  We consider the constructive ordinal notation system for the ordinal
${\epsilon_0}$ that were introduced by L.D. Beklemishev. There are fragments of
this system that are ordinal notation systems for the smaller ordinals
${\omega_n}$ (towers of ${\omega}$-exponentiations of the height $n$). This
systems are based on Japaridze's provability logic $\mathbf{GLP}$. They are
closely related with the technique of ordinal analysis of $\mathbf{PA}$ and
fragments of $\mathbf{PA}$ based on iterated reflection principles. We consider
this notation system and it's fragments as structures with the signatures
selected in a natural way. We prove that the full notation system and it's
fragments, for ordinals ${\ge\omega_4}$, have undecidable elementary theories.
We also prove that the fragments of the full system, for ordinals
${\le\omega_3}$, have decidable elementary theories. We obtain some results
about decidability of elementary theory, for the ordinal notation systems with
weaker signatures.
