Abstract
We present a dynamic logic for inductive learning from partial observations by a “rational” learner, that obeys AGM postulates for belief revision. We apply our logic to an example, showing how various concrete properties can be learnt with certainty or inductively by such an AGM learner. We present a sound and complete axiomatization, based on a combination of relational and neighbourhood version of the canonical model method.
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Notes
- 1.
Indeed, our observational events can be seen as corresponding to a special type of (single-agent) epistemic events in the so-called BMS style.
- 2.
A total preorder \(\le \) on X is a reflexive and transitive binary relation such that every two points are comparable: for all \(x, y\in X\), either \(x\le y\) or \(y\le x\) (or both).
- 3.
Since \(\le \) is a total preorder, this definition coincides with the standard definition of maximal elements as \(Max_\le (O):=\{x\in O : \forall y\in O(x\le y \text{ implies } x\le y)\}\).
- 4.
This is a standard method in Dynamic Epistemic Logic and we refer the reader to [18, Chap. 7.4] for further details.
- 5.
When we quantify over learners, learnability with certainty (by some learners) matches the standard concept of “finite identifiability” from Formal Learning Theory.
- 6.
When we quantify over learners, inductive learnability (by some learners) matches the standard concept of “identifiability in the limit” from Formal Learning Theory, see e.g. [13].
- 7.
We use \(\preceq \) to denote the plausibility order in this frame, to distinguish it from the natural order on \(X\subseteq R\).
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Baltag, A., Özgün, A., Vargas-Sandoval, A.L. (2020). The Logic of AGM Learning from Partial Observations. In: Soares Barbosa, L., Baltag, A. (eds) Dynamic Logic. New Trends and Applications. DALI 2019. Lecture Notes in Computer Science(), vol 12005. Springer, Cham. https://doi.org/10.1007/978-3-030-38808-9_3
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