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Provability logic GL is one of the normal modal logics, which is obtained from the

smallest normal modal logic K by adding L\"ob's axiom. The name ``provability logic''

derives from Solovay's theorem. He proved that GL is complete for the formal provability

interpretation in Peano arithmetic PA. So, GL has been considered as one of the most

important modal logics.

Interpretability logics were introduced by Albert Visser in 1990 as extensions of

the provability logic GL with a binary modality \rhd. The arithmetic realization of

A \rhd B in a theory T will be that T plus the realization of B is interpretable in T

plus the realization of A (T+A interprets T+B). More precisely, there exists a function

f(the relative interpretation) on the formulas of the language of T such that T+B \vdash C

implies T+A \vdash f(C).

Here, we describe some results on these logics in a recent few years.

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