Coinductive proof search for polarized logic with applications to full intuitionistic propositional logic

Detalhes bibliográficos
Autor(a) principal: Espírito Santo, José
Data de Publicação: 2021
Outros Autores: Matthes, Ralph, Pinto, Luís F.
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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spelling 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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