where I stands for {1, . . . , n} \ I, and 6≺ is the negation of ≺.

For instance, if we consider aggregate (7), the conjunctive terms in (9) correspond to the cases when the sum of weights is 1, that is, when I = {1} and I = {2}. The two implications are q → p and p → q respectively, so that (7) is

$$(q\to p)\land(p\to q).$$
$$(10)$$
$$s u m\langle\{p=1,q=1\}\rangle=1$$
$$(11)$$
$$(p\lor q)\land\lnot(p\land q).$$
$\left(12\right)^{2}$
(q → p) ∧ (p → q). (10)
Similarly,
sumh{p = 1, q = 1}i = 1 (11)
is
(p ∨ q) ∧ ¬(p ∧ q). (12)
Even though (11) can be seen as the negation of (7), the negation of (12) is not strongly equivalent to (10) (although they are classically equivalent). This shows that it is generally incorrect to "move" a negation from a binary relation symbol
(such as 6=) in front of the aggregate as the unary connective ¬, and vice versa.

Next proposition shows that this understanding of aggregates as propositional formulas is equivalent to the semantics for theories with aggregates of the previous section. Two formulas with aggregates are *classically equivalent* to each other if they are satisfied by the same sets of atoms.

Proposition 12. Let A be an aggregate of the form (6) and let G *be the corresponding formula (9). Then*
(a) G is classically equivalent to A, and

$$r\;a n y\;s e t\;X\;o f\;a t o m s,\;G^{X}\;i s\;\epsilon$$

(b) for any set X of atoms, GX *is classically equivalent to* AX.

Treating aggregates as propositional formulas allows us to apply many properties of propositional theories presented in Section 2.4 to theories with aggregates also. Consequently, we have the concept of an head atom, of strong equivalence, we can use the completion lemma and so on. We will use several of those properties to prove Proposition 14 below. In the rest of the paper we will often make no distinctions between the two ways of defining the semantics of aggregates discussed here.

Notice that replacing, in a theory, an aggregate of the form (6) with a formula that is not strongly equivalent to the corresponding formula (9) may lead to different stable models. This shows that there is no other way (modulo strong equivalence) of representing our aggregates as propositional formulas.

$$\mathbf{\partial}\mathbf{\partial}A^{X}.$$
