# Comparing the Reasoning Capabilities of Equilibrium Theories and Answer Set Programs
## 6 Projective Concepts
### Forks and Projective \(B\)-Entailment for Theories  
We have seen that there is good agreement between equivalence and entailment concepts defined in terms of equilibrium models and their analogous counterparts couched in terms of consequence or query answering. As expected, in the case of (projective) strong entailment, the agreement is with the sceptical version of strong \(B\)-consequence. For the standard model-theoretic concepts, many characterisation results are known (see Section 8). In the case of projective entailment and equivalence, the main results are those of [3], which apply to disjunctive logic programs (see also the recent work [28]). For the remainder of the article, we consider projective concepts for programs and propositional theories and also make use of the concept of _fork_. In [20], the language of logic programs was extended to include a new construct'', called _fork_, whose intuitive meaning is that the stable models of \(\Pi_{1}\mid\Pi_{2}\) correspond to the union of stable models from \(\Pi_{1}\) and \(\Pi_{2}\) in any context \(\Pi^{\prime}\), that is \(SM[(\Pi_{1}\mid\Pi_{2})\wedge\Pi^{\prime}]=SM[\Pi_{1}\wedge\Pi^{\prime}]\cup SM [\Pi_{2}\wedge\Pi^{\prime}]\). (Remark: \(SM[\Pi]\) denotes the collection of stable models of \(\Pi\). Because a program (or theory) is in our case finite, we can also regard it as a conjunction of its formulas. This allows us to write expressions such as \(\Pi\wedge\Gamma\) or \(\Pi\mid\Gamma\) with the obvious meaning). Using the construct of fork, [20] studied the property of projective strong equivalence (PSE) for forks: two forks satisfy PSE for a vocabulary \(V\) iff they yield the same stable models projected on \(V\) for any context over \(V\). This property corresponds to the one defined in Definition 13 for the case \(A=B\). [20] also provides a semantic characterisation of PSE that allows one to prove that it is always possible to forget (under strong persistence) an auxiliary atom in a fork--something shown to be false in standard **HT**. Now, we recall some definitions from [19; 20]. **Definition 15**. : _Given \(T\subseteq At\), a \(T\)-support \(\mathcal{H}\) is a set of subsets of \(T\), that is \(\mathcal{H}\subseteq 2^{T}\), satisfying \(\mathcal{H}\
eq\emptyset\) iff \(T\in\mathcal{H}\)._  
To increase the readability of examples, we can write a support as a sequence of interpretations between square brackets. For instance, possible supports for \(T=\{a,b\}\) are \([\{a,b\}\ \{a\}]\), \([\{a,b\}\ \{b\}\ \emptyset]\) or the empty support \([\,]\). It is well-known that, given a propositional formula \(\varphi\), the set:  
\[\{H\subseteq T\,\mid\,\langle H,T\rangle\models\varphi\}\]  
is always a \(T\)-support denoted by \([\![\,\varphi\,]\!]^{T}\). Moreover, in [3], it was shown that \(\mathcal{H}\subseteq 2^{T}\) is a \(T\)-support iff there exists a propositional formula \(\varphi\) such that \(\mathcal{H}=[\![\,\varphi\,]\!]^{T}\). **Example 1**. : \[\left[\
eg p\to q\right]^{T}=\left\{ \begin{array}{ll}\left[\,\right]&\text{if }T= \emptyset\\ \left[\,\emptyset,\{p\}\,\right]&\text{if }T=\{p\}\\ \left[\,\{q\}\,\right]&\text{if }T=\{q\}\\ 2^{T}&\text{if }T=\{p,q\}\end{array} \right.\quad\left[\,\left[\,p\lor q\,\right]^{T}= \left\{ \begin{array}{ll}\left[\,\right]&\text{if }T=\emptyset\\ \left[\,\{p\}\,\right]&\text{if }T=\{p\}\\ \left[\,\{q\}\,\right]&\text{if }T=\{q\}\\ \left[\,\{p\},\{q\},\{p,q\}\,\right]&\text{if }T=\{p,q\}\end{array} \right.\]  
If \(A\subseteq At\), we say that a \(T\)-support \(\mathcal{H}\) is \(A\)_-feasible_ iff there is no \(H\subset T\) in \(\mathcal{H}\) satisfying \(H\cap A=T\cap A\). (Remark: If \(\mathcal{H}=\left[\,\varphi\,\right]^{T}\), for some \(\varphi\) and \(T\subseteq V\), suppose that \(\mathcal{H}\) is V-unfeasible. Then, there exists \(H\subset T\) with \(H\cap V=T\cap V=T\) such that \(H\in\mathcal{H}\). In this case, \(T\) would never be a stable model of \(\varphi\wedge\lambda\) if \(\lambda\in\mathcal{L}(V)\).