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Outline

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_3

Abstract

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.

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  27. 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 .