where WI stands for the multiset {wi: i ∈ I}, and G(I1,I2)stands for

$$\bigwedge_{i\in I_{1}}l_{i},\quad\bigwedge_{i\in\{1,\ldots,n\}\setminus I_{2}}\overline{{{l_{i}}}}.$$

For instance, for the PDB-aggregate A = sumh{p = −1, q = 1**}i ≥** 0, if we take F1 = p, F2 = q then the pairs (I1, I2) that "contribute" to the disjunction in Atr are

$(\emptyset,\emptyset)$$(\{2\},\{2\})$$(\{1,2\},\{1,2\})$$(\emptyset,\{2\})$$(\{2\},\{1,2\})$.  
The corresponding nested expressions G(I1,I2) are

$$\neg p\land\neg q\qquad q\land\neg p\qquad p\land q\qquad\neg p\qquad q.$$

It can be shown, using strong equivalent transformations (see Proposition 5) that the disjunction of such nested expressions can be rewritten as ¬p ∨ q.

In case of monotone and antimonotone PDB-aggregates and in the absence of negation as failure, the semantics of Pelov *et al.* is equivalent to ours.

Proposition 18. For any monotone or antimonotone PDB-aggregates A *of the* form (6) where F1, . . . , Fn are atoms, Atr *is strongly equivalent to (9).*
The claim above is generally not true when either the aggregates are not monotone or antimonotone, or when some formula in the aggregate is a negative literal. Relatively to aggregates that are neither monotone nor antimonotone, the semantics of [Pelov *et al.*, 2003] seems to have the same unintuitive behaviour of [Niemel¨a and Simons, 2000]: for instance, according to [Pelov *et al.*, 2003], (27)
has no stable models while p ← sumh{p = 1**}i ≥** 0 has stable model {p}.

To illustrate the problem with negative literals, consider the following Π:

$$\begin{array}{l}{{p\gets s u m\langle\{q=1\}\rangle<1}}\\ {{q\gets\neg p}}\end{array}$$
and Π′:
$$p\gets sum\langle\{\neg p=1\}\rangle<1\tag{30}$$  $q\gets\neg p$
Intuitively, the two programs should have the same stable models. Indeed, the operation of replacing q with ¬p in the first rule of Π should not affect the stable

$$(29)$$
