models since the second rule "defines" q as ¬p: it is the only rule with q in the head.

However, under the semantics of [Pelov *et al.*, 2003], Π has stable model {p} only and Π′ has stable model {q} also. Under our semantics, both (29) and (30) have stable models {p} and {q}.

Note that already the first rule of (30) has different stable models under the two semantics. Under ours, they are ∅ and {p}. According to [Pelov *et al.*, 2003], only the empty set is a stable model; it couldn't have both stable models because stable models as defined in [Pelov *et al.*, 2003] have the anti-chain property.

## 3.6.3 Flp-Aggregates

An *FLP-aggregate* is an expression of the form (6) where each of F1, . . . , Fn is a conjunction of literals. A *program with FLP-aggregates* is a set of rules of the form a1 ∨ · · · ∨ an ← A1 ∧ · · · ∧ Am ∧ ¬Am+1 **∧ · · · ∧ ¬**Ap (31)
where n ≥ 0, 0 ≤ m ≤ p, a1, . . . , an are atoms and A1, . . . , Ap are FLPaggregates.

A program with FLP-aggregates is a generalization of a disjunctive program, by identifying each atom a in the bodies of disjunctive rules by aggregate sumh{a =
1**}i ≥** 1.

The semantics of [Faber *et al.*, 2004] defines when a set of atoms is a stable model for a program with FLP-aggregates. The definition of satisfaction of an aggregate is identical to ours. The reduct, however, is computed differently. The reduct Π
X of a program Π with FLP-aggregates relative to a set X of atoms consists of the rules of the form (31) such that X satisfies its body. Set X is a stable model for Π if X is a minimal set satisfying ΠX.

For instance, let Π be the FLP-program p ← sumh{p = 2**}i ≥** 1.

The only stable model of Π is the empty set. Indeed, since the empty set doesn't satisfy the aggregate, Π
∅ = ∅, which has ∅ as the unique minimal model; we can conclude that ∅ is a stable model of Π. On the other hand, Π
{p}= Π because {p}
satisfies the aggregate in Π. Since ∅ |= Π, {p} is not a minimal model of Π
{p}and then it is not a stable model of Π.

This definition of a reduct is different from all other definitions of a reduct described in this paper (and also from many other definitions), in the sense that it may leave negation ¬ in the body of a rule. For instance, the reduct of a ← ¬b relative to {a} is according to those definitions the fact a. In the theory of FLPaggregates, the reduct doesn't modify the rule. On the other hand, this definition