Observe that by means of the autonomous fixpoints of a given SOLP **program**
P **we represent the mental states of the corresponding agent, assuming that every** social condition in P **is not taken into account.** Example 7 Consider the following SOLP program P:
okay(a) ← b, [1, ]{c}
b ← [2, 4]{d}
It is easy to see that AFP(P) = {{b}, {a, b}}**, i.e. the interpretations** I1 = {b}
and I2 = {a, b} are the autonomous fixpoints of P, since it holds that ATP (I1) = I1 and ATP (I2) = I2.

## Definition 6

Given a SOLP collection C = {P1, · · · ,Pn}, let Pi (1 ≤ i ≤ n**) be a SOLP program**
of C and L **be a set of atoms. The** labelled version of L w.r.t. Pi**, denoted by (**L)Pi is the set {aPi| a ∈ L}. Each element of (L)Pi**is referred to as a** labelled atom w.r.t. Pi.

## Example 8

Given a SOLP program P1 of a SOLP collection C, if L = {a, b, c}**, then (**L)P1 = {aP1
, bP1
, cP1 }, where the program identifier P1 **indicates the associated SOLP**
program.

Now we introduce the concept of social interpretation**, devoted to representing**
the mental states of the collectivity described by a given SOLP collection and then we give the definition of truth for both literals and SCs w.r.t. a given social interpretation. To this aim, the classical notion of interpretation is **extended by** means of program identifiers introducing a link between atoms of the **interpretation** and programs of the SOLP collection.

## Definition 7

Let C = {P1, · · · ,Pn} **be a SOLP collection. A** social interpretation for C **is a set**
¯I = (I
1)P1 **∪ · · · ∪** (I
n)Pn**, where** I
jis an interpretation for Pj (1 ≤ j ≤ n**) and**
(I
j)Pj**is the labelled version of** I
j w.r.t. Pj **(see Definition 6).**

## Example 9

Given C = {P1,P2,P3}, I
1 = {a, b, c}, I
2 = {a, d, e} and I
3 = {b, c, d}**, where**
I
jis an interpretation for Pj (1 ≤ j ≤ **3), then** ¯I = {aP1
, bP1
, cP1
, aP2
, dP2
,
eP2
, bP3
, cP3
, dP3 } **is a social interpretation for** C.

We define now the notion of truth for literals, SCs and social rules, respectively.

Let C = {P1, · · · ,Pn} be a SOLP collection. Given a social interpretation ¯I for C **and a positive literal** a ∈SP∈C Var (P), a **(resp.** not a**) is** true for Pj (1 ≤ j ≤ n)
w.r.t. ¯I if aPj ∈ ¯I **(resp.** aPj ∈/
¯I**); otherwise it is** false.

Because of the recursive nature of SCs, before giving the definition of truth for a SC s, we introduce a way to identify s (and also every SC nested in s) occurring in