Abbreviations

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Model theory for extensions of modal logic

Many formal languages arising in computer science can be seen as extensions of modal logic. What we mean by "extension" can be categorized as follows: i. Axiomatic extensions (same language, more theorems, less models). ii. Expressive extensions (adding new syntactic constructs, more formulas). iii. Semantic, or signature extensions (enriching the type of mathematical structures described, more models). We explore these directions and their interaction, focusing on the model theory of the obtained languages. As a case study, we consider the expressivity and definability of a number of extended modal languages interpreted on topological spaces. After completing this course the students will have learned about various ways of extending modal logic; about expressivity and definability in modal logic; about first order topological language and topological model theory; about using the model-theoretic ideas and techniques in modal logic. ROUGH COURSE OUTLINE: Mon: Basic modal logic and three ways of extending it. - i. Axiomatic extensions of modal logic: a brief survey. Tue: - ii. Language enrichments: a survey (case study: hybrid languages) Wed: - iii. Richer types of structures (eg. polyadic Kripke structures, topological models). - Zooming in on the case of topological models: McKinsey-Tarski theorem, topo-bisimulations. Thu & Fri: - Mixed extensions - Case study: hybrid languages interpreted on (various classes of) topological spaces: expressivity, definability, axiomatization and complexity PREREQUISITS: Basic knowledge of modal logic, corresponding to chapters 1-3 of Blackburn-et-al. In particular, understanding the meaning of Van Benthem and/or Goldblatt-Thomason theorem (without necessarily understanding their proofs).