Coalgebraic Modal Logic: Fixpoints and Nested Modalities

模态逻辑是对变化进行推理的形式框架。模态逻辑在计算机科学和相关学科中的应用非常广泛，并且在各种应用环境中研究了许多不同的逻辑。除了在并发、移动和概率系统领域的经典应用之外，模态逻辑还用于人工智能，例如在不确定性推理的背景下，以及在知识表示领域中以描述逻辑的形式。模态逻辑也被用于推理多智能体系统中的博弈和联盟力量。在经济学中，它们被用来描述相互作用的主体共享的概率信息，而在哲学中则研究义务逻辑、义务逻辑和许可逻辑。所有这些逻辑的研究都围绕着一些反复出现的问题，包括完整性（“所有有效的陈述都是形式上可推导的吗？”）)，可判定性(“逻辑是否适用于自动推理？”)和复杂性(“机械化逻辑需要什么资源？”)，通常对每个逻辑分别进行研究。与其在每个逻辑的基础上解决这些问题，不如在一个统一的框架中进行研究，这个框架包含了尽可能多的具体给定逻辑作为特殊情况。提出的研究调查并扩展了一个通用框架——共代数模态逻辑——它包含了一大类模态逻辑作为具体示例，包括上述所有实例。这种方法的通用性将导致软件工具不仅更可靠，而且更容易设计、实现和维护。

Modal logic is a formal framework for reasoning about change. Applications of modal logics are abundant in computer science and related disciplines, and a multitude of different logics have been studied in a variety of application contexts. Apart from classical applications in the field of concurrent, mobile and probabilistic systems, modal logics are used in artificial intelligence, e.g. in the context of reasoning with uncertainty and -- in the shape ofdescription logics -- in the field of knowledge representation.Modal logics are also employed to reason about games and coalitional power in multi-agent systems. In economics, they have been used to describe probabilistic information shared by interacting agents whereas e.g. deontic logic, the logic of obligation and permission is studied in philosophy.The study of all of these logics centres around a number of recurring questions, including completeness (``are all valid statements formally derivable?''), decidability (``is the logic amenable to automated reasoning?'') and complexity (``what resources are required to mechanise the logic?''), which are usually studied for each logic individually.Rather than addressing these questions in a per-logic basis, it is clearly desirable to conduct the studies in a uniform framework that encompasses the greatest possible number ofconcretely given logics as special cases.The proposed research investigates and extends a generic framework -- coalgebraic modal logic -- that encompasses a large class of modal logics as specific examples, including all instances mentioned above. The genericity of the approach will lead to software tools that are not only more reliable but also and easier to design, implement and to maintain.