is satisfied by the sets of atoms that satisfy both p and q or none of them.

As usual, we say that X satisfies a theory Γ with aggregates if X satisfies all formulas in Γ. We extend the concept of classical equivalence to formulas/theories with aggregates.

We extend the definition of a stable models of propositional theories (Section 2)
to cover aggregates, in a very natural way. Let X be a set of atoms. The *reduct* F
X
of a formula F with aggregates relative to X is again the result of replacing each maximal formula not satisfied by X with ⊥. That is, it is sufficient to add a clause relative to aggregates to the recursive definition of a reduct: for an aggregate A of the form (6),

$$A^{X}={\begin{cases}o p\langle\{F_{1}^{X}=w_{1},\ldots,F_{n}^{X}=w_{n}\}\rangle\prec N,&{\text{if}X\models A,\\ \bot,&{\text{otherwise.}}\end{cases}}$$

This is similar to the clause for binary connectives:

$$(F\otimes G)^{X}={\begin{cases}F^{X}\otimes G^{X},&{\mathrm{if~}}X\models F\otimes G,\\ \bot,&{\mathrm{otherwise.}}\end{cases}}$$

The rest of the definition of a stable model remains the same: the *reduct* Γ
X of a theory Γ with aggregates is {F
X : F ∈ Γ}, and X is a *stable model* of Γ if X is a minimal model of Γ
X.

Consider, for instance, the theory Γ consisting of one formula

$$s u m\langle\{p=-1,q=1\}\rangle\geq0\to q.$$
$\eqref{eq:walpha}$
sumh{p = −1, q = 1**}i ≥** 0 → q. (8)
Set {q} is a stable model of Γ. Indeed, since both the antecedent and consequent of (8) are satisfied by {q}, Γ
{q}is

$$s u m\langle\{\bot=-1,q=1\}\rangle\geq0\to q.$$

The antecedent of the implication above is satisfied by every set of atoms, so the whole formula is equivalent to q. Consequently, {q} is the minimal model of Γ
{q},
and then a stable model of Γ.

## 3.2 Aggregates As Propositional Formulas

A formula/theory with aggregates can also be seen as a normal propositional formula/theory, by identifying (6) with the formula

$$I\subseteq\{1,...,n\}:op(\{w_{i}:i\in I\})\not\times N\quad i\in I\quad i\in T\tag{9}$$
$$15$$
