# From Surprise Exams to Topological Mu-Calculus

## Alexandru Baltag, University of Amsterdam

I present a topological epistemic logic, motivated by a famous epistemic puzzle: the Surprise Exam Paradox. It is a fixed-point modal logic, with modalities for knowledge (modelled as the universal modality), knowability of a proposition (represented by the topological interior operator), and (un)knowability of the actual world. The last notion has a non-self-referential reading (modelled by Cantor derivative: the set of limit points of a given set) and a self-referential one (modelled by the so-called perfect core of a given set: its largest subset which is a fixed point of relativized derivative). I completely axiomatize this logic, showing that it is decidable and PSPACE-complete, as well as briefly explain how the same model-theoretic method can be elaborated to prove the completeness and decidability of the full topological mu-calculus. Finally, I apply it to the analysis of the Surprise Exam Paradox and other puzzles.

References:

- A. Baltag, N. Bezhanishvili, D. Fernández-Duque. The Topology of Surprise. Proceedings of the International Conference on Principles of Knowledge Representation and Reasoning. Vol. 19 (1), 33-42, 2022. Available online in ILLC Prepublication (PP) series PP-2022-06.
- A. Baltag, N. Bezhanishvili, D. Fernández-Duque. The Topological Mu-Calculus: Completeness and Decidability. LICS ‘21: Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science, vol 89: 1-13, 2021. doi:10.1109/lics52264.2021.9470560. Available online in ILLC Prepublication (PP) series PP-2021-07.