A coinductive approach to proof search
Autor(a) principal: | |
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Data de Publicação: | 2013 |
Outros Autores: | , |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/1822/25226 |
Resumo: | We propose to study proof search from a coinductive point of view. In this paper, we consider intuitionistic logic and a focused system based on Herbelin’s LJT for the implicational fragment. We introduce a variant of lambda calculus with potentially infinitely deep terms and a means of expressing alternatives for the description of the “solution spaces” (called Böhm forests), which are a representation of all (not necessarily well-founded but still locally well-formed) proofs of a given formula (more generally: of a given sequent). As main result we obtain, for each given formula, the reduction of a coinductive definition of the solution space to a effective coinductive description in a finitary term calculus with a formal greatest fixed-point operator. This reduction works in a quite direct manner for the case of Horn formulas. For the general case, the naive extension would not even be true. We need to study “co-contraction” of contexts (contraction bottom-up) for dealing with the varying contexts needed beyond the Horn fragment, and we point out the appropriate finitary calculus, where fixed-point variables are typed with sequents. Co-contraction enters the interpretation of the formal greatest fixed points - curiously in the semantic interpretation of fixed-point variables and not of the fixed-point operator. |
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A coinductive approach to proof searchProof searchCoinductionIntuitionistic logicLambda-calculusScience & TechnologyWe propose to study proof search from a coinductive point of view. In this paper, we consider intuitionistic logic and a focused system based on Herbelin’s LJT for the implicational fragment. We introduce a variant of lambda calculus with potentially infinitely deep terms and a means of expressing alternatives for the description of the “solution spaces” (called Böhm forests), which are a representation of all (not necessarily well-founded but still locally well-formed) proofs of a given formula (more generally: of a given sequent). As main result we obtain, for each given formula, the reduction of a coinductive definition of the solution space to a effective coinductive description in a finitary term calculus with a formal greatest fixed-point operator. This reduction works in a quite direct manner for the case of Horn formulas. For the general case, the naive extension would not even be true. We need to study “co-contraction” of contexts (contraction bottom-up) for dealing with the varying contexts needed beyond the Horn fragment, and we point out the appropriate finitary calculus, where fixed-point variables are typed with sequents. Co-contraction enters the interpretation of the formal greatest fixed points - curiously in the semantic interpretation of fixed-point variables and not of the fixed-point operator.FEDER funds through “Programa Operacional Factores de Competitividade– COMPETE”FCT – “Fundação para a Ciência e a Tecnologia”, within the project PEst-C/MAT/UI0013/2011Climt project (ANR-11-BS02-016 of the French Agence Nationale de la Recherche)Open Publishing AssociationUniversidade do MinhoEspírito Santo, JoséMatthes, RalphPinto, Luís F.2013-08-282013-08-28T00:00:00Zconference paperinfo:eu-repo/semantics/publishedVersionapplication/pdfhttp://hdl.handle.net/1822/25226eng2075-218010.4204/EPTCS.126.3http://rvg.web.cse.unsw.edu.au/eptcs/paper.cgi?FICS13.3info: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:RCAAP2024-05-11T05:40:52Zoai:repositorium.sdum.uminho.pt:1822/25226Portal AgregadorONGhttps://www.rcaap.pt/oai/openairemluisa.alvim@gmail.comopendoar:71602024-05-11T05:40:52Repositório 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 |
A coinductive approach to proof search |
title |
A coinductive approach to proof search |
spellingShingle |
A coinductive approach to proof search Espírito Santo, José Proof search Coinduction Intuitionistic logic Lambda-calculus Science & Technology |
title_short |
A coinductive approach to proof search |
title_full |
A coinductive approach to proof search |
title_fullStr |
A coinductive approach to proof search |
title_full_unstemmed |
A coinductive approach to proof search |
title_sort |
A coinductive approach to proof search |
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 |
Proof search Coinduction Intuitionistic logic Lambda-calculus Science & Technology |
topic |
Proof search Coinduction Intuitionistic logic Lambda-calculus Science & Technology |
description |
We propose to study proof search from a coinductive point of view. In this paper, we consider intuitionistic logic and a focused system based on Herbelin’s LJT for the implicational fragment. We introduce a variant of lambda calculus with potentially infinitely deep terms and a means of expressing alternatives for the description of the “solution spaces” (called Böhm forests), which are a representation of all (not necessarily well-founded but still locally well-formed) proofs of a given formula (more generally: of a given sequent). As main result we obtain, for each given formula, the reduction of a coinductive definition of the solution space to a effective coinductive description in a finitary term calculus with a formal greatest fixed-point operator. This reduction works in a quite direct manner for the case of Horn formulas. For the general case, the naive extension would not even be true. We need to study “co-contraction” of contexts (contraction bottom-up) for dealing with the varying contexts needed beyond the Horn fragment, and we point out the appropriate finitary calculus, where fixed-point variables are typed with sequents. Co-contraction enters the interpretation of the formal greatest fixed points - curiously in the semantic interpretation of fixed-point variables and not of the fixed-point operator. |
publishDate |
2013 |
dc.date.none.fl_str_mv |
2013-08-28 2013-08-28T00:00:00Z |
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/1822/25226 |
url |
http://hdl.handle.net/1822/25226 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
2075-2180 10.4204/EPTCS.126.3 http://rvg.web.cse.unsw.edu.au/eptcs/paper.cgi?FICS13.3 |
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 |
Open Publishing Association |
publisher.none.fl_str_mv |
Open Publishing Association |
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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1817544702734368768 |