# Comparing the Reasoning Capabilities of Equilibrium Theories and Answer Set Programs
## 6 Projective Concepts
### Forks and Projective \(B\)-Entailment for Theories
Completing the picture, there are authorisations \(\mathcal{A}\), usually statements saying whether a subject can/cannot perform the operation on the object--for instance, whether [PERSON] can access the Web area. It may be 2- or 3-valued, depending on the context. Using a logic-based language such as ASP allows us to analyse in a straightforward manner different kinds of reasoning problems that may arise. As [PERSON] observes, the most basic problem is one of _entailment_. Is an authorisation \(\varphi\) granted by \(\Pi\) and \(\mathcal{C}\)? This is the case if \(\Pi\cup C\models\varphi\), where \(\models\) is a suitable nonmonotonic inference relation, such as the relation \(\models_{s}\) associated with stable model semantics. The second problem is in fact an _abductive_, _satisfaction_ problem. Roughly speaking, given an authorisation request, the problem is to deliver a set of conditions (credentials) that are sufficient to answer the authorisation positively, if such a set exists. Thus, given a set \(\mathcal{D}\) of digital credentials, the abduction problem is to find a subset \(\mathcal{D}^{\prime}\subseteq\mathcal{D}\) of credentials for a given authorisation \(\varphi\) and context \(C\), such that  
\[\Pi\cup\mathcal{D}^{\prime}\cup C\models_{s}\varphi\]  
A solution to the abduction problem can provide a suitable _explanation_. Suppose that [PERSON] has only recently joined the faculty but is not yet registered in the appropriate database. He is denied access with the explanation that a registering process is required first, i.e., he is informed of a missing credential that will grant him access. Third, there is the _conservative extension_ problem. Suppose that conditions have changed and now a new type of user may be admitted, e.g., some students can now access the area providing they belong to a specific committee. The program \(\Pi\) is enlarged to a new program, \(\Pi^{\prime}\), specifying the new conditions. The context and the set of credentials is also enlarged. However, we want to be sure that all authorisations that were valid previously continue to hold in the new situation and also that no loopholes in the system have been created that would allow unintended authorisations that were previously barred. In other words, the new program _conservatively_ extends the previous one. Fourth, there is a related problem of _relative strength_. We can say that in a given context \(\mathcal{C}\), a policy \(\Pi\) is at least as strong as \(\Pi^{\prime}\) if every authorisation request accepted by \(\Pi\) is also accepted by \(\Pi^{\prime}\). So, if \(\Pi^{\prime}\) rejects authorisation \(A\), then so does \(\Pi\). For simplicity, let us suppress contexts for the moment and consider a _policy framework_\(\mathcal{P}\) to be a triple \((\Pi,\mathcal{D},\mathcal{A})\), where \(\Pi\) is a theory, possibly in the form of a set of program rules in language \(\mathcal{L}\), \(\mathcal{D}\) is a set of credentials, comprising certain atomic sentences of \(\mathcal{L}\), and \(\mathcal{A}\) are authorisations. Let \(\mathcal{P}_{1}=(\Pi_{1},\mathcal{D},\mathcal{A})\), and \(\mathcal{P}_{2}=(\Pi_{2},\mathcal{D},\mathcal{A})\) be policy frameworks. Then, we can say that \(\mathcal{P}_{1}\) is at least as strong as \(\mathcal{P}_{2}\) if for any \(\varphi\in\mathcal{A}\), and \(D\subseteq\mathcal{D}\):  
\[\Pi_{2}\cup D\
ot\models_{s}A\Rightarrow\Pi_{1}\cup D\
ot\models_{s}A \tag{4}\]  
Fifth, there is the problem of policy _equivalence_ which may come in different degrees. Two policies that admit exactly the same authorisations and rejections in a given context can be said to be equivalent in that context. A stronger property is that they are equivalent in all contexts. Furthermore, a still stronger property is that they remain equivalent when they are extended by adding new policy rules.