# Comparing the Reasoning Capabilities of Equilibrium Theories and Answer Set Programs
## 6 Projective Concepts
### Forks and Projective \(B\)-Entailment for Theories
Notice that \(\langle H,T\rangle\models\varphi\) and \(\langle H,T\rangle\models\lambda\)). **Lemma 3** (Lemma 7 from [20]). : _Given \(T\subseteq A\subseteq At\) and any \(T\)-support \(\mathcal{H}\), there is a propositional formula \(\varphi_{T}\in\mathcal{L}_{A}\) such that \(\left[\,\varphi_{T}\,\right]^{T}=\mathcal{H}\) and \(\left[\,\varphi_{T}\,\right]^{Y}=\left[\,\right]\) for any \(Y\subseteq A\) and \(Y\
eq T\)._  
**Definition 16**. : _Given \(T\), \(A\subseteq At\), we say that a \(T\)-support \(\mathcal{H}\) is \(A\)-respectful, if for any \(H,H^{\prime}\subseteq T\), with \(H\cap A=H^{\prime}\cap A\), it follows that \(H\in\mathcal{H}\) iff \(H^{\prime}\in\mathcal{H}\)._  
Notice that, when \(\varphi\in\mathcal{L}_{A}\), then, for any \(T\subseteq At\), \(\left[\,\varphi\,\right]^{T}\) is \(A\)-respectful. **Lemma 4** (Lemma 13 from [20]). : _Let \(T,A\subseteq At\) be two sets of atoms and \(\mathcal{H},\mathcal{H}^{\prime}\) be a pair of \(T\)-supports. Then:_  
1. \((\mathcal{H}\cap\mathcal{H}^{\prime})_{A}\subseteq\mathcal{H}_{A}\cap \mathcal{H}^{\prime}_{A}\)_,_
2. _In addition, if_ \(\mathcal{H}\) _is_ \(A\)_-respectful, then_ \[(\mathcal{H}\cap\mathcal{H}^{\prime})_{A}=\mathcal{H}_{A}\cap\mathcal{H}^{ \prime}_{A}.\]  
We can define an order relation \(\preceq\) between \(T\)-supports by saying that, given two \(T\)-supports, \(\mathcal{H}\) and \(\mathcal{H}^{\prime}\), \(\mathcal{H}\preceq\mathcal{H}^{\prime}\) iff either \(\mathcal{H}=\left[\,\right]\) or \(\left[\,\right]\
eq\mathcal{H}^{\prime}\subseteq\mathcal{H}\). It is clear that \(\left[\,\right]\) and \(\left[T\right]\) are the bottom and top elements, respectively, in the class of all \(T\)-supports. Going back to Example 1, it is clear that \(\left[\,\
eg p\to q\,\right]^{T}\preceq\left[\,p\lor q\,\right]^{T}\), for any \(T\subseteq\{p,q\}\). Given a \(T\)-support \(\mathcal{H}\), we define its complementary support \(\overline{\mathcal{H}}\) as:  
\[\overline{\mathcal{H}}\stackrel{{\text{def}}}{{=}}\left\{ \begin{array}{ll}\left[\,\right]&\text{if }\mathcal{H}=2^{T}\\ \left[\,\,T\,\right]\cup\{H\subseteq T\mid H\
otin\mathcal{H}\}&\text{ otherwise}\end{array} \right.\]  
We also consider the _ideal_ of \(\mathcal{H}\):  
\[\downarrow\mathcal{H}=\{\mathcal{H}^{\prime}\mid\mathcal{H}^{\prime} \preceq\mathcal{H}\}\setminus\{\,\left[\,\right]\,\}.\]  
Note that the empty support \(\left[\,\right]\) is not included in the ideal, so \(\downarrow\left[\,\right]=\emptyset\). If \(\Delta\) is any set of supports:  
\[\downarrow\Delta\stackrel{{\text{def}}}{{=}}\bigcup_{\mathcal{H }\in\Delta}\downarrow\mathcal{H}=\{\,\mathcal{H}^{\prime}\mid\mathcal{H}^{ \prime}\preceq\mathcal{H},\mathcal{H}\in\Delta\,\}\setminus\{\,\left[\,\right]\,\}\]  
**Definition 17**. : _A \(T\)-view is a set of \(T\)-supports \(\Delta\subseteq\mathbf{H}_{T}\) that is \(\preceq\)-closed, i.e., \(\downarrow\Delta=\Delta\)._  
A _fork_ is defined using the grammar:  
\[F\ \ \ ::=\  
**Definition 21**.: _Let \(F\) and \(G\) be forks and \(A,B\subseteq At\) two sets. We say that \(F\) and \(G\) are strongly equivalent relative to \(A\) projected onto \(B\), in symbols \(F\equiv_{B}^{s,A}G\), if for any fork \(L\) in \(\mathcal{L}_{A}\), \(E(F\wedge L)|B=E(G\wedge L)|B\)._  
In [20], it was shown that, in case \(A=B\), we have:  
**Theorem 3**.: _Given \(F\) and \(G\) two forks and \(A\subseteq At\), the following holds:_  
\[F\sim_{A}^{s,A}G,\,\text{iff, for any }Y\subseteq A,\,\langle\!\langle\,F\, \rangle\!\rangle_{A}^{Y}\subseteq\langle\!\langle\,G\,\rangle\!\rangle_{A}^{Y}\]  
We recall from [20] that when \(F\) is a fork and \  
are contexts \(\mathcal{C}\) that express additional facts that hold at some times; perhaps this particular Web area allows different types of access in different periods. Then, there are credentials \(\mathcal{D}\) that are also (atomic) facts. Let us say that, in general, only faculty members are allowed to access the restricted area, so a credential might be \(faculty\_member([PERSON])\).