On provability logics of Niebergall arithmetic

K. G. Niebergall suggested a simple example of a non-gödelean arithmeticaltheory $\mathrm{NA}$, in which a natural formalization of its consistencyis derivable. In the present paper we consider the provability logicof $\mathrm{NA}$ with respect to Peano arithmetic. We describe the class of itsfinite Kripke frames and establish the corresponding completeness theorem.For a conservative extension of this logic in the language with an additionalpropositional constant, we obtain a finite axiomatization. We also considerthe truth provability logic of $\mathrm{NA}$ and the provability logic of $\mathrm{NA}$ with respect to $\mathrm{NA}$ itself. We describe the classes of Kripkemodels with respect to which these logics are complete. We establish$\mathrm{PSpace}$-completeness of the derivability problem in these logicsand describe their variable free fragments. We also prove thatthe provability logic of $\mathrm{NA}$ with respect to Peano arithmeticdoes not have the Craig interpolation property.