A coinductive approach to proof search

Detalhes bibliográficos
Autor(a) principal: Espírito Santo, José
Data de Publicação: 2013
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/1822/25226
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.
id RCAP_db0b5b01d29dda98e9d0c1497e4e81a9
oai_identifier_str oai:repositorium.sdum.uminho.pt:1822/25226
network_acronym_str RCAP
network_name_str Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
repository_id_str 7160
spelling 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
_version_ 1817544702734368768