A Category Theoretic View of Contextual Types: From Simple Types to Dependent Types

IF 0.7 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS
Jason Z. S. Hu, B. Pientka, Ulrich Schöpp
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引用次数: 2

Abstract

We describe the categorical semantics for a simply typed variant and a simplified dependently typed variant of Cocon, a contextual modal type theory where the box modality mediates between the weak function space that is used to represent higher-order abstract syntax (HOAS) trees and the strong function space that describes (recursive) computations about them. What makes Cocon different from standard type theories is the presence of first-class contexts and contextual objects to describe syntax trees that are closed with respect to a given context of assumptions. Following M. Hofmann’s work, we use a presheaf model to characterise HOAS trees. Surprisingly, this model already provides the necessary structure to also model Cocon. In particular, we can capture the contextual objects of Cocon using a comonad ♭ that restricts presheaves to their closed elements. This gives a simple semantic characterisation of the invariants of contextual types (e.g. substitution invariance) and identifies Cocon as a type-theoretic syntax of presheaf models. We further extend this characterisation to dependent types using categories with families and show that we can model a fragment of Cocon without recursor in the Fitch-style dependent modal type theory presented by Birkedal et al.
语境类型的范畴论视角:从简单类型到依赖类型
我们描述了Cocon的简单类型变体和简化依赖类型变体的分类语义,Cocon是一种上下文模态类型理论,其中盒模态介于用于表示高阶抽象语法(HOAS)树的弱函数空间和描述关于它们的(递归)计算的强函数空间之间。Cocon与标准类型理论的不同之处在于,它存在一级上下文和上下文对象来描述相对于给定假设上下文闭合的语法树。根据M.Hofmann的工作,我们使用预剪切模型来表征HOAS树。令人惊讶的是,该模型已经提供了对Cocon进行建模所需的结构。特别是,我们可以使用comonad捕获Cocon的上下文对象♭ 这将预应力限制在其封闭元件上。这给出了上下文类型不变量(例如替换不变性)的简单语义表征,并将Cocon识别为预学习模型的类型论语法。我们使用带有族的类别将这种表征进一步扩展到依赖类型,并表明我们可以在Birkedal等人提出的Fitch风格依赖模态类型理论中对没有递归的Cocon片段进行建模。
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来源期刊
ACM Transactions on Computational Logic
ACM Transactions on Computational Logic 工程技术-计算机:理论方法
CiteScore
2.30
自引率
0.00%
发文量
37
审稿时长
>12 weeks
期刊介绍: TOCL welcomes submissions related to all aspects of logic as it pertains to topics in computer science. This area has a great tradition in computer science. Several researchers who earned the ACM Turing award have also contributed to this field, namely Edgar Codd (relational database systems), Stephen Cook (complexity of logical theories), Edsger W. Dijkstra, Robert W. Floyd, Tony Hoare, Amir Pnueli, Dana Scott, Edmond M. Clarke, Allen E. Emerson, and Joseph Sifakis (program logics, program derivation and verification, programming languages semantics), Robin Milner (interactive theorem proving, concurrency calculi, and functional programming), and John McCarthy (functional programming and logics in AI). Logic continues to play an important role in computer science and has permeated several of its areas, including artificial intelligence, computational complexity, database systems, and programming languages. The Editorial Board of this journal seeks and hopes to attract high-quality submissions in all the above-mentioned areas of computational logic so that TOCL becomes the standard reference in the field. Both theoretical and applied papers are sought. Submissions showing novel use of logic in computer science are especially welcome.
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