proof systems ranging from natural deduction (with and without general elimination rules) to a fragment of sequent calculus with an iterable left-introduction rule, and which can still be recognized as a call-by-name lambda-calculus. In this context, naturality consists of a certain restricted use of generalized applications. We consider the further restriction obtained by the combination of naturality with normality w.r.t. the commutative conversion engendered by generalized applications. This combination sheds light on the interpretation of naturality as a vectorization mechanism, allowing a multitude of different ways of structuring lambda-terms, and the structuring of a multitude of interesting fragments of the systems under study. We also consider a relaxation of naturality, called weak naturality: this not only brings similar structural benefits, but also suggests a new “weak” system of natural deduction with generalized applications which is exempt from commutative conversions. In the end, we use all of this evidence as a stepping stone to propose a computational interpretation of generalized application (whether multiary or not, and without any restriction): it includes, alongside the argument(s) for the function, a general list – a new, very general, vectorization mechanism, that structures the continuation of the computation.The authors were financed by Portuguese Funds through FCT (Fundação para a Ciência e Tecnologia) within the projects UIDB/00013/2020, UIDP/00013/2020 and UIDB/50014/2020.ElsevierUniversidade do MinhoEspírito Santo, JoséFrade, M. J.Pinto, Luís F.20232023-01-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/1822/87622engEspírito Santo, J., Frade, M. J., & Pinto, L. (2023, February). Variations and interpretations of naturality in call-by-name lambda-calculi with generalized applications. Journal of Logical and Algebraic Methods in Programming. Elsevier BV. http://doi.org/10.1016/j.jlamp.2022.1008302352-22082352-221610.1016/j.jlamp.2022.100830100830https://doi.org/10.1016/j.jlamp.2022.100830info:eu-repo/semantics/embargoedAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2024-02-03T01:20:09Zoai:repositorium.sdum.uminho.pt:1822/87622Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-20T00:55:36.742958Repositório 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 |
Variations and interpretations of naturality in call-by-name lambda-calculi with generalized applications |
title |
Variations and interpretations of naturality in call-by-name lambda-calculi with generalized applications |
spellingShingle |
Variations and interpretations of naturality in call-by-name lambda-calculi with generalized applications Espírito Santo, José Commutative conversion Natural deduction Normal and natural proofs Permutative conversion Sequent calculus Vector notation Science & Technology |
title_short |
Variations and interpretations of naturality in call-by-name lambda-calculi with generalized applications |
title_full |
Variations and interpretations of naturality in call-by-name lambda-calculi with generalized applications |
title_fullStr |
Variations and interpretations of naturality in call-by-name lambda-calculi with generalized applications |
title_full_unstemmed |
Variations and interpretations of naturality in call-by-name lambda-calculi with generalized applications |
title_sort |
Variations and interpretations of naturality in call-by-name lambda-calculi with generalized applications |
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 |
Commutative conversion Natural deduction Normal and natural proofs Permutative conversion Sequent calculus Vector notation Science & Technology |
topic |
Commutative conversion Natural deduction Normal and natural proofs Permutative conversion Sequent calculus Vector notation Science & Technology |
description |
In the context of intuitionistic sequent calculus, “naturality” means permutation-freeness (the terminology is essentially due to Mints). We study naturality in the context of the lambda-calculus with generalized applications and its multiary extension, to cover, under the Curry-Howard correspondence, proof systems ranging from natural deduction (with and without general elimination rules) to a fragment of sequent calculus with an iterable left-introduction rule, and which can still be recognized as a call-by-name lambda-calculus. In this context, naturality consists of a certain restricted use of generalized applications. We consider the further restriction obtained by the combination of naturality with normality w.r.t. the commutative conversion engendered by generalized applications. This combination sheds light on the interpretation of naturality as a vectorization mechanism, allowing a multitude of different ways of structuring lambda-terms, and the structuring of a multitude of interesting fragments of the systems under study. We also consider a relaxation of naturality, called weak naturality: this not only brings similar structural benefits, but also suggests a new “weak” system of natural deduction with generalized applications which is exempt from commutative conversions. In the end, we use all of this evidence as a stepping stone to propose a computational interpretation of generalized application (whether multiary or not, and without any restriction): it includes, alongside the argument(s) for the function, a general list – a new, very general, vectorization mechanism, that structures the continuation of the computation. |
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/1822/87622 |
url |
https://hdl.handle.net/1822/87622 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
Espírito Santo, J., Frade, M. J., & Pinto, L. (2023, February). Variations and interpretations of naturality in call-by-name lambda-calculi with generalized applications. Journal of Logical and Algebraic Methods in Programming. Elsevier BV. http://doi.org/10.1016/j.jlamp.2022.100830 2352-2208 2352-2216 10.1016/j.jlamp.2022.100830 100830 https://doi.org/10.1016/j.jlamp.2022.100830 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/embargoedAccess |
eu_rights_str_mv |
embargoedAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.publisher.none.fl_str_mv |
Elsevier |
publisher.none.fl_str_mv |
Elsevier |
dc.source.none.fl_str_mv |
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Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
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RCAAP |
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RCAAP |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
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