Publicação
A coinductive approach to proof search through typed lambda-calculi
| Resumo: | In reductive proof search, proofs are naturally generalized by solutions, comprising all (possibly infinite) structures generated by locally correct, bottom-up application of inference rules. We propose a rather natural extension of the Curry-Howard paradigm of representation, from proofs to solutions: to represent solutions by (possibly infinite) terms of the coinductive variant of the typed lambda-calculus that represents proofs. We take this as a starting point for a new, comprehensive approach to proof search; our case study is proof search in the sequent calculus LIT for intuitionistic implication logic. A second, finitary representation is proposed, comprising a syntax of lambda-terms extended with a formal greatest fixed point, and a type system that can be seen as a logic of coinductive proofs. In the finitary system, fixed-point variables enjoy a relaxed form of binding that allows the detection of cycles through the type system. Formal sums are used in both representations to express alternatives in the search process, so that not only individual solutions but actually solution spaces are expressed. Moreover, formal sums are used in the coinductive syntax to define "decontraction" (contraction bottom-up)-an operation whose theory we initiate in this paper. A semantics is defined assigning a coinductive lambda-term to each finitary term, making use of decontraction as a semantical match to the relaxed form of binding of fixed-point variables present in the finitary system. The main result is the existence of an equivalent finitary representation for any full solution space expressed coinductively. This result is the main ingredient in the proof that our logic of coinductive proofs is sound and complete with respect to the coinductive semantics. These results are the foundation for an original approach to proof search, where the search builds the finitary representation of the full solution space, and the a posteriori analysis typically consisting in applying a syntax-directed procedure or function. The paper illustrates the potential of the methodology to the study of proof search and inhabitation problems in the simply-typed lambda-calculus, reviewing results detailed elsewhere, and including new results that obtain extensive generalizations of the so-called monatomic theorem. |
|---|---|
| Autores principais: | Espírito Santo, José |
| Outros Autores: | Matthes, Ralph; Pinto, Luís F. |
| Assunto: | Curry-Howard isomorphism Proof search Coinductive methods Solution spaces |
| Ano: | 2021 |
| País: | Portugal |
| Tipo de documento: | artigo |
| Tipo de acesso: | acesso aberto |
| Instituição associada: | Universidade do Minho |
| Idioma: | inglês |
| Origem: | RepositóriUM - Universidade do Minho |
| _version_ | 1866877426221449216 |
|---|---|
| author | Espírito Santo, José |
| author2 | Matthes, Ralph Pinto, Luís F. |
| author2_role | author author |
| author_facet | Espírito Santo, José Matthes, Ralph 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\":\"Matthes, Ralph\"},{\"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 Matthes, Ralph Pinto, Luís F. |
| datacite.date.Accepted.fl_str_mv | 2021-12-01T00:00:00Z |
| datacite.date.available.fl_str_mv | 2024-01-01T07:01:07Z |
| datacite.date.embargoed.fl_str_mv | 2024-01-01T07:01:07Z |
| datacite.rights.fl_str_mv | http://purl.org/coar/access_right/c_abf2 |
| datacite.subjects.subject.fl_str_mv | Curry-Howard isomorphism Proof search Coinductive methods Solution spaces |
| datacite.titles.title.fl_str_mv | A coinductive approach to proof search through typed lambda-calculi |
| dc.contributor.none.fl_str_mv | Universidade do Minho |
| dc.creator.none.fl_str_mv | Espírito Santo, José Matthes, Ralph Pinto, Luís F. |
| dc.date.Accepted.fl_str_mv | 2021-12-01T00:00:00Z |
| dc.date.available.fl_str_mv | 2024-01-01T07:01:07Z |
| dc.date.embargoed.fl_str_mv | 2024-01-01T07:01:07Z |
| dc.format.none.fl_str_mv | application/pdf |
| dc.identifier.none.fl_str_mv | https://hdl.handle.net/1822/75251 |
| dc.language.none.fl_str_mv | eng |
| dc.publisher.none.fl_str_mv | Elsevier |
| dc.rights.none.fl_str_mv | http://purl.org/coar/access_right/c_abf2 |
| dc.subject.none.fl_str_mv | Curry-Howard isomorphism Proof search Coinductive methods Solution spaces |
| dc.title.fl_str_mv | A coinductive approach to proof search through typed lambda-calculi |
| dc.type.none.fl_str_mv | http://purl.org/coar/resource_type/c_6501 |
| description | In reductive proof search, proofs are naturally generalized by solutions, comprising all (possibly infinite) structures generated by locally correct, bottom-up application of inference rules. We propose a rather natural extension of the Curry-Howard paradigm of representation, from proofs to solutions: to represent solutions by (possibly infinite) terms of the coinductive variant of the typed lambda-calculus that represents proofs. We take this as a starting point for a new, comprehensive approach to proof search; our case study is proof search in the sequent calculus LIT for intuitionistic implication logic. A second, finitary representation is proposed, comprising a syntax of lambda-terms extended with a formal greatest fixed point, and a type system that can be seen as a logic of coinductive proofs. In the finitary system, fixed-point variables enjoy a relaxed form of binding that allows the detection of cycles through the type system. Formal sums are used in both representations to express alternatives in the search process, so that not only individual solutions but actually solution spaces are expressed. Moreover, formal sums are used in the coinductive syntax to define "decontraction" (contraction bottom-up)-an operation whose theory we initiate in this paper. A semantics is defined assigning a coinductive lambda-term to each finitary term, making use of decontraction as a semantical match to the relaxed form of binding of fixed-point variables present in the finitary system. The main result is the existence of an equivalent finitary representation for any full solution space expressed coinductively. This result is the main ingredient in the proof that our logic of coinductive proofs is sound and complete with respect to the coinductive semantics. These results are the foundation for an original approach to proof search, where the search builds the finitary representation of the full solution space, and the a posteriori analysis typically consisting in applying a syntax-directed procedure or function. The paper illustrates the potential of the methodology to the study of proof search and inhabitation problems in the simply-typed lambda-calculus, reviewing results detailed elsewhere, and including new results that obtain extensive generalizations of the so-called monatomic theorem. |
| dirty | 0 |
| eu_rights_str_mv | openAccess |
| format | article |
| fulltext.url.fl_str_mv | https://prod-dspace.uminho.pt/bitstreams/e5ab51f0-8734-46d4-bc17-0a9f20cb2c30/download |
| id | rum_43f41aba0ea1e7c5def2cb981f0331ec |
| identifier.url.fl_str_mv | https://hdl.handle.net/1822/75251 |
| instacron_str | repositorium |
| institution | Universidade do Minho |
| instname_str | Universidade do Minho |
| language | eng |
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| network_name_str | RepositóriUM - Universidade do Minho |
| oai_identifier_str | oai:repositorium.uminho.pt:1822/75251 |
| organization_str_mv | urn:organizationAcronym:repositorium |
| person_str_mv | Espírito Santo, José Matthes, Ralph Pinto, Luís F. |
| publishDate | 2021 |
| publisher.none.fl_str_mv | Elsevier |
| reponame_str | RepositóriUM - Universidade do Minho |
| repository_id_str | urn:repositoryAcronym:rum |
| service_str_mv | urn:repositoryAcronym:rum |
| spelling | engElsevierporIn reductive proof search, proofs are naturally generalized by solutions, comprising all (possibly infinite) structures generated by locally correct, bottom-up application of inference rules. We propose a rather natural extension of the Curry-Howard paradigm of representation, from proofs to solutions: to represent solutions by (possibly infinite) terms of the coinductive variant of the typed lambda-calculus that represents proofs. We take this as a starting point for a new, comprehensive approach to proof search; our case study is proof search in the sequent calculus LIT for intuitionistic implication logic. A second, finitary representation is proposed, comprising a syntax of lambda-terms extended with a formal greatest fixed point, and a type system that can be seen as a logic of coinductive proofs. In the finitary system, fixed-point variables enjoy a relaxed form of binding that allows the detection of cycles through the type system. Formal sums are used in both representations to express alternatives in the search process, so that not only individual solutions but actually solution spaces are expressed. Moreover, formal sums are used in the coinductive syntax to define "decontraction" (contraction bottom-up)-an operation whose theory we initiate in this paper. A semantics is defined assigning a coinductive lambda-term to each finitary term, making use of decontraction as a semantical match to the relaxed form of binding of fixed-point variables present in the finitary system. The main result is the existence of an equivalent finitary representation for any full solution space expressed coinductively. This result is the main ingredient in the proof that our logic of coinductive proofs is sound and complete with respect to the coinductive semantics. These results are the foundation for an original approach to proof search, where the search builds the finitary representation of the full solution space, and the a posteriori analysis typically consisting in applying a syntax-directed procedure or function. The paper illustrates the potential of the methodology to the study of proof search and inhabitation problems in the simply-typed lambda-calculus, reviewing results detailed elsewhere, and including new results that obtain extensive generalizations of the so-called monatomic theorem.application/pdfporA coinductive approach to proof search through typed lambda-calculiEspírito Santo, JoséMatthes, RalphPinto, Luís F.HostingInstitutionOrganizationalUniversidade do Minhoe-mailmailto:repositorium@usdb.uminho.ptrepositorium@usdb.uminho.ptISSNIsPartOf0168-0072DOIIsPartOf10.1016/j.apal.2021.1030262024-01-01T07:01:07Z2021-122021-12-29T18:17:29Z2021-12-01T00:00:00ZHandlehttps://hdl.handle.net/1822/75251http://purl.org/coar/access_right/c_abf2open accessCurry-Howard isomorphismProof searchCoinductive methodsSolution spaces635642 bytesliteraturehttp://purl.org/coar/resource_type/c_6501journal articlehttp://purl.org/coar/access_right/c_abf2application/pdffulltexthttps://prod-dspace.uminho.pt/bitstreams/e5ab51f0-8734-46d4-bc17-0a9f20cb2c30/download |
| spellingShingle | A coinductive approach to proof search through typed lambda-calculi Espírito Santo, José Curry-Howard isomorphism Proof search Coinductive methods Solution spaces |
| status | SINGLETON |
| subject.fl_str_mv | Curry-Howard isomorphism Proof search Coinductive methods Solution spaces |
| title | A coinductive approach to proof search through typed lambda-calculi |
| title_full | A coinductive approach to proof search through typed lambda-calculi |
| title_fullStr | A coinductive approach to proof search through typed lambda-calculi |
| title_full_unstemmed | A coinductive approach to proof search through typed lambda-calculi |
| title_short | A coinductive approach to proof search through typed lambda-calculi |
| title_sort | A coinductive approach to proof search through typed lambda-calculi |
| topic | Curry-Howard isomorphism Proof search Coinductive methods Solution spaces |
| topic_facet | Curry-Howard isomorphism Proof search Coinductive methods Solution spaces |
| url | https://hdl.handle.net/1822/75251 |
| visible | 1 |