The concept of a stable model, or answer set, is used to define a declarative semantics for logic programs with negation as failure. This is one of several standard approaches to the meaning of negation in logic programming, along with program completion and the well-founded semantics. The stable model semantics is the basis of answer set programming.

Research on the declarative semantics of negation in logic programming was motivated by the fact that the behavior of SLDNF resolution—the generalization of SLD resolution used by Prolog in the presence of negation in the bodies of rules—does not fully match the truth tables familiar from classical propositional logic. Consider, for instance, the program

Given this program, the query p will succeed, because the program includes p as a fact; the query q will fail, because it does not occur in the head of any of the rules. The query r will fail also, because the only rule with r in the head contains the subgoal q in its body; as we have seen, that subgoal fails. Finally, the query s succeeds, because each of the subgoals p, ${\displaystyle \operatorname {not} q}$ succeeds. (The latter succeeds because the corresponding positive goal q fails.) To sum up, the behavior of SLDNF resolution on the given program can be represented by the following truth assignment:

On the other hand, the rules of the given program can be viewed as propositional formulas if we identify the comma with conjunction ${\displaystyle \land }$, the symbol ${\displaystyle \operatorname {not} }$ with negation ${\displaystyle \neg }$, and agree to treat ${\displaystyle F\leftarrow G}$ as the implication ${\displaystyle G\rightarrow F}$ written backwards. For instance, the last rule of the given program is, from this point of view, alternative notation for the propositional formula

If we calculate the truth values of the rules of the program for the truth assignment shown above then we will see that each rule gets the value T. In other words, that assignment is a model of the program. But this program has also other models, for instance

Thus one of the models of the given program is special in the sense that it correctly represents the behavior of SLDNF resolution. What are the mathematical properties of that model that make it special? An answer to this question is provided by the definition of a stable model.

The meaning of negation in logic programs is closely related to two theories of nonmonotonic reasoning—autoepistemic logic and default logic. The discovery of these relationships was a key step towards the invention of the stable model semantics.

The syntax of autoepistemic logic uses a modal operator that allows us to distinguish between what is true and what is known. Michael Gelfond [1987] proposed to read ${\displaystyle \operatorname {not} p}$ in the body of a rule as "${\displaystyle p}$ is not known", and to understand a rule with negation as the corresponding formula of autoepistemic logic. The stable model semantics, in its basic form, can be viewed as a reformulation of this idea that avoids explicit references to autoepistemic logic.