Coalgebraic Logic: Expanding the Scope

代数逻辑：扩大范围

• 批准号: EP/G041296/1
• 负责人: Alexander Kurz
• 金额: $ 46.02万
• 依托单位: University of Leicester
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2009
• 资助国家: 英国
• 起止时间: 2009 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FG041296%2F1
• 关键词: Coalgebraic; Logic; Expanding; Scope

COALGEBRAIC LOGICLogic plays a fundamental role in Computer Science. At the most basiclevel, Boolean logic is used to design the circuits we use every day inour computers. At the higher end, the tasks that computers perform need toconform to specifications expressed in logics suitable for programmers,analysts or even other computational devices.Such specification logics have to be able to express many differentconcepts such as time, knowledge, space, mobility, communication,probability, conditionals etc. Bespoke logics for each of these conceptsexist and are studied under the umbrella of Modal Logic.In any substantial application of Modal Logic to the specification ofa system, the need to combine different logics will arise, each logicaccounting for, eg, one of the aspects mentioned above. The need thenarises to deal with these logics in a uniform and modular way.Not all of these logics have a standard Kripke semantics, but in allcases, the semantics can be considered to be coalgebraic. Coalgebrasgeneralise the standard Kripke semantics of modal logic to encompassnotions such as neighbourhood frames, Markov chains, topologicalspaces, etc.Moreover, Coalgebra is a concept from Category Theory. Category Theoryis an area of mathematics which describes mathematical constructionsin abstract terms that make these constructions available to manydifferent areas of mathematics, logic, and computer science. Inparticular, the category theoretic nature of Coalgebras allows us totackle the modularity problem using category theoreticconstructions. One of the benefits of category theory is that theseconstructions, because of their generality, apply to specificationlanguages and to their semantic models.To summarise, Coalgebraic Logic combines Modal Logic withCoalgebra. This generalises modal logics from Kripke frames tocoalgebras and makes category theoretic methods and constructionsavailable in Modal Logic.EXPANDING THE SCOPECoalgebraic Logic can be traced back to 1997 when the first draft ofMoss's paper with the same title was circulated. Since then, it hasbeen developed by a number of researchers. Just now, Coalgebraic Logicis about to establish itself as an own area. Whereas much of thecurrent work in Coalgebraic Logic aims at exploiting the currentachievements towards more applications, this project starts from thefollowing two observations:First, Coalgebraic logic did not yet make use of many of the importantdevelopments that have taken place in Modal Logic. Two of thesedevelopments are:1) the relationship between Modal Logic and First-Order Logic and2) the uniform treatment of classes of modal logics.Second, there exist many parallel developments in Modal Logic andDomain Theory. Some of the relationships have only recently becomeclear, through the connection of both areas with Coalgebra. Wetherefore plan to3) generalise methods from Modal Logic so that they can be applied tothe logics arising in Domain Theory (this will include the work doneunder 1 and 2 above)

组合代数逻辑在计算机科学中起着基础性的作用。在最基本的层面上，布尔逻辑被用来设计我们每天在计算机中使用的电路。在更高的层次上，计算机执行的任务需要符合以适合程序员、分析师甚至其他计算设备的逻辑表达的规范。这样的规范逻辑必须能够表达许多不同的概念，如时间、知识、空间、移动性、通信、概率，这些概念的定制逻辑都存在，并在模态逻辑的保护伞下进行研究。在模态逻辑的任何实质性应用中，对于一个系统的规范来说，将出现对联合收割机不同逻辑的需要，每个逻辑例如负责上述方面之一。这就需要用统一的、模块化的方法来处理这些逻辑，虽然不是所有的逻辑都有标准的Kripke语义，但是在所有的情况下，这些语义都可以被认为是余代数的。余代数将模态逻辑的标准Kripke语义推广到邻域框架、马尔可夫链、拓扑空间等概念上，而且余代数是范畴论中的一个概念。范畴论是数学的一个领域，它用抽象的术语描述数学构造，使这些构造适用于数学、逻辑和计算机科学的许多不同领域。特别地，余代数的范畴论性质允许我们使用范畴论构造来处理模块性问题。范畴论的好处之一是，这些构造由于其通用性而适用于规约语言及其语义模型。这将模态逻辑从Kripke框架推广到余代数，并使范畴论方法和构造在模态逻辑中可用。扩展范围余代数逻辑可以追溯到1997年，当时Moss的同名论文的第一稿被传阅。从那时起，它被许多研究人员开发出来。就在不久前，余代数逻辑学即将建立自己的领域。鉴于目前在余代数逻辑中的大部分工作旨在利用当前的成就走向更多的应用，本项目从以下两个观察开始：首先，余代数逻辑尚未利用模态逻辑中发生的许多重要发展。其中两个发展是：1）模态逻辑与一阶逻辑的关系; 2）模态逻辑类的统一处理;一些关系最近才变得清晰，通过这两个领域的连接与余代数。因此，我们计划将模态逻辑的方法加以推广，使它们能够应用于域理论中的逻辑（这将包括上面1和2中所做的工作）。

项目成果

Bitopological duality for distributive lattices and Heyting algebras

• DOI: 10.1017/s0960129509990302
• 发表时间: 2010-01
• 期刊: Mathematical Structures in Computer Science
• 影响因子: 0.5
• 作者: G. Bezhanishvili;N. Bezhanishvili;D. Gabelaia;A. Kurz

RELATION LIFTING, WITH AN APPLICATION TO THE MANY-VALUED COVER MODALITY

• DOI: 10.2168/lmcs-9(4:8)2013
• 发表时间: 2013-01-01
• 期刊: LOGICAL METHODS IN COMPUTER SCIENCE
• 影响因子: 0.6
• 作者: Bilkova, Marta;Kurz, Alexander;Velebil, Jiri
• 通讯作者: Velebil, Jiri

Advances in Modal Logic 8

模态逻辑的进展 8

• 发表时间: 2010
• 作者: A Kurz;Y Venema
• 通讯作者: Y Venema

On a Categorical Framework for Coalgebraic Modal Logic

代数模态逻辑的分类框架

• DOI: 10.1016/j.entcs.2014.10.007
• 发表时间: 2014
• 期刊: Electronic Notes in Theoretical Computer Science
• 作者: Chen L

On Coalgebras over Algebras

论代数之上的余代数

• DOI: 10.1016/j.entcs.2010.07.013
• 发表时间: 2010
• 期刊: Electronic Notes in Theoretical Computer Science
• 作者: Balan A