condition is falsified and h**I, J**i is a model of r. Thus, if S is the set of SE-models of a rule r, then every atom p ∈ B(r)
− satisfies

$$s\ \langle I,J\rangle\in{\mathcal{S}}$$
$$(C_{B^{-}})$$

$$p\in J$$

p ∈ J implies hI, J**i ∈ S** (CB− )
and every atom p ∈ H(r)
+ satisfies

$$p\in I\ \mathrm{implies}\ \langle I,J\rangle\in{\mathcal{S}}\ .$$
$$(C_{H^{+}})$$
p ∈ I implies hI, J**i ∈ S** . (CH+ )
If we restrict ourselves to canonical rules different from ε, we find that these conditions are not only necessary, but, when combined properly, also sufficient to decide what atoms belong to the negative body and positive head of the rule.

For the rest of this Subsection, we assume that r is a canonical rule different from ε and S is the set of SE-models of r. Keeping in mind that every atom that satisfies condition (CB− ) also satisfies condition (CH+ ) (because I is a subset of J), and that B(r)
− is by definition disjoint from H(r)
+, we arrive at the following results:
Lemma 9. An atom p *belongs to* B(r)
− *if and only if for all*hI, J**i ∈ I**SE*, the condition*
(CB− ) is satisfied. An atom p *belongs to* H(r)
+ *if and only if it does not belong to* B(r)
− *and for all* hI, J**i ∈ I**SE, the condition (CH+ ) *is satisfied.*
Proof. See Appendix A, page 19. ⊓⊔
As can be seen from Lemma 8, the role of positive body and negative head atoms is dual to that of negative body and positive head atoms. Intuitively, their absence in J, and sometimes also in I, implies that h**I, J**i is an SE-model of r. It follows from the first condition of Lemma 8 that if p belongs to H(r)
− ∪ B(r)
+, then the following condition is satisfied:
p /∈ J implies hI, J**i ∈ S** . (CH− )
Furthermore, the second condition in Lemma 8 implies that every p ∈ B(r)
+ satisfies the following condition:

$$[,J)\in{\mathcal{S}}$$
$$\mathbf{\hat{\Sigma}}_{+}$$

p /∈ I and J ∩ H(r)
+ 6= ∅ implies hI, J**i ∈ S** . (CB+ )
These observations lead to the following results:
Lemma 10. An atom p *belongs to* B(r)
+ *if and only if for all* hI, J**i ∈ I**SE, the conditions (CH− ) and (CB+ ) are satisfied. An atom p *belongs to* H(r)
− if and only if it does not belong to B(r)
+ *and for all* hI, J**i ∈ I**SE, the condition (CH− ) is satisfied.

Proof. See Appendix A, page 21. ⊓⊔
Together, the two Lemmas above are sufficient to reconstruct a canonical rule from its set of SE-models. The following definition sums up these results by introducing the notion of a rule induced by a set of SE-interpretations: