On decidability of the theory $\mathrm{Th}(\omega,0,1,<,+,f_0,\dots,f_n)$

In the paper we consider theories which are obtained from the Semenov arithmetics introducing functions $f_i$, $i>0$. They are called “hyperfunctions” and they are obtained when we iterate an addition-connected function. We have proved, that such theories are model complete. It is also shown, that these theories are decidable when the condition of effective periodicity is satisfied for hyperfunctions.