Hierarchies in independence and inclusion logic with strict semantics

We study the expressive power of fragments of inclusion and independence logic defined by restricting the number k of universal quantifiers in formulas. Assuming the so-called strict semantics for these logics, we relate these fragments of inclusion and independence logic to sublogics
${\mathrm ESO}_f(k\forall)$
of existential second-order logic, which in turn are known to capture the complexity classes
${\mathrm NTIME_{RAM}}(n^k)$
.