定点逻辑和可定义拓扑特性

IF 0.4 4区 计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS Mathematical Structures in Computer Science Pub Date : 2023-12-13 DOI:10.1017/s0960129523000385
David Fernández-Duque, Quentin Gougeon
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引用次数: 0

摘要

模态逻辑的拓扑语义可以追溯到麦肯锡和塔尔斯基,通过模态公理对拓扑空间进行分类是一个活跃的研究领域。在过去的二十年里,人们对将拓扑模态逻辑扩展到缪算子语言产生了兴趣,但在此之前,人们还不知道有哪一类拓扑空间是可以用缪算子定义的,而不是已经可以用模态定义的。在本文中,我们证明了完整的mu-calculus确实比标准模态逻辑更具表现力,因为有一些拓扑空间类(以及弱传递克里普克帧)是可以用mu-calculus定义的,但却不能用模态定义。我们展示的这些类在其完美核心之外满足模态可定义的性质,因此我们称它们为不完美空间。我们证明,对于这些类来说,μ算式是健全而完整的。我们的例子是最小的,因为它们只使用了最大定点的一个实例,而且我们证明,仅凭最小定点不足以定义任何一类尚未模态定义的空间。
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Fixed point logics and definable topological properties

Modal logic enjoys topological semantics that may be traced back to McKinsey and Tarski, and the classification of topological spaces via modal axioms is a lively area of research. In the past two decades, there has been interest in extending topological modal logic to the language of the mu-calculus, but previously no class of topological spaces was known to be mu-calculus definable that was not already modally definable. In this paper, we show that the full mu-calculus is indeed more expressive than standard modal logic, in the sense that there are classes of topological spaces (and weakly transitive Kripke frames), which are mu-definable but not modally definable. The classes we exhibit satisfy a modally definable property outside of their perfect core, and thus we dub them imperfect spaces. We show that the mu-calculus is sound and complete for these classes. Our examples are minimal in the sense that they use a single instance of a greatest fixed point, and we show that least fixed points alone do not suffice to define any class of spaces that is not already modally definable.

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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.
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