Publicação
Permutability in proof terms for intuitionistic sequent calculus with cuts
| Resumo: | This paper gives a comprehensive and coherent view on permutability in the intuitionistic sequent calculus with cuts. Specifically we show that, once permutability is packaged into appropriate global reduction procedures, it organizes the internal structure of the system and determines fragments with computational interest, both for the computation-as-proof-normalization and the computation-as-proof-search paradigms. The vehicle of the study is a lambda-calculus of multiary proof terms with generalized application, previously developed by the authors (the paper argues this system represents the simplest fragment of ordinary sequent calculus that does not fall into mere natural deduction). We start by adapting to our setting the concept of normal proof, developed by Mints, Dyckhoff, and Pinto, and by defining natural proofs, so that a proof is normal iff it is natural and cut-free. Natural proofs form a subsystem with a transparent Curry-Howard interpretation (a kind of formal vector notation for lambda-terms with vectors consisting of lists of lists of arguments), while searching for normal proofs corresponds to a slight relaxation of focusing (in the sense of LJT). Next, we define a process of permutative conversion to natural form, and show that its combination with cut elimination gives a concept of normalization for the sequent calculus. We derive a systematic picture of the full system comprehending a rich set of reduction procedures (cut elimination, flattening, permutative conversion, normalization, focalization), organizing the relevant subsystems and the important subclasses of cut-free, normal, and focused proofs. |
|---|---|
| Autores principais: | Espírito Santo, José |
| Outros Autores: | Frade, M. J.; Pinto, Luís F. |
| Assunto: | Sequent calculus Permutative conversion Curry-Howard isomorphism Vector of arguments Generalized application Normal proof Natural proof Cut elimination Flatenning Normalization Focalization |
| Ano: | 2018 |
| País: | Portugal |
| Tipo de documento: | comunicação em conferência |
| Tipo de acesso: | acesso aberto |
| Instituição associada: | Universidade do Minho |
| Idioma: | inglês |
| Origem: | RepositóriUM - Universidade do Minho |
| _version_ | 1866877781816639488 |
|---|---|
| author | Espírito Santo, José |
| author2 | Frade, M. J. Pinto, Luís F. |
| author2_role | author author |
| author_facet | Espírito Santo, José Frade, M. J. Pinto, Luís F. |
| author_role | author |
| contributor_name_str_mv | Universidade do Minho |
| country_str | PT |
| creators_json_txt | [{\"Person.name\":\"Espírito Santo, José\"},{\"Person.name\":\"Frade, M. J.\"},{\"Person.name\":\"Pinto, Luís F.\"}] |
| datacite.contributors.contributor.contributorName.fl_str_mv | Universidade do Minho |
| datacite.creators.creator.creatorName.fl_str_mv | Espírito Santo, José Frade, M. J. Pinto, Luís F. |
| datacite.date.Accepted.fl_str_mv | 2018-01-01T00:00:00Z |
| datacite.date.available.fl_str_mv | 2019-01-10T14:19:44Z |
| datacite.date.embargoed.fl_str_mv | 2019-01-10T14:19:44Z |
| datacite.rights.fl_str_mv | http://purl.org/coar/access_right/c_abf2 |
| datacite.subjects.subject.fl_str_mv | Sequent calculus Permutative conversion Curry-Howard isomorphism Vector of arguments Generalized application Normal proof Natural proof Cut elimination Flatenning Normalization Focalization |
| datacite.titles.title.fl_str_mv | Permutability in proof terms for intuitionistic sequent calculus with cuts |
| dc.contributor.none.fl_str_mv | Universidade do Minho |
| dc.creator.none.fl_str_mv | Espírito Santo, José Frade, M. J. Pinto, Luís F. |
| dc.date.Accepted.fl_str_mv | 2018-01-01T00:00:00Z |
| dc.date.available.fl_str_mv | 2019-01-10T14:19:44Z |
| dc.date.embargoed.fl_str_mv | 2019-01-10T14:19:44Z |
| dc.format.none.fl_str_mv | application/pdf |
| dc.identifier.none.fl_str_mv | https://hdl.handle.net/1822/58009 |
| dc.language.none.fl_str_mv | eng |
| dc.publisher.none.fl_str_mv | Schloss Dagstuhl – Leibniz-Zentrum für Informatik GmbH |
| dc.rights.cclincense.fl_str_mv | http://creativecommons.org/licenses/by/4.0/ |
| dc.rights.none.fl_str_mv | http://purl.org/coar/access_right/c_abf2 |
| dc.rights.rights.copyright.fl_str_mv | openAccess |
| dc.subject.none.fl_str_mv | Sequent calculus Permutative conversion Curry-Howard isomorphism Vector of arguments Generalized application Normal proof Natural proof Cut elimination Flatenning Normalization Focalization |
| dc.title.fl_str_mv | Permutability in proof terms for intuitionistic sequent calculus with cuts |
| dc.type.none.fl_str_mv | http://purl.org/coar/resource_type/c_5794 |
| description | This paper gives a comprehensive and coherent view on permutability in the intuitionistic sequent calculus with cuts. Specifically we show that, once permutability is packaged into appropriate global reduction procedures, it organizes the internal structure of the system and determines fragments with computational interest, both for the computation-as-proof-normalization and the computation-as-proof-search paradigms. The vehicle of the study is a lambda-calculus of multiary proof terms with generalized application, previously developed by the authors (the paper argues this system represents the simplest fragment of ordinary sequent calculus that does not fall into mere natural deduction). We start by adapting to our setting the concept of normal proof, developed by Mints, Dyckhoff, and Pinto, and by defining natural proofs, so that a proof is normal iff it is natural and cut-free. Natural proofs form a subsystem with a transparent Curry-Howard interpretation (a kind of formal vector notation for lambda-terms with vectors consisting of lists of lists of arguments), while searching for normal proofs corresponds to a slight relaxation of focusing (in the sense of LJT). Next, we define a process of permutative conversion to natural form, and show that its combination with cut elimination gives a concept of normalization for the sequent calculus. We derive a systematic picture of the full system comprehending a rich set of reduction procedures (cut elimination, flattening, permutative conversion, normalization, focalization), organizing the relevant subsystems and the important subclasses of cut-free, normal, and focused proofs. |
| dirty | 0 |
| eu_rights_str_mv | openAccess |
| format | conferencePaper |
| fulltext.url.fl_str_mv | https://prod-dspace.uminho.pt/bitstreams/bf07ff20-5cb8-4dcd-bdd8-6b6fbb1fcbb1/download |
| id | rum_ca0119b418517fc654f2b2adb671f06e |
| identifier.url.fl_str_mv | https://hdl.handle.net/1822/58009 |
| instacron_str | repositorium |
| institution | Universidade do Minho |
| instname_str | Universidade do Minho |
| language | eng |
| network_acronym_str | rum |
| network_name_str | RepositóriUM - Universidade do Minho |
| oai_identifier_str | oai:repositorium.uminho.pt:1822/58009 |
| organization_str_mv | urn:organizationAcronym:repositorium |
| person_str_mv | Espírito Santo, José Frade, M. J. Pinto, Luís F. |
| publishDate | 2018 |
| publisher.none.fl_str_mv | Schloss Dagstuhl – Leibniz-Zentrum für Informatik GmbH |
| reponame_str | RepositóriUM - Universidade do Minho |
| repository_id_str | urn:repositoryAcronym:rum |
| service_str_mv | urn:repositoryAcronym:rum |
| spelling | engSchloss Dagstuhl – Leibniz-Zentrum für Informatik GmbHporThis paper gives a comprehensive and coherent view on permutability in the intuitionistic sequent calculus with cuts. Specifically we show that, once permutability is packaged into appropriate global reduction procedures, it organizes the internal structure of the system and determines fragments with computational interest, both for the computation-as-proof-normalization and the computation-as-proof-search paradigms. The vehicle of the study is a lambda-calculus of multiary proof terms with generalized application, previously developed by the authors (the paper argues this system represents the simplest fragment of ordinary sequent calculus that does not fall into mere natural deduction). We start by adapting to our setting the concept of normal proof, developed by Mints, Dyckhoff, and Pinto, and by defining natural proofs, so that a proof is normal iff it is natural and cut-free. Natural proofs form a subsystem with a transparent Curry-Howard interpretation (a kind of formal vector notation for lambda-terms with vectors consisting of lists of lists of arguments), while searching for normal proofs corresponds to a slight relaxation of focusing (in the sense of LJT). Next, we define a process of permutative conversion to natural form, and show that its combination with cut elimination gives a concept of normalization for the sequent calculus. We derive a systematic picture of the full system comprehending a rich set of reduction procedures (cut elimination, flattening, permutative conversion, normalization, focalization), organizing the relevant subsystems and the important subclasses of cut-free, normal, and focused proofs.application/pdfporPermutability in proof terms for intuitionistic sequent calculus with cutsEspírito Santo, JoséFrade, M. J.Pinto, Luís F.HostingInstitutionOrganizationalUniversidade do Minhoe-mailmailto:repositorium@usdb.uminho.ptrepositorium@usdb.uminho.ptISBNIsPartOf9783959770651ISSNIsPartOf1868-8969DOIIsPartOf10.4230/LIPIcs.TYPES.2016.102019-01-10T14:19:44Z20182018-01-01T00:00:00ZHandlehttps://hdl.handle.net/1822/58009http://purl.org/coar/access_right/c_abf2open accessSequent calculusPermutative conversionCurry-Howard isomorphismVector of argumentsGeneralized applicationNormal proofNatural proofCut eliminationFlatenningNormalizationFocalization638759 bytesother research producthttp://purl.org/coar/resource_type/c_5794conference paper2018http://creativecommons.org/licenses/by/4.0/openAccesshttp://purl.org/coar/access_right/c_abf2application/pdffulltexthttps://prod-dspace.uminho.pt/bitstreams/bf07ff20-5cb8-4dcd-bdd8-6b6fbb1fcbb1/download |
| spellingShingle | Permutability in proof terms for intuitionistic sequent calculus with cuts Espírito Santo, José Sequent calculus Permutative conversion Curry-Howard isomorphism Vector of arguments Generalized application Normal proof Natural proof Cut elimination Flatenning Normalization Focalization |
| status | SINGLETON |
| subject.fl_str_mv | Sequent calculus Permutative conversion Curry-Howard isomorphism Vector of arguments Generalized application Normal proof Natural proof Cut elimination Flatenning Normalization Focalization |
| title | Permutability in proof terms for intuitionistic sequent calculus with cuts |
| title_full | Permutability in proof terms for intuitionistic sequent calculus with cuts |
| title_fullStr | Permutability in proof terms for intuitionistic sequent calculus with cuts |
| title_full_unstemmed | Permutability in proof terms for intuitionistic sequent calculus with cuts |
| title_short | Permutability in proof terms for intuitionistic sequent calculus with cuts |
| title_sort | Permutability in proof terms for intuitionistic sequent calculus with cuts |
| topic | Sequent calculus Permutative conversion Curry-Howard isomorphism Vector of arguments Generalized application Normal proof Natural proof Cut elimination Flatenning Normalization Focalization |
| topic_facet | Sequent calculus Permutative conversion Curry-Howard isomorphism Vector of arguments Generalized application Normal proof Natural proof Cut elimination Flatenning Normalization Focalization |
| url | https://hdl.handle.net/1822/58009 |
| visible | 1 |