\[sec:proofdisj\] for disjunctive programs, and in Section \[sec:proofnorm\] for normal programs. In Section \[sec:embedding\] we show “completeness” of [$\rm ASP^{\rm sc}$]{} via simulating reasoning tasks over [$\rm ASP$]{} by tasks over [$\rm ASP^{\rm sc}$]{}. In Section \[sec:impl\] we briefly discuss the relation to equivalence problems before concluding the work in Section \[sec:concl\].

Preliminaries {#sec:preliminaries}
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In this paper we consider propositional programs, so an atom $p$ is a member of a countable set ${{\cal U}}$. A [*literal*]{} is either an atom $p$ (a positive literal), or an atom preceded by the [ *negation as failure*]{} symbol ${{\mathit not}}$ (a negative literal). A [*rule*]{} ${\ensuremath{r}}$ is of the form $$\begin{array}{l}
    p_1 \ \lor\ \cdots \ \lor\ p_n {\leftarrow}q_1,\ \ldots,\ q_j,\ {{\mathit not}}~q_{j+1},\ \ldots,\ {{\mathit not}}~q_m  
\end{array}$$ where $p_1,\ \ldots,\ p_n,\ q_1,\ \ldots,\ q_m$ are atoms and $n\geq 0,$ $m\geq j\geq 0$. The disjunction $p_1\ \lor\ \cdots \ \lor\ p_n$ is the [*head*]{} of [$r$]{}, while the conjunction $q_1,\ \ldots,\ q_j,\ {{\mathit not}}~q_{j+1},\ \ldots,\ {{\mathit not}}~q_m$ is the [*body*]{} of [$r$]{}. Moreover, ${\ensuremath{H({\ensuremath{r}})}}$ denotes the set of head atoms, while ${\ensuremath{B({\ensuremath{r}})}}$ denotes the set of body literals. We also use ${\ensuremath{B^+({\ensuremath{r}})}}$ and ${\ensuremath{B^-({\ensuremath{r}})}}$ for denoting the set of atoms appearing in positive 