Learning, topology, and modal logic
Realistic modeling of intelligent behavior requires that agents are endowed with methods of integrating new information into their prior beliefs and knowledge. Such learning methods allow belief change on the basis of assessing new information. But how effective is a learning method in eventually finding the truth? Does such reliability requirement agree with the rationality postulates customarily imposed on knowledge and belief change?
The two questions have been addressed in several lines of research. Combining belief revision procedures with learning-theoretic notions led to many interesting observations about reliability and rationality within different formal frameworks (see, e.g., [8, 9]). However, the setting of possible world semantics is particularly well-suited for the more recent controversies surrounding the logical notions of knowledge and belief [5, 1, 2, 6]. Such treatment of learnability allows a smooth transition to general topology, where the notion of reliable, limiting learning can be elegantly characterized and generalized [3]. This does not come as a great surprise since the connections between learnability in the limit and topology have been previously studied (see, e.g., [4, 7]). However, the particular way of viewing learnable epistemic spaces through the lens of topological semantics of modal logic informs the (dynamic) epistemic logic axiomatizations of knowledge and belief. It indicates which properties of belief operator are adequate for characterizing learnable spaces.
In conclusion, the main purpose of this invited talk is to present a topological bridge between learnability in the limit and (dynamic) epistemic logic.
References
[1] A. Baltag, N. Gierasimczuk, and S. Smets. Belief revision as a truth- tracking process. In K. Apt, editor, Proceedings of TARK'11, pages 187-190. ACM, 2011.
[2] A. Baltag, N. Gierasimczuk, and S. Smets. Truth tracking by belief revision. ILLC Prepublication Series PP-2014-20 (to appear in Studia Logica), 2014.
[3] A. Baltag, N. Gierasimczuk, and S. Smets. On the solvability of inductive problems: A study in epistemic topology. ILLC Prepublication Series PP-2015-13 (to appear in Proceedings of TARK'15), 2015.
[4] M. de Brecht and A. Yamamoto. Topological properties of concept spaces. Information and Computation, 208(4):327-340, 2010.
[5] N. Gierasimczuk. Knowing One's Limits. Logical Analysis of Inductive Inference. PhD thesis, Universiteit van Amsterdam, The Netherlands, 2010.
[6] N. Gierasimczuk, D. de Jongh, and V. F. Hendricks. Logic and learning. In A. Baltag and S. Smets, editors, Johan van Benthem on Logical and Informational Dynamics. Springer, 2014.
[7] K. T. Kelly. The Logic of Reliable Inquiry. Oxford University Press, 1996.
[8] K. T. Kelly. Iterated belief revision, reliability, and inductive amnesia. Erkenntnis, 50:11-58, 1998.
[9] E. Martin and D. Osherson. Scientific discovery based on belief revision. The Journal of Symbolic Logic, 62(4):1352-1370, 1997.