Modal Logic for Granularity

这个项目的目的是开发一种逻辑，它允许语句的表示，如“在所有级别的细节…”和“在某些级别的细节…”。主要应用领域是空间信息推理，特别是在离散空间中，如表示网络的图形，或由具有邻接性建模的图形的像素构建的图像。该项目建立在使用非经典模态逻辑BISKT的基础上，用于描述离散空间中单个细节级别的空间关系。该项目将研究BISKT逻辑的双模拟和Hennessy-Milner属性，因为这是尚未探索的潜在模态理论的一部分。为了研究粒度，将采用协代数的视角。现有的BISKT语义可以用T-余代数来表示，对于一个合适的内函子T，在预序集和保序函数的范畴上。这种设置为开发BISKT语义提供了一个非常自然的空间，并开辟了一条新的研究路线，其目标是提供BISKT语义和句法属性的共代数解释。

The aim of this project is to develop a logic which allows the representation of statements such as "At all levels of detail ..." and "At some level of detail ...". The main area of application is to reasoning about spatial information, especially in a discrete space such as a graph representing a network, or an image built of pixels with a graph modelling adjacency. The project builds on work using a non-classical modal logic, BISKT, for describing spatial relations at a single level of detail in a discrete space. The project will investigate bisimulation and the Hennessy-Milner property for the BISKT logic, as this is part of the underlying modal theory that has not been explored. In order to study granularity, the perspective of co-algebras will be applied. The existing BISKT semantics can be expressed in terms of T-coalgebras for a suitable endofunctor, T, on the category of pre-ordered sets and order-preserving functions. This setting represents a very natural space to develop the BISKT semantics and opens up a new line of research where, the objective would be to provide a coalgebraic interpretation of semantic and syntactic properties of BISKT.