The lambda-calculus and the unity of structural proof theory
Autor(a) principal: | |
---|---|
Data de Publicação: | 2009 |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/1822/11174 |
Resumo: | In the context of intuitionistic implicational logic, we achieve a perfect correspondence (technically an isomorphism) between sequent calculus and natural deduction, based on perfect correspondences between left-introduction and elimination, cut and substitution, and cut-elimination and normalisation. This requires an enlarged system of natural deduction that refines von Plato's calculus. It is a calculus with modus ponens and primitive substitution; it is also a ``coercion calculus'', in the sense of Cervesato and Pfenning. Both sequent calculus and natural deduction are presented as typing systems for appropriate extensions of the $\lambda$-calculus. The whole difference between the two calculi is reduced to the associativity of applicative terms (sequent calculus = right associative, natural deduction = left associative), and in fact the achieved isomorphism may be described as the mere inversion of that associativity. The novel natural deduction system is a ``multiary'' calculus, because ``applicative terms'' may exhibit a list of several arguments. But the combination of ``multiarity'' and left-associativity seems simply wrong, leading necessarily to non-local reduction rules (reason: nomalisation, like cut-elimination, acts at the head of applicative terms, but natural deduction focuses at the tail of such terms). A solution is to extend natural deduction even further to a calculus that unifies sequent calculus and natural deduction, based on the unification of cut and substitution. In the unified calculus, a sequent term behaves like in the sequent calculus, whereas the reduction steps of a natural deduction term are interleaved with explicit steps for bringing heads to focus. A variant of the calculus has the symmetric role of improving sequent calculus in dealing with tail-active permutative conversions. |
id |
RCAP_7e27b09ad6afbd2036e62496e85ba877 |
---|---|
oai_identifier_str |
oai:repositorium.sdum.uminho.pt:1822/11174 |
network_acronym_str |
RCAP |
network_name_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
repository_id_str |
7160 |
spelling |
The lambda-calculus and the unity of structural proof theoryScience & TechnologyIn the context of intuitionistic implicational logic, we achieve a perfect correspondence (technically an isomorphism) between sequent calculus and natural deduction, based on perfect correspondences between left-introduction and elimination, cut and substitution, and cut-elimination and normalisation. This requires an enlarged system of natural deduction that refines von Plato's calculus. It is a calculus with modus ponens and primitive substitution; it is also a ``coercion calculus'', in the sense of Cervesato and Pfenning. Both sequent calculus and natural deduction are presented as typing systems for appropriate extensions of the $\lambda$-calculus. The whole difference between the two calculi is reduced to the associativity of applicative terms (sequent calculus = right associative, natural deduction = left associative), and in fact the achieved isomorphism may be described as the mere inversion of that associativity. The novel natural deduction system is a ``multiary'' calculus, because ``applicative terms'' may exhibit a list of several arguments. But the combination of ``multiarity'' and left-associativity seems simply wrong, leading necessarily to non-local reduction rules (reason: nomalisation, like cut-elimination, acts at the head of applicative terms, but natural deduction focuses at the tail of such terms). A solution is to extend natural deduction even further to a calculus that unifies sequent calculus and natural deduction, based on the unification of cut and substitution. In the unified calculus, a sequent term behaves like in the sequent calculus, whereas the reduction steps of a natural deduction term are interleaved with explicit steps for bringing heads to focus. A variant of the calculus has the symmetric role of improving sequent calculus in dealing with tail-active permutative conversions.Fundação para a Ciência e Tecnologia (FCT)SpringerUniversidade do MinhoEspírito Santo, José2009-02-052009-02-05T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/1822/11174eng"Theory of Computing Systems". ISSN 1432-4350. 45 (Febr. 2009) 963-994.1432-435010.1007/s00224-009-9183-9www.springerlink.cominfo: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:RCAAP2024-05-11T05:35:50Zoai:repositorium.sdum.uminho.pt:1822/11174Portal AgregadorONGhttps://www.rcaap.pt/oai/openairemluisa.alvim@gmail.comopendoar:71602024-05-11T05:35:50Repositório 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 |
The lambda-calculus and the unity of structural proof theory |
title |
The lambda-calculus and the unity of structural proof theory |
spellingShingle |
The lambda-calculus and the unity of structural proof theory Espírito Santo, José Science & Technology |
title_short |
The lambda-calculus and the unity of structural proof theory |
title_full |
The lambda-calculus and the unity of structural proof theory |
title_fullStr |
The lambda-calculus and the unity of structural proof theory |
title_full_unstemmed |
The lambda-calculus and the unity of structural proof theory |
title_sort |
The lambda-calculus and the unity of structural proof theory |
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 |
Science & Technology |
topic |
Science & Technology |
description |
In the context of intuitionistic implicational logic, we achieve a perfect correspondence (technically an isomorphism) between sequent calculus and natural deduction, based on perfect correspondences between left-introduction and elimination, cut and substitution, and cut-elimination and normalisation. This requires an enlarged system of natural deduction that refines von Plato's calculus. It is a calculus with modus ponens and primitive substitution; it is also a ``coercion calculus'', in the sense of Cervesato and Pfenning. Both sequent calculus and natural deduction are presented as typing systems for appropriate extensions of the $\lambda$-calculus. The whole difference between the two calculi is reduced to the associativity of applicative terms (sequent calculus = right associative, natural deduction = left associative), and in fact the achieved isomorphism may be described as the mere inversion of that associativity. The novel natural deduction system is a ``multiary'' calculus, because ``applicative terms'' may exhibit a list of several arguments. But the combination of ``multiarity'' and left-associativity seems simply wrong, leading necessarily to non-local reduction rules (reason: nomalisation, like cut-elimination, acts at the head of applicative terms, but natural deduction focuses at the tail of such terms). A solution is to extend natural deduction even further to a calculus that unifies sequent calculus and natural deduction, based on the unification of cut and substitution. In the unified calculus, a sequent term behaves like in the sequent calculus, whereas the reduction steps of a natural deduction term are interleaved with explicit steps for bringing heads to focus. A variant of the calculus has the symmetric role of improving sequent calculus in dealing with tail-active permutative conversions. |
publishDate |
2009 |
dc.date.none.fl_str_mv |
2009-02-05 2009-02-05T00:00:00Z |
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 |
http://hdl.handle.net/1822/11174 |
url |
http://hdl.handle.net/1822/11174 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
"Theory of Computing Systems". ISSN 1432-4350. 45 (Febr. 2009) 963-994. 1432-4350 10.1007/s00224-009-9183-9 www.springerlink.com |
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 |
publisher.none.fl_str_mv |
Springer |
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 |
_version_ |
1817544673680424960 |