For instance, if X contains p but not q then

$$\begin{array}{c}{{(p\leftarrow\neg q)^{X}=(p\leftarrow(q\to\bot))^{X}=p\leftarrow(\bot\to\bot)=p\leftarrow\top}}\\ {{(q\leftarrow\neg p)^{X}=(q\leftarrow(p\to\bot))^{X}=\bot\leftarrow\bot}}\\ {{((p\to q)\vee(q\to p))^{X}=\bot\vee(\bot\to p)}}\end{array}$$
$$\left(2\right)$$

The *reduct* Γ
X of a propositional theory Γ relative to X is {F

 $\circ X\text{is}\{F^X:F\in\Gamma\}\\ \circ\circ\circ\circ\circ\circ=F^X$
set X of atoms is a *stable model* of Γ if X is a minimal set satisfying Γ
X.

For instance, let Γ be the theory consisting of

$$p\leftarrow\neg q$$
$$q\leftarrow\neg p$$
$$({\boldsymbol{3}})$$

Theory Γ is actually a traditional program, a logic program in the sense of [Gelfond and Lifschitz, 1988]
(more details in the next section). Set {p} is a stable model of Γ; indeed, by looking at the first two lines of (2) we can see that Γ
{p}is {p ← ⊤, **⊥ ← ⊥}**, which is satisfied by {p} but not by its unique proper subset ∅. It is easy to verify that {q}
is the only other stable model of Γ. Similarly, it is not difficult to see that {p} is the only stable model of the theory

$$(p\to q)\lor(q\to p)$$
p
$$\left(4\right)$$

(The reduct relative to {p} is {⊥ ∨ (⊥ → p), p}).

As the name suggests, a stable model of a propositional theory Γ is a model —
in the sense of classical logic - of Γ. Indeed, it follows from the easily verifiable fact that, for each set X of atoms, X |= ΓX iff X |= Γ. On the other hand, formulas that are equivalent in classical logic may have different stable models: for instance,
{¬¬p} has no stable models, while {p} has stable model {p}. Proposition 5 below will give some characterizations of transformations that preserves stable models.

Notice that classically equivalent transformations can be applied to the reduct of a theory, as the sets of atoms that are minimal don't change.

Finally, a note about a second kind of negation in propositional theories. In
[Ferraris and Lifschitz, 2005a, Section 3.9], atoms were divided into two groups:
"positive" and "negative", so that each negative atom has the form ∼a, where a is a positive atom. Symbol ∼ is called "strong negation", to distinguish it from the connective ¬, which is called *negation as failure*.

6In presence of strong negation, 7