Coinductive proof search for polarized logic with applications to full intuitionistic propositional logic
Autor(a) principal: | |
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Data de Publicação: | 2021 |
Outros Autores: | , |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/1822/75253 |
Resumo: | The approach to proof search dubbed “coinductive proof search”, and previously developed by the authors for implicational intuitionistic logic, is in this paper extended to LJP, a focused sequent-calculus presentation of polarized intuitionistic logic, including an array of positive and negative connectives. As before, this includes developing a coinductive description of the search space generated by a sequent, an equivalent inductive syntax describing the same space, and decision procedures for inhabitation problems in the form of predicates defined by recursion on the inductive syntax. We prove the decidability of existence of focused inhabitants, and of finiteness of the number of focused inhabitants for polarized intuitionistic logic, by means of such recursive procedures. Moreover, the polarized logic can be used as a platform from which proof search for other logics is understood. We illustrate the technique with LJT, a focused sequent calculus for full intuitionistic propositional logic (including disjunction). For that, we have to work out the “negative translation” of LJT into LJP (that sees all intuitionistic types as negative types), and verify that the translation gives a faithful representation of proof search in LJT as proof search in the polarized logic. We therefore inherit decidability of both problems studied for LJP and thus get new proofs of these results for LJT. |
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Coinductive proof search for polarized logic with applications to full intuitionistic propositional logicCoinductionInhabitation problemsLambda-calculusPolarized logicCiências Naturais::Ciências da Computação e da InformaçãoThe approach to proof search dubbed “coinductive proof search”, and previously developed by the authors for implicational intuitionistic logic, is in this paper extended to LJP, a focused sequent-calculus presentation of polarized intuitionistic logic, including an array of positive and negative connectives. As before, this includes developing a coinductive description of the search space generated by a sequent, an equivalent inductive syntax describing the same space, and decision procedures for inhabitation problems in the form of predicates defined by recursion on the inductive syntax. We prove the decidability of existence of focused inhabitants, and of finiteness of the number of focused inhabitants for polarized intuitionistic logic, by means of such recursive procedures. Moreover, the polarized logic can be used as a platform from which proof search for other logics is understood. We illustrate the technique with LJT, a focused sequent calculus for full intuitionistic propositional logic (including disjunction). For that, we have to work out the “negative translation” of LJT into LJP (that sees all intuitionistic types as negative types), and verify that the translation gives a faithful representation of proof search in LJT as proof search in the polarized logic. We therefore inherit decidability of both problems studied for LJP and thus get new proofs of these results for LJT.COST - European Cooperation in Science and TechnologySchloss Dagstuhl – Leibniz-Zentrum für Informatik GmbHUniversidade do MinhoEspírito Santo, JoséMatthes, RalphPinto, Luís F.2021-06-012021-06-01T00:00:00Zconference paperinfo:eu-repo/semantics/publishedVersionapplication/pdfhttp://hdl.handle.net/1822/75253eng978-3-95977-182-51868-896910.4230/LIPIcs.TYPES.2020.4https://doi.org/ 10.4230/LIPIcs.TYPES.2020.4info: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:RCAAP2024-05-11T04:50:08Zoai:repositorium.sdum.uminho.pt:1822/75253Portal AgregadorONGhttps://www.rcaap.pt/oai/openairemluisa.alvim@gmail.comopendoar:71602024-05-11T04:50:08Repositório 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 |
Coinductive proof search for polarized logic with applications to full intuitionistic propositional logic |
title |
Coinductive proof search for polarized logic with applications to full intuitionistic propositional logic |
spellingShingle |
Coinductive proof search for polarized logic with applications to full intuitionistic propositional logic Espírito Santo, José Coinduction Inhabitation problems Lambda-calculus Polarized logic Ciências Naturais::Ciências da Computação e da Informação |
title_short |
Coinductive proof search for polarized logic with applications to full intuitionistic propositional logic |
title_full |
Coinductive proof search for polarized logic with applications to full intuitionistic propositional logic |
title_fullStr |
Coinductive proof search for polarized logic with applications to full intuitionistic propositional logic |
title_full_unstemmed |
Coinductive proof search for polarized logic with applications to full intuitionistic propositional logic |
title_sort |
Coinductive proof search for polarized logic with applications to full intuitionistic propositional logic |
author |
Espírito Santo, José |
author_facet |
Espírito Santo, José Matthes, Ralph Pinto, Luís F. |
author_role |
author |
author2 |
Matthes, Ralph 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 Matthes, Ralph Pinto, Luís F. |
dc.subject.por.fl_str_mv |
Coinduction Inhabitation problems Lambda-calculus Polarized logic Ciências Naturais::Ciências da Computação e da Informação |
topic |
Coinduction Inhabitation problems Lambda-calculus Polarized logic Ciências Naturais::Ciências da Computação e da Informação |
description |
The approach to proof search dubbed “coinductive proof search”, and previously developed by the authors for implicational intuitionistic logic, is in this paper extended to LJP, a focused sequent-calculus presentation of polarized intuitionistic logic, including an array of positive and negative connectives. As before, this includes developing a coinductive description of the search space generated by a sequent, an equivalent inductive syntax describing the same space, and decision procedures for inhabitation problems in the form of predicates defined by recursion on the inductive syntax. We prove the decidability of existence of focused inhabitants, and of finiteness of the number of focused inhabitants for polarized intuitionistic logic, by means of such recursive procedures. Moreover, the polarized logic can be used as a platform from which proof search for other logics is understood. We illustrate the technique with LJT, a focused sequent calculus for full intuitionistic propositional logic (including disjunction). For that, we have to work out the “negative translation” of LJT into LJP (that sees all intuitionistic types as negative types), and verify that the translation gives a faithful representation of proof search in LJT as proof search in the polarized logic. We therefore inherit decidability of both problems studied for LJP and thus get new proofs of these results for LJT. |
publishDate |
2021 |
dc.date.none.fl_str_mv |
2021-06-01 2021-06-01T00:00:00Z |
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/1822/75253 |
url |
http://hdl.handle.net/1822/75253 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
978-3-95977-182-5 1868-8969 10.4230/LIPIcs.TYPES.2020.4 https://doi.org/ 10.4230/LIPIcs.TYPES.2020.4 |
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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1817544433255579648 |