k-Provability in PA

Detalhes bibliográficos
Autor(a) principal: Santos, Paulo Guilherme
Data de Publicação: 2021
Outros Autores: Kahle, Reinhard
Tipo de documento: Artigo
Idioma: eng
Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
Texto Completo: http://hdl.handle.net/10362/124149
Resumo: We study the decidability of k-provability in PA —the relation ‘being provable in PA with at most k steps’—and the decidability of the proof-skeleton problem—the problem of deciding if a given formula has a proof that has a given skeleton (the list of axioms and rules that were used). The decidability of k-provability for the usual Hilbert-style formalisation of PA is still an open problem, but it is known that the proof-skeleton problem is undecidable for that theory. Using new methods, we present a characterisation of some numbers k for which k-provability is decidable, and we present a characterisation of some proof-skeletons for which one can decide whether a formula has a proof whose skeleton is the considered one. These characterisations are natural and parameterised by unification algorithms.
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spelling k-Provability in PADecidabilityk-provabilityPeano arithmeticProof-skeleton problemLogicApplied MathematicsWe study the decidability of k-provability in PA —the relation ‘being provable in PA with at most k steps’—and the decidability of the proof-skeleton problem—the problem of deciding if a given formula has a proof that has a given skeleton (the list of axioms and rules that were used). The decidability of k-provability for the usual Hilbert-style formalisation of PA is still an open problem, but it is known that the proof-skeleton problem is undecidable for that theory. Using new methods, we present a characterisation of some numbers k for which k-provability is decidable, and we present a characterisation of some proof-skeletons for which one can decide whether a formula has a proof whose skeleton is the considered one. These characterisations are natural and parameterised by unification algorithms.CMA - Centro de Matemática e AplicaçõesRUNSantos, Paulo GuilhermeKahle, Reinhard2021-09-08T00:19:23Z2021-122021-12-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10362/124149eng1661-8297PURE: 32227129https://doi.org/10.1007/s11787-021-00278-1info: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:RCAAP2023-07-10T16:02:32ZPortal AgregadorONG
dc.title.none.fl_str_mv k-Provability in PA
title k-Provability in PA
spellingShingle k-Provability in PA
Santos, Paulo Guilherme
Decidability
k-provability
Peano arithmetic
Proof-skeleton problem
Logic
Applied Mathematics
title_short k-Provability in PA
title_full k-Provability in PA
title_fullStr k-Provability in PA
title_full_unstemmed k-Provability in PA
title_sort k-Provability in PA
author Santos, Paulo Guilherme
author_facet Santos, Paulo Guilherme
Kahle, Reinhard
author_role author
author2 Kahle, Reinhard
author2_role author
dc.contributor.none.fl_str_mv CMA - Centro de Matemática e Aplicações
RUN
dc.contributor.author.fl_str_mv Santos, Paulo Guilherme
Kahle, Reinhard
dc.subject.por.fl_str_mv Decidability
k-provability
Peano arithmetic
Proof-skeleton problem
Logic
Applied Mathematics
topic Decidability
k-provability
Peano arithmetic
Proof-skeleton problem
Logic
Applied Mathematics
description We study the decidability of k-provability in PA —the relation ‘being provable in PA with at most k steps’—and the decidability of the proof-skeleton problem—the problem of deciding if a given formula has a proof that has a given skeleton (the list of axioms and rules that were used). The decidability of k-provability for the usual Hilbert-style formalisation of PA is still an open problem, but it is known that the proof-skeleton problem is undecidable for that theory. Using new methods, we present a characterisation of some numbers k for which k-provability is decidable, and we present a characterisation of some proof-skeletons for which one can decide whether a formula has a proof whose skeleton is the considered one. These characterisations are natural and parameterised by unification algorithms.
publishDate 2021
dc.date.none.fl_str_mv 2021-09-08T00:19:23Z
2021-12
2021-12-01T00:00:00Z
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 http://hdl.handle.net/10362/124149
url http://hdl.handle.net/10362/124149
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv 1661-8297
PURE: 32227129
https://doi.org/10.1007/s11787-021-00278-1
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.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
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