{"title":"A concrete model for a typed linear algebraic lambda calculus","authors":"Alejandro Díaz-Caro, Octavio Malherbe","doi":"10.1017/s0960129523000361","DOIUrl":null,"url":null,"abstract":"We give an adequate, concrete, categorical-based model for Lambda-<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000361_inline1.png\" /> <jats:tex-math> ${\\mathcal S}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, which is a typed version of a linear-algebraic lambda calculus, extended with measurements. Lambda-<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000361_inline2.png\" /> <jats:tex-math> ${\\mathcal S}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is an extension to first-order lambda calculus unifying two approaches of non-cloning in quantum lambda-calculi: to forbid duplication of variables and to consider all lambda-terms as algebraic linear functions. The type system of Lambda-<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000361_inline3.png\" /> <jats:tex-math> ${\\mathcal S}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> has a superposition constructor <jats:italic>S</jats:italic> such that a type <jats:italic>A</jats:italic> is considered as the base of a vector space, while <jats:italic>SA</jats:italic> is its span. Our model considers <jats:italic>S</jats:italic> as the composition of two functors in an adjunction relation between the category of sets and the category of vector spaces over <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000361_inline4.png\" /> <jats:tex-math> $\\mathbb C$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. The right adjoint is a forgetful functor <jats:italic>U</jats:italic>, which is hidden in the language, and plays a central role in the computational reasoning.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2023-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"8","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Structures in Computer Science","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1017/s0960129523000361","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 8
Abstract
We give an adequate, concrete, categorical-based model for Lambda- ${\mathcal S}$ , which is a typed version of a linear-algebraic lambda calculus, extended with measurements. Lambda- ${\mathcal S}$ is an extension to first-order lambda calculus unifying two approaches of non-cloning in quantum lambda-calculi: to forbid duplication of variables and to consider all lambda-terms as algebraic linear functions. The type system of Lambda- ${\mathcal S}$ has a superposition constructor S such that a type A is considered as the base of a vector space, while SA is its span. Our model considers S as the composition of two functors in an adjunction relation between the category of sets and the category of vector spaces over $\mathbb C$ . The right adjoint is a forgetful functor U, which is hidden in the language, and plays a central role in the computational reasoning.
期刊介绍:
Mathematical Structures in Computer Science is a journal of theoretical computer science which focuses on the application of ideas from the structural side of mathematics and mathematical logic to computer science. The journal aims to bridge the gap between theoretical contributions and software design, publishing original papers of a high standard and broad surveys with original perspectives in all areas of computing, provided that ideas or results from logic, algebra, geometry, category theory or other areas of logic and mathematics form a basis for the work. The journal welcomes applications to computing based on the use of specific mathematical structures (e.g. topological and order-theoretic structures) as well as on proof-theoretic notions or results.