A categorical construction for the computational definition of vector spaces

Autores
Díaz Caro, Alejandro; Malherbe, Octavio
Año de publicación
2020
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
Lambda-S is an extension to first-order lambda calculus unifying two approaches of non-cloning in quantum lambda-calculi. One is to forbid duplication of variables, while the other is to consider all lambda-terms as algebraic linear functions. The type system of Lambda-S has a constructor S such that a type A is considered as the base of a vector space while S(A) is its span. Lambda-S can also be seen as a language for the computational manipulation of vector spaces: The vector spaces axioms are given as a rewrite system, describing the computational steps to be performed. In this paper we give an abstract categorical semantics of Lambda-S∗ (a fragment of Lambda-S), showing that S can be interpreted as the composition of two functors in an adjunction relation between a Cartesian category and an additive symmetric monoidal category. The right adjoint is a forgetful functor U, which is hidden in the language, and plays a central role in the computational reasoning.
Fil: Díaz Caro, Alejandro. Universidad Nacional de Quilmes; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; Argentina
Fil: Malherbe, Octavio. Universidad de la Republica. Facultad de Ingeniería; Uruguay
Materia
ALGEBRAIC LAMBDA-CALCULUS
CATEGORICAL SEMANTICS
QUANTUM COMPUTING
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/141616
Acceder
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network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling A categorical construction for the computational definition of vector spacesDíaz Caro, AlejandroMalherbe, OctavioALGEBRAIC LAMBDA-CALCULUSCATEGORICAL SEMANTICSQUANTUM COMPUTINGhttps://purl.org/becyt/ford/1.2https://purl.org/becyt/ford/1Lambda-S is an extension to first-order lambda calculus unifying two approaches of non-cloning in quantum lambda-calculi. One is to forbid duplication of variables, while the other is to consider all lambda-terms as algebraic linear functions. The type system of Lambda-S has a constructor S such that a type A is considered as the base of a vector space while S(A) is its span. Lambda-S can also be seen as a language for the computational manipulation of vector spaces: The vector spaces axioms are given as a rewrite system, describing the computational steps to be performed. In this paper we give an abstract categorical semantics of Lambda-S∗ (a fragment of Lambda-S), showing that S can be interpreted as the composition of two functors in an adjunction relation between a Cartesian category and an additive symmetric monoidal category. The right adjoint is a forgetful functor U, which is hidden in the language, and plays a central role in the computational reasoning.Fil: Díaz Caro, Alejandro. Universidad Nacional de Quilmes; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; ArgentinaFil: Malherbe, Octavio. Universidad de la Republica. Facultad de Ingeniería; UruguaySpringer2020-10info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/141616Díaz Caro, Alejandro; Malherbe, Octavio; A categorical construction for the computational definition of vector spaces; Springer; Applied Categorical Structures; 28; 5; 10-2020; 807-8440927-2852CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1007/s10485-020-09598-7info:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007%2Fs10485-020-09598-7info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1905.01305info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-10T13:22:26Zoai:ri.conicet.gov.ar:11336/141616instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-10 13:22:27.052CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv A categorical construction for the computational definition of vector spaces
title A categorical construction for the computational definition of vector spaces
spellingShingle A categorical construction for the computational definition of vector spaces
Díaz Caro, Alejandro
ALGEBRAIC LAMBDA-CALCULUS
CATEGORICAL SEMANTICS
QUANTUM COMPUTING
title_short A categorical construction for the computational definition of vector spaces
title_full A categorical construction for the computational definition of vector spaces
title_fullStr A categorical construction for the computational definition of vector spaces
title_full_unstemmed A categorical construction for the computational definition of vector spaces
title_sort A categorical construction for the computational definition of vector spaces
dc.creator.none.fl_str_mv Díaz Caro, Alejandro
Malherbe, Octavio
author Díaz Caro, Alejandro
author_facet Díaz Caro, Alejandro
Malherbe, Octavio
author_role author
author2 Malherbe, Octavio
author2_role author
dc.subject.none.fl_str_mv ALGEBRAIC LAMBDA-CALCULUS
CATEGORICAL SEMANTICS
QUANTUM COMPUTING
topic ALGEBRAIC LAMBDA-CALCULUS
CATEGORICAL SEMANTICS
QUANTUM COMPUTING
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.2
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv Lambda-S is an extension to first-order lambda calculus unifying two approaches of non-cloning in quantum lambda-calculi. One is to forbid duplication of variables, while the other is to consider all lambda-terms as algebraic linear functions. The type system of Lambda-S has a constructor S such that a type A is considered as the base of a vector space while S(A) is its span. Lambda-S can also be seen as a language for the computational manipulation of vector spaces: The vector spaces axioms are given as a rewrite system, describing the computational steps to be performed. In this paper we give an abstract categorical semantics of Lambda-S∗ (a fragment of Lambda-S), showing that S can be interpreted as the composition of two functors in an adjunction relation between a Cartesian category and an additive symmetric monoidal category. The right adjoint is a forgetful functor U, which is hidden in the language, and plays a central role in the computational reasoning.
Fil: Díaz Caro, Alejandro. Universidad Nacional de Quilmes; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; Argentina
Fil: Malherbe, Octavio. Universidad de la Republica. Facultad de Ingeniería; Uruguay
description Lambda-S is an extension to first-order lambda calculus unifying two approaches of non-cloning in quantum lambda-calculi. One is to forbid duplication of variables, while the other is to consider all lambda-terms as algebraic linear functions. The type system of Lambda-S has a constructor S such that a type A is considered as the base of a vector space while S(A) is its span. Lambda-S can also be seen as a language for the computational manipulation of vector spaces: The vector spaces axioms are given as a rewrite system, describing the computational steps to be performed. In this paper we give an abstract categorical semantics of Lambda-S∗ (a fragment of Lambda-S), showing that S can be interpreted as the composition of two functors in an adjunction relation between a Cartesian category and an additive symmetric monoidal category. The right adjoint is a forgetful functor U, which is hidden in the language, and plays a central role in the computational reasoning.
publishDate 2020
dc.date.none.fl_str_mv 2020-10
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/141616
Díaz Caro, Alejandro; Malherbe, Octavio; A categorical construction for the computational definition of vector spaces; Springer; Applied Categorical Structures; 28; 5; 10-2020; 807-844
0927-2852
CONICET Digital
CONICET
url http://hdl.handle.net/11336/141616
identifier_str_mv Díaz Caro, Alejandro; Malherbe, Octavio; A categorical construction for the computational definition of vector spaces; Springer; Applied Categorical Structures; 28; 5; 10-2020; 807-844
0927-2852
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1007/s10485-020-09598-7
info:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007%2Fs10485-020-09598-7
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1905.01305
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
application/pdf
dc.publisher.none.fl_str_mv Springer
publisher.none.fl_str_mv Springer
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 12.48226

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