A Dynamic Logic for Learning Theory
2019, Journal of Logical and Algebraic Methods in Programming
https://doi.org/10.1007/978-3-319-73579-5_3Abstract
Building on previous work [4, 5] that bridged Formal Learning Theory and Dynamic Epistemic Logic in a topological setting, we introduce a Dynamic Logic for Learning Theory (DLLT), extending Subset Space Logics [17, 9] with dynamic observation modalities [o]ϕ, as well as with a learning operator L(#» o), which encodes the learner's conjecture after observing a finite sequence of data #» o. We completely axiomatise DLLT, study its expressivity and use it to characterise various notions of knowledge, belief, and learning.
References (27)
- Aiello, M., Pratt-Hartmann, I., van Benthem, J., et al. (2007) Handbook of spatial logics, vol. 4. Springer.
- Balbiani, P., Baltag, A., van Ditmarsch, H., Herzig, A., Hoshi, T., and Lima, T. D. (2008) 'Knowable' as 'Known after an announcement'. The Review of Symbolic Logic, 1, 305-334.
- Balbiani, P., van Ditmarsch, H., and Kudinov, A. (2013) Subset space logic with arbitrary announcements. Proc. of the 5th ICLA, pp. 233-244, Springer.
- Baltag, A., Gierasimczuk, N., and Smets, S. (2011) Belief revision as a truth-tracking pro- cess. Proceedings of the 13th Conference on Theoretical Aspects of Rationality and Knowl- edge, pp. 187-190, ACM.
- Baltag, A., Gierasimczuk, N., and Smets, S. (2015) On the solvability of inductive problems: A study in epistemic topology. Ramanujam, R. (ed.), Proceedings of the 15th conference TARK, also available as a technical report in ILLC Prepublication Series PP-2015-13.
- Baltag, A., Moss, L. S., and Solecki, S. (1998) The logic of public announcements, common knowledge, and private suspicions. Proceedings of the 7th Conference TARK, pp. 43-56, Morgan Kaufmann Publishers Inc.
- Baltag, A., Özgün, A., and Vargas-Sandoval, A. L. (2017) Topo-logic as a dynamic-epistemic logic. Proceedings of the 6th International Workshop on Logic, Rationality and Interaction (LORI 2017), to appear.
- Bjorndahl, A. (2016) Topological subset space models for public announcements. To appear in Trends in Logic, Outstanding Contributions: Jaakko Hintikka.
- Dabrowski, A., Moss, L. S., and Parikh, R. (1996) Topological reasoning and the logic of knowledge. Annals of Pure and Applied Logic, 78, 73-110.
- Dégremont, C. and Gierasimczuk, N. (2011) Finite identification from the viewpoint of epis- temic update. Information and Computation, 209, 383-396.
- Georgatos, K. (1994) Knowledge theoretic properties of topological spaces. Masuch, M. and Pólos, L. (eds.), Knowledge Representation and Reasoning Under Uncertainty: Logic at Work, vol. 808 of Lecture Notes in Artificial Intelligence, pp. 147-159, Springer.
- Georgatos, K. (1997) Knowledge on treelike spaces. Studia Logica, 59, 271-301.
- Gierasimczuk, N. (2010) Knowing One's Limits. Logical Analysis of Inductive Inference. Ph.D. thesis, Universiteit van Amsterdam, The Netherlands.
- Gierasimczuk, N. and de Jongh, D. (2013) On the complexity of conclusive update. The Computer Journal, 56, 365-377.
- Gold, E. M. (1967) Language identification in the limit. Information and control, 10, 447- 474.
- Kelly, K. T. (1996) The Logic of Reliable Inquiry. Oxford University Press.
- Moss, L. S. and Parikh, R. (1992) Topological reasoning and the logic of knowledge. Pro- ceedings of the 4th conference TARK, pp. 95-105, Morgan Kaufmann.
- Parikh, R., Moss, L. S., and Steinsvold, C. (2007) Topology and epistemic logic. Handbook of Spatial Logics, pp. 299-341.
- van Benthem, J. (2011) Logical Dynamics of Information and Interaction. Cambridge Uni- versity Press, New York, NY, USA.
- van Ditmarsch, H., Knight, S., and Özgün, A. (2014) Arbitrary announcements on topolog- ical subset spaces. Proceedings of the 12th European Conference on Multi-Agent Systems (EUMAS), pp. 252-266, Springer.
- van Ditmarsch, H., Knight, S., and Özgün, A. (2015) Announcement as effort on topological spaces. Proc. of the 15th TARK, pp. 95-102.
- van Ditmarsch, H., Knight, S., and Özgün, A. (2015) Announce- ment as effort on topological spaces. Extended version, Submitted. http://www.lix.polytechnique.fr/ sophia/papers/effort.pdf.
- van Ditmarsch, H., van der Hoek, W., and Kooi, B. (2007) Dynamic Epistemic Logic. Springer Publishing Company, 1st edn.
- Vickers, S. (1989) Topology via logic. Cambridge Tracts in Theoretical Computer Science, Cambridge University Press.
- Wang, Y. N. and Ågotnes, T. (2013) Multi-agent subset space logic. Proceedings of the 23rd IJCAI, pp. 1155-1161, IJCAI/AAAI.
- Weiss, M. A. and Parikh, R. (2002) Completeness of certain bimodal logics for subset spaces. Studia Logica, 71, 1-30. {θ : Kθ ∈ T } [s][o]¬ϕ for all o ∈ O. Then, by normality of K, T K[s][o]¬ϕ for all o ∈ O. Since K[s][o]¬ϕ := [K, s][o]¬ϕ is a necessity form and T is O-witnessed, we obtain T [K, s] ¬ϕ, i.e., T K[s] ¬ϕ. As T is maximal, we have K[s] ¬ϕ ∈ T , thus [s] ¬ϕ ∈ {θ : Kθ ∈ T }.
- Proof (Lemma 8 and Corolary 1). From left-to-right follows directly from the definition of X c and ∼ K . For the right-to-left direction, we prove the contrapositive: Let ϕ ∈ L such that Kϕ T . Then, by Lemma 7 and Lemma 4, we obtain that {ψ : Kψ ∈ T } ∪ {¬ϕ} is an O-witnessed theory. We can then apply Lindenbaum's Lemma (Lemma 5) and extend it to a maximal O-witnessed theory S such that ϕ S .