T. Kurahashi, "Rosser Provability and Normal Modal Logics"
https://arxiv.org/abs/1812.09735
In this paper, we investigate Rosser provability predicates whose provability logics are normal modal logics. First, we prove that there exists a Rosser provability predicate whose provability logic is exactly the normal modal logic $ \sf KD（正規様相論理KD）. Secondly, we introduce a new normal modal logic $ \sf KDR which is a proper extension of KD, and prove that there exists a Rosser provability predicate whose provability logic includes $ \sf KDR.