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We consider some DL-features denoted by (I) (inverse), (O) (nominal), (F) (functionality), (N) (unqualified number restriction), (Q) (qualified number restriction), (U) (universal role), (mathsf {Self}) (local reflexivity of a role).
We will consider DL-features denoted by I (inverse roles), (Q_k) (qualified number restrictions with numbers bounded by a constant k), and (mathsf {Self}) (local reflexivity of a role).
Proof By Lemma 4.2, for each i ∈ N, ( I − λ i B i ) U i = ( I 2 − δ i L i ) are averaged and I 2 − U i is k i inverse strongly monotone for some k i ≥ 1 2. Following the same argument as in Theorem 3.1 [24], we can show that for each i ∈ N, A i ∗ ( I 2 − U i ) A i, is k i R i -inverse strong monotone.
We have proved that any concept in any description logic that extends (mathcal {ALC}) with some features amongst I (inverse roles), (Q_k) (qualified number restrictions with numbers bounded by a constant k), and (mathsf {Self}) (local reflexivity of a role) can be learned if the training information system (specified as a finite interpretation) is good enough.
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Let Ψ i : C → X (i = 1, 2) be β ̃ i -inverse-strongly accretive and Φ i : C → X (i = 1, 2) be γ ̃ i -inverse-strongly accretive.
Let A : C → E ∗ be a κ i -inverse-strongly mappinge mapping.
Let the mapping B i : C → X be α i -inverse-strongly accretive.
Let Q C be a sunny nonexpansive retraction from X onto C. Let Ψ i : C → H (i = 1, 2) be β ̃ i -inverse-strongly accretive and Φ i : C → H (i = 1, 2) be γ ̃ i -inverse-strongly accretive.
For each i = 1, 2, let κ i > 0 and let B i be a κ i -inverse-strongly mofotone mapping of C into H.
For every i = 1, 2, …, N, A i : H → H be α i -inverse strongly monotone mapping with η = min i = 1, 2, …, N { α i }.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com