Among many interpretations of modal logic the one pertaining to knowledge and belief has been especially buoyant in recent years. The framework of epistemic logic offers a platform for a systematic study of knowledge and belief. Dynamic epistemic logic further covers many kinds of transformations knowledge undergoes in communication, and under other informational events. Such iterated changes can be given a long-term horizon of learning, i.e., they can be seen as ways to acquire a desirable kind of epistemic state. Thus, the question arises: Can modal logic contribute to our understanding of learning processes in general?
In my course I will overview the modal-logical and topological perspective on learnability. The link between dynamic epistemic logic and computational learning theory was introduced in [6,7,8], where it was shown that exact learning in finite time (also known as finite identification, see [10,11]) can be modelled in dynamic epistemic logic, and that the elimination process of learning by erasing [9] can be seen as iterated upgrade of dynamic doxastic logic. This bridge opened a way to study truth-tracking properties of doxastic upgrade methods on positive, negative, and erroneous input [2,4]. Switching from relational to topological semantics for modal logic allowed characterising favourable conditions for learning in the limit in terms of general topology [3] and culminated in proposing a Dynamic Logic for Learning Theory, which extends Subset Space Logics [5] with dynamic observation modalities and a learning operator [1].
[1] Baltag, Gierasimczuk, Özgün, Vargas-Sandoval, and Smets, A dynamic logic for learning theory, Journal of Logical and Algebraic Methods in Programming, vol. 109 (2019), pp. 100485.
[2] Baltag, Gierasimczuk, and Smets, Belief revision as a truth-tracking process, Proceedings of TARK’11, ACM, New York, 2011, pp. 187-190.
[3] Baltag, Gierasimczuk, and Smets, On the solvability of inductive problems: A study in epistemic topology, Proceedings of TARK’15, vol. 215, EPTCS, 2016, pp. 81-98.
[4] Baltag, Gierasimczuk, and Smets, Truth-tracking by belief revision, Studia Logica, vol. 107 (2019), no. 5, pp. 917-947.
[5] Dabrowski, Moss, and Parikh, Topological reasoning and the logic of knowledge, Annals of Pure and Applied Logic, vol. 78 (1996), no. 1, pp. 73-110.
[6] Gierasimczuk, Bridging learning theory and dynamic epistemic logic, Synthese, vol. 169 (2009), no. 2, pp. 371-384.
[7] Gierasimczuk, Learning by erasing in dynamic epistemic logic, Proceedings of LATA’09, vol. 5457, LNCS, Springer, 2009, pp. 362-373.
[8] Gierasimczuk, Knowing One’s Limits. Logical Analysis of Inductive Inference, PhD thesis, Universiteit van Amsterdam, The Netherlands, 2010.
[9] Lange, Wiehagen, and Zeugmann, Learning by erasing, ALT 1996: Algorithmic Learning Theory, vol. 1160, LNCS, Springer, 1996, pp. 228-241.
[10] Lange and Zeugmann, Types of monotonic language learning and their characterization, Proceedings of COLT’92, ACM, New York, 1992, pp. 377-390.
[11] Mukouchi, Characterization of finite identi cation, Proceedings AII’92, vol. 642, LNCS, Springer, 1992, pp. 260-267.
