A categorical construction for the computational definition of vector spaces
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 tha...
| Autores: | ; |
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| Tipo de documento: | Artigo |
| Estado: | Published version |
| Data de publicação: | 2020 |
| País: | Argentina |
| Recursos: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositório: | CONICET Digital (CONICET) |
| Idioma: | Inglês |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/141616 |
| Acesso em linha: | http://hdl.handle.net/11336/141616 |
| Nível de acesso: | Acesso aberto |
| Palavra-chave: | ALGEBRAIC LAMBDA-CALCULUS CATEGORICAL SEMANTICS QUANTUM COMPUTING https://purl.org/becyt/ford/1.2 https://purl.org/becyt/ford/1 |
| Resumo: | 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. |
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