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)

Colgebraic Logiclogic在计算机科学中起着基本作用。在最大的基础上，布尔逻辑用于设计我们每天使用INOUR计算机的电路。在高端，计算机执行的任务需要符合适用于程序员，分析师甚至其他计算设备的逻辑表达的规范。规范逻辑必须能够表达许多不同的概念，例如时间，知识，空间，交流，交流，概率，概率，有条件，条件，条件性等等的逻辑等于这些概念的逻辑和模式下的逻辑效果。系统的规范，结合不同逻辑的需求将出现，每个逻辑审计，例如，例如上述方面之一。当时需要以统一和模块化的方式处理这些逻辑。所有这些逻辑都不具有标准的Kripke语义，但在所有cass中，可以认为语义被认为是calgebraic。 ColgeBraseneralise将模态逻辑的标准kripke语义学到包括邻里框架，马尔可夫链，拓扑间等等诸如诸如诸如邻里框架之类的概念。此外，Calgebra是类别理论的概念。类别理论是一个数学领域，该领域描述了数学构造的抽象术语，这些术语使这些构造可用于数学，逻辑和计算机科学的许多不同领域。围栏内，山结构的类别理论性质使我们使用类别理论构造解决了模块化问题。类别理论的好处之一是，由于其通用性，该构造适用于规范语言和其语义模型。总而言，colgebraic Logic将模态逻辑与Coalgebra结合在一起。这将概括从kripke框架tocoalgebras中进行概括，并在模态逻辑中制造类别理论方法和构造方法。扩展scopecoalgebraic逻辑可以追溯到1997年，当时具有相同标题的莫姆斯的初稿时，可以追溯到1997年。从那时起，它是由许多研究人员开发的。刚才，占地逻辑将成为自己的区域。尽管煤层逻辑中的大部分工作旨在利用当前方面的应用程序，但该项目始于遵循两个观察结果：首先，煤层逻辑尚未利用在模态逻辑中发生的许多重要开发项目。第两个heesdevermations是：1）模态逻辑与一阶逻辑和2）模态逻辑类别的统一处理之间的关系。第二，模态逻辑和域理论中存在许多平行的发展。通过两个地区与山地的联系，一些关系才最近才成为现实。因此，计划对3）从模态逻辑中概括方法，以便可以应用它们在域理论中产生的逻辑（这将包括上面的上限1和2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

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

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

Families of Symmetries as Efficient Models of Resource Binding

作为资源绑定有效模型的对称族

• DOI: 10.1016/j.entcs.2010.07.014
• 发表时间: 2010
• 期刊: Electronic Notes in Theoretical Computer Science
• 作者: Ciancia V