Order-topological and model-theoretic methods for modal logics

研究内容涉及模态逻辑的基本理论和其他一些非经典逻辑。使用模型论、拓扑论、代数、序论和范畴论工具，使用增强的Sahlqvist理论和规范公式证明一般完备性定理，以及一些逻辑系统的特定完备性定理。本研究还将发展Fine的正则性定理和代数补全理论，并将双模拟理论扩展到描述框架（赋予附加拓扑结构的Kripke框架）。描述性框架将是一个重要的工具，研究将有助于对其性质的一般理解。将开发新的工具和技术来处理这些问题，这些工具和技术将得到更广泛的应用。这项研究是将代数和拓扑方法应用于非经典逻辑的一个新兴领域的一部分。

The research concerns the fundamental theory of modal logic and some other non-classical logics. Using model-theoretic, topological, algebraic, order-theoretic, and category-theoretic tools, general completeness theorems using enhanced Sahlqvist theory and canonical formulas will be proved, as well as specific completeness theorems for a number of logical systems. The research will also develop Fine's canonicity theorem and the theory of algebraic completions, and extend the theory of bisimulations to descriptive frames (Kripke frames endowed with additional topological structure). Descriptive frames will be an important tool, and the research will contribute to general understanding of their properties. New tools and techniques will be developed for dealing with them, and these will be of wider use. The research is part of a growing field that applies algebraic and topological methods in non-classical logics.