Modal structures in groups and vector spaces

IF 0.7 4区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Journal of Logic and Computation Pub Date : 2023-02-07 DOI:10.1093/logcom/exac105

Johan van Benthem, Nick Bezhanishvili

引用次数: 0

Abstract

Abstract Vector spaces contain a number of general structures that invite analysis in modal languages. The resulting logical systems provide an interesting counterpart to the much better-studied modal logics of topological spaces. In this programmatic paper, we investigate issues of definability and axiomatization using standard techniques for modal and hybrid languages. The analysis proceeds in stages. We first present a modal analysis of commutative groups that establishes our main techniques, next we introduce a new modal logic of linear dependence and independence in vector spaces and, finally, we study a modal logic for describing full-fledged vector spaces. While still far from covering every basic aspect of linear algebra, our discussion identifies several leads for more systematic research.

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群和向量空间中的模态结构

向量空间包含许多一般结构，需要在模态语言中进行分析。由此产生的逻辑系统提供了一个有趣的对应物，以更好地研究拓扑空间的模态逻辑。在这篇程序化的论文中，我们研究了使用模态和混合语言的标准技术的可定义性和公理化问题。分析分阶段进行。我们首先提出了交换群的模态分析，建立了我们的主要技术，然后我们引入了向量空间中线性相关和独立的新模态逻辑，最后我们研究了描述成熟向量空间的模态逻辑。虽然还远远没有涵盖线性代数的每一个基本方面，但我们的讨论为更系统的研究指明了几个方向。

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 Logic and Computation 工程技术-计算机：理论方法

CiteScore

1.90

自引率

14.30%

发文量

审稿时长

6-12 weeks

期刊介绍： Logic has found application in virtually all aspects of Information Technology, from software engineering and hardware to programming and artificial intelligence. Indeed, logic, artificial intelligence and theoretical computing are influencing each other to the extent that a new interdisciplinary area of Logic and Computation is emerging. The Journal of Logic and Computation aims to promote the growth of logic and computing, including, among others, the following areas of interest: Logical Systems, such as classical and non-classical logic, constructive logic, categorical logic, modal logic, type theory, feasible maths.... Logical issues in logic programming, knowledge-based systems and automated reasoning; logical issues in knowledge representation, such as non-monotonic reasoning and systems of knowledge and belief; logics and semantics of programming; specification and verification of programs and systems; applications of logic in hardware and VLSI, natural language, concurrent computation, planning, and databases. The bulk of the content is technical scientific papers, although letters, reviews, and discussions, as well as relevant conference reviews, are included.