  The class of all $\ast$-continuous Kleene algebras, whose description
includes an infinitary condition on the iteration operator, plays an important
role in computer science. The complexity of reasoning in such algebras -
ranging from the equational theory to the Horn one, with restricted fragments
of the latter in between - was analyzed by Kozen (2002). This paper deals with
similar problems for $\ast$-continuous residuated Kleene lattices, also called
$\ast$-continuous action lattices, where the product operation is augmented by
adding residuals. We prove that in the presence of residuals the fragment of
the corresponding Horn theory with $\ast$-free hypotheses has the same
complexity as the $\omega^\omega$ iteration of the halting problem, and hence
is properly hyperarithmetical. We also prove that if only commutativity
conditions are allowed as hypotheses, then the complexity drops down to
$\Pi^0_1$ (i.e. the complement of the halting problem), which is the same as
that for $\ast$-continuous Kleene algebras. In fact, we get stronger upper
bound results: the fragments under consideration are translated into suitable
fragments of infinitary action logic with exponentiation, and the upper bounds
are obtained for the latter ones.
