Complexity of the problem of being equivalent to Horn formulas

We look at the complexity of the existence problem for a Horn sentence (identity, quasi-identity, $\forall$-sentence, $\exists$-sentence) equivalent to a given one. It is proved that if the signature contains at least one symbol of arity $k\geqslant 2$, then each of the problems mentioned is an $m$-complete $\Sigma^0_1$ set.