A concrete model for a typed linear algebraic lambda calculus

IF 0.4 4区 计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS Mathematical Structures in Computer Science Pub Date : 2023-11-21 DOI:10.1017/s0960129523000361
Alejandro Díaz-Caro, Octavio Malherbe
{"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":"38 13","pages":""},"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.
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
一类线性代数λ演算的具体模型
我们给出了Lambda- ${\mathcal S}$的一个充分的,具体的,基于分类的模型,它是线性代数Lambda演算的一个类型版本,扩展了测量。λ - ${\mathcal S}$是对一阶λ演算的扩展,统一了量子λ演算中的两种非克隆方法:禁止变量重复和将所有λ项视为代数线性函数。λ - ${\mathcal S}$的类型系统有一个叠加构造函数S,使得类型a被认为是向量空间的基,而SA是它的张成空间。我们的模型认为S是在$\mathbb C$上集合范畴与向量空间范畴之间的附加关系中的两个函子的复合。右伴随是一个遗忘的函子U,它隐藏在语言中,在计算推理中起着核心作用。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 去求助
来源期刊
Mathematical Structures in Computer Science
Mathematical Structures in Computer Science 工程技术-计算机:理论方法
CiteScore
1.50
自引率
0.00%
发文量
30
审稿时长
12 months
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.
期刊最新文献
On Hofmann–Streicher universes T0-spaces and the lower topology GADTs are not (Even partial) functors A linear linear lambda-calculus Countability constraints in order-theoretic approaches to computability
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1