Structural proof theory as rewriting

Detalhes bibliográficos
Autor(a) principal: Frade, M. J.
Data de Publicação: 2006
Outros Autores: Espírito Santo, José, Pinto, L.
Idioma: eng
Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
Texto Completo: https://hdl.handle.net/1822/35976
Resumo: The multiary version of the $\lambda$-calculus with generalized applications integrates smoothly both a fragment of sequent calculus and the system of natural deduction of von Plato. It is equipped with reduction rules (corresponding to cut-elimination/normalisation rules) and permutation rules, typical of sequent calculus and of natural deduction with generalised elimination rules. We argue that this system is a suitable tool for doing structural proof theory as rewriting. As an illustration, we investigate combinations of reduction and permutation rules and whether these combinations induce rewriting systems which are confluent and terminating. In some cases, the combination allows the simulation of non-terminating reduction sequences known from explicit substitution calculi. In other cases, we succeed in capturing interesting classes of derivations as the normal forms w.r.t. well-behaved combinations of rules. We identify six of these ``combined'' normal forms, among which are two classes, due to Herbelin and Mints, in bijection with normal, ordinary natural deductions. A computational explanation for the variety of ``combined'' normal forms is the existence of three ways of expressing multiple application in the calculus.
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spelling Structural proof theory as rewritingScience & TechnologyThe multiary version of the $\lambda$-calculus with generalized applications integrates smoothly both a fragment of sequent calculus and the system of natural deduction of von Plato. It is equipped with reduction rules (corresponding to cut-elimination/normalisation rules) and permutation rules, typical of sequent calculus and of natural deduction with generalised elimination rules. We argue that this system is a suitable tool for doing structural proof theory as rewriting. As an illustration, we investigate combinations of reduction and permutation rules and whether these combinations induce rewriting systems which are confluent and terminating. In some cases, the combination allows the simulation of non-terminating reduction sequences known from explicit substitution calculi. In other cases, we succeed in capturing interesting classes of derivations as the normal forms w.r.t. well-behaved combinations of rules. We identify six of these ``combined'' normal forms, among which are two classes, due to Herbelin and Mints, in bijection with normal, ordinary natural deductions. A computational explanation for the variety of ``combined'' normal forms is the existence of three ways of expressing multiple application in the calculus.(undefined)SpringerSpringerUniversidade do MinhoFrade, M. J.Espírito Santo, JoséPinto, L.20062006-01-01T00:00:00Zconference paperinfo:eu-repo/semantics/publishedVersionapplication/pdfhttps://hdl.handle.net/1822/35976engSanto, J.E., Frade, M.J., Pinto, L. (2006). Structural Proof Theory as Rewriting. In: Pfenning, F. (eds) Term Rewriting and Applications. RTA 2006. Lecture Notes in Computer Science, vol 4098. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11805618_15978-3-540-36834-20302-974310.1007/11805618_15https://link.springer.com/chapter/10.1007/11805618_15info: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:RCAAP2024-07-27T01:22:20Zoai:repositorium.sdum.uminho.pt:1822/35976Portal AgregadorONGhttps://www.rcaap.pt/oai/openairemluisa.alvim@gmail.comopendoar:71602024-07-27T01:22:20Repositório 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 Structural proof theory as rewriting
title Structural proof theory as rewriting
spellingShingle Structural proof theory as rewriting
Frade, M. J.
Science & Technology
title_short Structural proof theory as rewriting
title_full Structural proof theory as rewriting
title_fullStr Structural proof theory as rewriting
title_full_unstemmed Structural proof theory as rewriting
title_sort Structural proof theory as rewriting
author Frade, M. J.
author_facet Frade, M. J.
Espírito Santo, José
Pinto, L.
author_role author
author2 Espírito Santo, José
Pinto, L.
author2_role author
author
dc.contributor.none.fl_str_mv Universidade do Minho
dc.contributor.author.fl_str_mv Frade, M. J.
Espírito Santo, José
Pinto, L.
dc.subject.por.fl_str_mv Science & Technology
topic Science & Technology
description The multiary version of the $\lambda$-calculus with generalized applications integrates smoothly both a fragment of sequent calculus and the system of natural deduction of von Plato. It is equipped with reduction rules (corresponding to cut-elimination/normalisation rules) and permutation rules, typical of sequent calculus and of natural deduction with generalised elimination rules. We argue that this system is a suitable tool for doing structural proof theory as rewriting. As an illustration, we investigate combinations of reduction and permutation rules and whether these combinations induce rewriting systems which are confluent and terminating. In some cases, the combination allows the simulation of non-terminating reduction sequences known from explicit substitution calculi. In other cases, we succeed in capturing interesting classes of derivations as the normal forms w.r.t. well-behaved combinations of rules. We identify six of these ``combined'' normal forms, among which are two classes, due to Herbelin and Mints, in bijection with normal, ordinary natural deductions. A computational explanation for the variety of ``combined'' normal forms is the existence of three ways of expressing multiple application in the calculus.
publishDate 2006
dc.date.none.fl_str_mv 2006
2006-01-01T00:00:00Z
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 https://hdl.handle.net/1822/35976
url https://hdl.handle.net/1822/35976
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv Santo, J.E., Frade, M.J., Pinto, L. (2006). Structural Proof Theory as Rewriting. In: Pfenning, F. (eds) Term Rewriting and Applications. RTA 2006. Lecture Notes in Computer Science, vol 4098. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11805618_15
978-3-540-36834-2
0302-9743
10.1007/11805618_15
https://link.springer.com/chapter/10.1007/11805618_15
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 Springer
Springer
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dc.source.none.fl_str_mv reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
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instacron:RCAAP
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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
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