Finitary Type Theories With and Without Contexts

IF 0.9 3区计算机科学 Q4 COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE

Journal of Automated Reasoning Pub Date : 2023-10-07 DOI:10.1007/s10817-023-09678-y

Philipp G. Haselwarter, Andrej Bauer

引用次数: 2

Abstract

Abstract We give a definition of finitary type theories that subsumes many examples of dependent type theories, such as variants of Martin–Löf type theory, simple type theories, first-order and higher-order logics, and homotopy type theory. We prove several general meta-theorems about finitary type theories: weakening, admissibility of substitution and instantiation of metavariables, derivability of presuppositions, uniqueness of typing, and inversion principles. We then give a second formulation of finitary type theories in which there are no explicit contexts. Instead, free variables are explicitly annotated with their types. We provide translations between finitary type theories with and without contexts, thereby showing that they have the same expressive power. The context-free type theory is implemented in the nucleus of the Andromeda 2 proof assistant.

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有和没有上下文的有限型理论

摘要给出了有限型论的定义，它包含了依赖型论的许多例子，如Martin-Löf型论的变体、简单型论、一阶和高阶逻辑以及同伦型论。我们证明了有限型理论的几个一般元定理:弱化、元变量的代换和实例化的可容许性、假设的可导性、类型的唯一性和反转原理。然后，我们给出了有限型理论的第二种表述，其中没有明确的上下文。相反，自由变量被显式地注释为它们的类型。我们提供了有限类型理论之间的翻译有和没有上下文，从而表明它们具有相同的表达能力。上下文无关类型理论在仙女座2证明助理的核心中实现。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Automated Reasoning 工程技术-计算机：人工智能

CiteScore

3.60

自引率

9.10%

发文量

审稿时长

>12 weeks

期刊介绍： The Journal of Automated Reasoning is an interdisciplinary journal that maintains a balance between theory, implementation and application. The spectrum of material published ranges from the presentation of a new inference rule with proof of its logical properties to a detailed account of a computer program designed to solve various problems in industry. The main fields covered are automated theorem proving, logic programming, expert systems, program synthesis and validation, artificial intelligence, computational logic, robotics, and various industrial applications. The papers share the common feature of focusing on several aspects of automated reasoning, a field whose objective is the design and implementation of a computer program that serves as an assistant in solving problems and in answering questions that require reasoning. The Journal of Automated Reasoning provides a forum and a means for exchanging information for those interested purely in theory, those interested primarily in implementation, and those interested in specific research and industrial applications.