being an answer set, that is, those reduct models which are proper subsets of the original model.

We just note at this point that the models of the reduct $P^U$ given in Item 5 are not specified for particular purposes, but are required to allow for a realization via disjunctive programs. In fact, these models are just an effect of property (P1) mentioned in Section \[sec:preliminaries\]. However, before showing a program satisfying the properties of a *$\Phi$-reduction*, we first show the rationale behind the concept of $\Phi$-reductions.

For any QBF $\Phi = \forall X\exists Y\forall Z\phi$ with $\phi$ in DNF, a $\Phi$-reduction is super-coherent if and only if $\Phi$ is true.

Suppose that $\Phi$ is false. Hence, there exists an ${{\mathcal I}}\subseteq X$ such that, for all $J\subseteq Y$, there is a $K_Y\subseteq Z$ with ${{\mathcal I}}\cup J \cup K_Y\not\models \phi$. Now let $P$ be any $\Phi$-reduction and $F_{{\mathcal I}}= {{\mathcal I}}\cup \overline{(X\setminus {{\mathcal I}})}$. We show that ${{\mathit AS}}(P\cup F_{{\mathcal I}})=\emptyset$, thus $P$ is not super-coherent. Let ${{\mathcal M}}$ be a model of $P \cup F_{{\mathcal I}}$. Since $P$ is a $\Phi$-reduction, the only candidates for ${{\mathcal M}}$ are $U$, $M[{{\mathcal I}},J]$, and $M'[{{\mathcal I}},J]$, where $J\subseteq Y$. Indeed, 