# Comparing the Reasoning Capabilities of Equilibrium Theories and Answer Set Programs
## 6 Projective Concepts  
### Basic Definitions  
Very often we are interested only in certain parts of answer sets or equilibrium models, and the output of solvers may suppress the unwanted parts. If our query is expressed in a sublanguage \(B\) of the theory, we need only consult the projection of equilibrium models onto that sublanguage, i.e., we deal with the \(B\)-reducts of the equilibrium models ( In the remainder of the article, we use upper-case Latin letters, \(A\), \(B\), etc., to denote (sub)languages, i.e., regarded as sets of atoms). This is justified by the next lemma. For notation, let \(M\) be a classical model for a language \(\mathcal{L}\) and \(B\) a sublanguage of \(\mathcal{L}\). We denote by \(M|B\) the \(B\)-reduct of \(M\), i.e., \(M\cap B\), and where now truth and falsity in \(M|B\) is defined only for formulas expressible in \(B\). Likewise, for any set \(X\) of classical models, \(X|B\) denotes the set of their \(B\)-reducts.  
Lemma 2: _Let \(M\) be an equilibrium model (of some theory in \(\mathcal{L}\)) and \(B\) a sublanguage of \(\mathcal{L}\). Then, for any \(B\)-formula \(\varphi\), \(M\models\varphi\Leftrightarrow M|B\models\varphi\)._  
Definition 11 (\(B\)-consequence): For \(\omega\in\{c,s,p\}\)  
1. [label=()]
2. We say that \(\Pi_{2}\) is a \(B\)-consequence of \(\Pi_{1}\), in symbols \(\Pi_{1}\vdash_{\omega,B}\Pi_{2}\), if for any \(B\)-formula \(\varphi\), \(\Pi_{2}\models_{\omega}\varphi\Rightarrow\Pi_{1}\models_{\omega}\varphi\).
3. We say that \(\Pi_{1}\) and \(\Pi_{2}\) are \(B\)-inseparable (for \(\omega\)), in symbols \(\Pi_{1}\equiv_{\omega,B}\Pi_{2}\), if \(\Pi_{1}\vdash_{\omega,B}\Pi_{2}\) and \(\Pi_{2}\vdash_{\omega,B}\Pi_{1}\).  
Strong versions of \(B\) consequence and inseparability are obtained in the obvious way.  
Definition 12 (strong \(B\)-consequence): For \(\omega\in\{c,s,p\}\):  
* We write \(\Pi_{1}\vdash_{\omega,B}^{s}\Pi_{2}\) (for strong \(B\)-consequence) if for any \(\Pi\) and \(B\)-formula \(\varphi\), \(\Pi_{2}\cup\Pi\models_{\omega}\varphi\Rightarrow\Pi_{1}\cup\Pi\models_{\omega}\varphi\).
* Similarly for strong \(B\)-inseparability: \(\Pi_{1}\equiv_{\omega,B}^{s}\Pi_{2}\) if for any \(\Pi\) and \(B\)-formula \(\varphi\), \(\Pi_{1}\cup\Pi\vdash_{\omega,B}\Pi_{2}\) and \(\Pi_{2}\cup\Pi\vdash_{\omega,B}\Pi_{1}\).
* Relativised versions are easily obtained. e.g., strong \(B\)-consequence, relative to \(A\), in symbols \(\Pi_{1}\vdash_{\omega,B}^{s,A}\Pi_{2}\), obtains when for any set \(\Pi\) of \(A\) formulas and \(B\)-formula \(\varphi\), \(\Pi_{2}\cup\Pi\models_{\omega}\varphi\Rightarrow\Pi_{1}\cup\Pi\models_{\omega}\varphi\).
* Similarly, strong inseparability relative to \(A\) is denoted by \(\Pi_{1}\equiv_{\omega,B}^{s,A}\Pi_{2}\).  
Our previous notion of relativised strong equivalence has a straightforward projective version.  
Definition 13: Let \(\Pi_{1}\) and \(\Pi_{2}\) be theories. \(\Pi_{1}\) and \(\Pi_{2}\) are _strongly equivalent relative to \(A\) projected onto \(B\), in symbols \(\Pi_{1}\equiv_{B}^{s,A}\Pi_{2}\), if for any (empty or non-empty) set \(\Sigma\) of \(A\) formulas, \(E(\Pi_{1}\cup\Sigma)|B=E(\Pi_{2}\cup\Sigma)|B\).  
Proposition 9: Two theories, \(\Pi_{1}\) and \(\Pi_{2}\), are strongly \(B\)-inseparable relative to \(A\) iff they are strongly equivalent relative to \(A\), projected onto \(B\).  
Proof.: See Appendix A.  
**Definition 14** (strong \(B\)-entailment).: _Let \(\Pi_{1}\) and \(\Pi_{2}\) be theories. We say that \(\Pi_{1}\) strongly entails \(\Pi_{2}\) relative to \(A\), projected onto \(B\), in symbols \(\Pi_{1}\mathbin{\vartriangle}_{B}^{s,A}\Pi_{2}\), if for any set \(\Sigma\) of \(A\) formulas, \(E(\Pi_{1}\cup\Sigma)|B\subseteq E(\Pi_{2}\cup\Sigma)|B\)._  
Strong \(B\)-entailment and (relativised) strong \(B\)-consequence coincide for \(\omega=s\).  
**Proposition 10**.: _For any theories \(\Pi_{1}\), \(\Pi_{2}\):_  
\[\Pi_{1}\mathbin{\vartriangle}_{B}^{s,A}\Pi_{2}\Leftrightarrow\Pi_{1} \mathbin{\vartriangle}_{s,B}^{s,A}\Pi_{2} \tag{3}\]  
Proof.: See Appendix A.