# Comparing the Reasoning Capabilities of Equilibrium Theories and Answer Set Programs
## 5 Entailment Relations  
In [19], various nonmonotonic entailment relations are defined in terms of (equilibrium) models. In particular:  
**Definition 8** (strong and uniform entailment). : \(\Pi_{1}\) _strongly entails \(\Pi_{2}\), in symbols \(\Pi_{1}\models_{S}\Pi_{2}\), (respective \(\Pi_{1}\) uniformly entails \(\Pi_{2}\), in symbols \(\Pi_{1}\models_{U}\Pi_{2}\)) if for any set \(\Gamma\) of formulas (respective atoms)_  
\[E(\Pi_{1}\cup\Gamma)\subseteq E(\Pi_{2}\cup\Gamma) \tag{1}\]  
We can also consider entailment relations attuned to query answering. **Definition 9** (weak theory entailment). : _Let \(\Pi_{1}\) and \(\Pi_{2}\) be theories and \(\omega\in\{c,s,p\}\). Then, we write \(\Pi_{1}\mathbin{\vDash}_{\omega}\Pi_{2}\) if for any formula \(\varphi\), if \(\Pi_{2}\models_{\omega}\varphi\) then, \(\Pi_{1}\models_{\omega}\varphi\)._  
As in the case of Definition 2, we can compare theories for their deductive strength. **Definition 10** (strong and uniform theory entailment). : _Let \(\Pi_{1}\) and \(\Pi_{2}\) be theories and \(\omega\in\{c,s,p\}\). Then,_  
* _we write_ \(\Pi_{1}\mathbin{\vDash}_{\omega}^{s}\Pi_{2}\) _if for any further theory_ \(\Pi\)__\((\Pi_{1}\cup\Pi)\mathbin{\vDash}_{\omega}(\Pi_{2}\cup\Pi)\)_;_
* _we write_ \(\Pi_{1}\mathbin{\vDash}_{\omega}^{\vec{u}}\Pi_{2}\) _if for any set_ \(X\) _of atoms,_ \((\Pi_{1}\cup X)\mathbin{\vDash}_{\omega}(\Pi_{2}\cup X)\)_._  
**Proposition 6**. : _For any theories \(\Pi\), \(\Pi_{2}\):_  
1. \(\Pi_{1}\mathbin{\vDash}_{s}^{s}\Pi_{2}\Leftrightarrow\Pi_{1}\models_{S}\Pi_{2}\)__
2. \(\Pi_{1}\mathbin{\vDash}_{s}^{u}\Pi_{2}\Leftrightarrow\Pi_{1}\models_{U}\Pi_{2}\)__  
Proof. : (1). The implication from right to left holds by inspection. For the other direction, suppose that \(\Pi_{1}\
ot\models_{S}\Pi_{2}\). Then, for some \(\Gamma\), \(E(\Pi_{1}\cup\Gamma)\
ot\subseteq E(\Pi_{2}\cup\Gamma)\). Let \(M\in E(\Pi_{1}\cup\Gamma)\setminus E(\Pi_{2}\cup\Gamma)\). Apply Lemma 1(ii) to conclude that there is a sentence \(\psi\) that holds in each equilibrium model of \(\Pi_{2}\cup\Gamma\) and so \(\Pi_{2}\cup\Gamma\models_{s}\psi\). Inspection \(M\
ot\models\psi\), so \(\Pi_{1}\cup\Gamma\
ot\models_{s}\psi\), and hence \(\Pi_{1}\mathbin{\vDash}_{s}^{s}\Pi_{2}\). For (2), the proof is entirely analogous. The relation \(\Pi_{1}\models_{S}\Pi_{2}\) has been characterised in terms of **HT**-models. **Proposition 7** ([19]). : \(\Pi_{1}\models_{S}\Pi_{2}\) _holds iff the following two conditions are satisfied:_  
1. \(\Pi_{1}\) _classically entails_ \(\Pi_{2}\)_._
2. _For any model_ \(\langle H,T\rangle\) _of_ \(\Pi_{2}\) _such that_ \(\langle T,T\rangle\models\Pi_{1}\)_,_ \(\langle H,T\rangle\models\Pi_{1}\)_._  
Although the (strong) equivalence concepts are captured in the monotonic logic **HT**, the same is not true for strong entailment, because \(\Pi_{1}\models\Pi_{2}\) does not imply \(\Pi_{1}\models_{S}\Pi_{2}\). However, it is clear that  
\[\Pi_{1}\models_{S}\Pi_{2}\And\Pi_{2}\models_{S}\Pi_{1}\Rightarrow\Pi_{1} \equiv_{S}^{s}\Pi_{2} \tag{2}\]  
By Propositions 1 and 2, we can see that \(\Pi_{1}\equiv_{S}^{s}\Pi_{2}\) if and only if \(\Pi_{1}\) and \(\Pi_{2}\) are strongly equivalent theories. In turn, by the well-known characterisation [2], this means that \(\Pi_{1}\) and \(\Pi_{2}\) are equivalent in **HT**. It follows that, whereas \(\Pi_{1}\models\Pi_{2}\) does not entail \(\Pi_{1}\models_{S}\Pi_{2}\), it is easy to see that \(\Pi_{1}\equiv_{HT}\Pi_{2}\) implies both \(\Pi_{1}\models_{S}\Pi_{2}\) and \(\Pi_{2}\models_{S}\Pi_{1}\). Although the strong equivalence concepts for credulous, sceptical, and prudent reasoning all agree, in the case of strong entailment, credulous and sceptical reasoning behave in a kind of dual form. **Proposition 8**. : \(\Pi_{1}\mathbin{\vDash}_{s}^{s}\Pi_{2}\Leftrightarrow\Pi_{2}\mathbin{\vDash}_{c}^{s }\Pi_{1}\)_._Proof. : From Proposition 6, we established that \(\Pi_{1}\vdash_{s}^{s}\Pi_{2}\) holds if and only if (1) is true. Clearly, if (1) holds then \(\Pi_{2}\vdash_{c}^{s}\Pi_{1}\) by inspection.