this paper, we prefer showing a translational, equivalent semantics of such programs from [Ferraris and Lifschitz, 2005b], that consists in replacing each weight constraint C with a nested expression [C], preserving the stable models of the program: if C is (21) then [C] is (I ⊆ {1, . . . , n})

$$\bigvee_{I:N\leq\sum_{i\in I}w_{i}}(\bigwedge_{i\in I}l_{i})$$
$$(24)$$
$$(25)$$
$$(22)$$

and if C is (22) then [C] is

$[C]$ is . 
$I:N$$<$$\sum_{i\in I}w_{i}$$i$$\in$$I$
It turns out that the way of understanding a weight constraint C of this paper is not different from [C] when all weights are nonnegative. Proposition 17. In presence of nonnegative weights only, [N ≤ S] *is strongly* equivalent to sumhSi ≥ N, and [S ≤ N] *is strongly equivalent to* sumhSi ≤ N.

From this proposition, Propositions 2 and 5 of this paper, and Theorem 1 from [Ferraris and Lifschitz, 2005b] it follows that our concept of an aggregate captures the concept of weight constraints defined in [Niemel¨a and Simons, 2000]
when all weights are nonnegative. It also captures the absence of the anti-chain property of its stable models: for instance,

$$p\leftarrow\{\neg p\}\leq0$$

has stable models ∅ and {p} in both formalisms.

When we consider negative weights, however, such correspondence doesn't hold. For instance, p ← 0 ≤ {p = 2, p = −1}, (26)
according to [Niemel¨a and Simons, 2000], has no stable models, while

$$p\leftarrow0\leq\{p=2,p=-1\},$$
$$p\leftarrow\,s u m\langle\{p=2,p=-1\}\rangle\geq0$$
p ← sumh{p = 2, p = −1**}i ≥** 0 (27)
has stable model ∅. An explanation of this difference can be seen in the preprocessing proposed by [Niemel¨a and Simons, 2000] that eliminates negative weights.

For us, weight constraint 0 ≤ {p = 2, p = −1}, and the result 1 ≤ {p = 2,¬p =
1} of eliminating its negative weight, are semantically different.12 Surprisingly, 12The fact that the process of eliminating negative weights is somehow unintuitive was already mentioned in [Ferraris and Lifschitz, 2005b] with the same example proposed in this section.

$$(27)$$
