New aspects of the mu-calculus

形式逻辑为我们提供了一种指定和描述系统的数学语言，并提供了强大的方法来推理在给定情况下逻辑语句是否为真，哪些逻辑语句遵循哪些逻辑语句，等等-这些方法通常可以自动进行。模型检查应用这些思想来提供强大的自动化工具来验证软件是否符合其规范。这可以为公司节省资金，并让全社会对软件的可靠性充满信心。传统的模型检查在工业上取得了巨大的成功，但它的大部分都认为时间是离散的滴答-0、1、2等等。对于某些计算机系统来说，物理学中使用的连续时间更合适。这个项目的一部分将研究在与实时模型有关的情况下进行模型检查的可能性，使用一种非常强大的逻辑--时态微积分。从长远来看，它可能会带来新的模型检验方法，更复杂的系统。除了时间，空间也是许多现代应用的一个重要方面，包括数据库、地理信息系统和几何推理。逻辑也可以用来发表关于空间的陈述，并与之进行推理。已经使用了许多不同的逻辑系统，但在这种情况下，还没有对非常强大的u演算进行很大的研究。这个项目的目的是研究空间语境中的情态微积分，试图确定它的表现力，以及需要什么机制来正确地与它进行推理。从长远来看，这项工作有可能提高我们指定和推理涉及空间的情况的能力。我们还将借此机会建立一些关于u演算定义情况类的能力的基本事实，这与更简单逻辑的所谓Goldblatt-Thomason定理相呼应。如果成功，这将为研究人员提供一个适用于许多需要u演算的领域的基本工具。这一雄心勃勃的项目只有三个月的研究时间，可能不会解决所有问题，但我们希望取得良好的进展。

Formal logic provides us with a mathematical language for specifying and describing systems, as well as powerful methods for reasoning about whether a logical statement is true in a given situation, which logical statements follow from which, and so on - and these methods can often be automated. Model checking applies these ideas to provide powerful automated tools for verifying that software meets its specification. This can save companies money and provide confidence to wide society of the reliability of software. Conventional model checking has been enormously successful industrially, but much of it has considered time as discretely ticking - 0, 1, 2, and so on. For some computer systems, continuous time as used in physics is more appropriate. Part of this project will study the possibilities for model checking in situations concerning real-time models, using a very powerful logic, the temporal mu-calculus. In the long term, it may lead to new ways of model checking more sophisticated systems.As well as time, space is an important aspect of many modern applications, including databases, geographic information systems, and geometrical reasoning. Logic can also be used to make statements about space, and reason with them. Many different logical systems have been used, but again the very powerful mu-calculus has not been greatly investigated in this context. This project aims to study the modal mu-calculus in spatial contexts, trying to ascertain its expressiveness, and what machinery is needed to reason correctly with it. The work has the potential in the long run to improve our ability to specify and reason about situations involving space.We will also take the opportunity to try to establish some fundamental facts about the mu-calculus's power to define classes of situations, echoing the so-called Goldblatt-Thomason theorem for simpler logics. If successful, this will provide researchers with a basic tool usable in many areas requiring the mu-calculus.Only three months is available for the research on this ambitious project, and it is likely that not all problems will be solved, but we hope that good progress will be made.