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2018 | OriginalPaper | Chapter

A Dynamic Logic for Learning Theory

Authors : Alexandru Baltag, Nina Gierasimczuk, Aybüke Özgün, Ana Lucia Vargas Sandoval, Sonja Smets

Published in: Dynamic Logic. New Trends and Applications

Publisher: Springer International Publishing

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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 [9, 17] with dynamic observation modalities \([o]\varphi \), as well as with a learning operator

https://static-content.springer.com/image/chp%3A10.1007%2F978-3-319-73579-5_3/461709_1_En_3_IEq2_HTML.gif

, which encodes the learner’s conjecture after observing a finite sequence of data

https://static-content.springer.com/image/chp%3A10.1007%2F978-3-319-73579-5_3/461709_1_En_3_IEq3_HTML.gif

. We completely axiomatise DLLT, study its expressivity and use it to characterise various notions of knowledge, belief, and learning.

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previous chapter Axiomatization and Computability of a Variant of Iteration-Free PDL with Fork

next chapter Layered Logics, Coalgebraically

From probabilistic and statistical formalisms based on Bayesian reasoning, Popper-style measures of corroboration, through default and non-monotonic logics, Carnap-style ‘inductive logic’, to AGM-style rational belief revision and theory change.

‘The Logic of Reliable Inquiry’ is the title of a classic text in FLT-based epistemology [16].

We will return to it, with complete definitions, later in the paper. Our DLLT is interpreted over such frames.

In the tautological information state X, the learner believes P iff \(\mathbb {L}(X)\subseteq P\).

This topology is T1 iff for every two distinct points \(x\not =y\) there exist an observation \(O\in \mathscr {O}\) with \(x\in O\) and \(y\not \in O\).

The observational topology is T0 iff points can be distinguished by observations; i.e. if x and y satisfy the same observable properties in \(\mathscr {O}\), then \(x=y\). Obviously, T0 is a minimally necessary condition for any kind of learnability of the real world from observations.

A set is locally closed if it is the intersection of a closed and an open set.

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Title: A Dynamic Logic for Learning Theory
Authors: Alexandru Baltag
Nina Gierasimczuk
Aybüke Özgün
Ana Lucia Vargas Sandoval
Sonja Smets
Publisher: Springer International Publishing
Book: Dynamic Logic. New Trends and Applications
Print ISBN: 978-3-319-73578-8

Electronic ISBN: 978-3-319-73579-5

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-3-319-73579-5_3

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