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Stable model semantics

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, 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: {| cellpadding=5 style="width:18em" |p |q |r

s
T
F
F
T.
}
On the other hand, the rules of the given program can be viewed as propositional formulas if we identify the comma with conjunction , the symbol with negation , and agree to treat as the implication 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
{
p
q
r
s
-
T
T
T
F.
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Negation as failure
Negation as failure (NAF, for short) is a non-monotonic inference rule in logic programming, used to derive (i.e. that is assumed not to hold) from failure to derive . Note that can be different from the statement of the logical negation of , depending on the completeness of the inference algorithm and thus also on the formal logic system. Negation as failure has been an important feature of logic programming since the earliest days of both Planner and Prolog. In Prolog, it is usually implemented using Prolog's extralogical constructs.
Closed-world assumption
The closed-world assumption (CWA), in a formal system of logic used for knowledge representation, is the presumption that a statement that is true is also known to be true. Therefore, conversely, what is not currently known to be true, is false. The same name also refers to a logical formalization of this assumption by Raymond Reiter. The opposite of the closed-world assumption is the open-world assumption (OWA), stating that lack of knowledge does not imply falsity. Decisions on CWA vs.
Autoepistemic logic
The autoepistemic logic is a formal logic for the representation and reasoning of knowledge about knowledge. While propositional logic can only express facts, autoepistemic logic can express knowledge and lack of knowledge about facts. The stable model semantics, which is used to give a semantics to logic programming with negation as failure, can be seen as a simplified form of autoepistemic logic. The syntax of autoepistemic logic extends that of propositional logic by a modal operator indicating knowledge: if is a formula, indicates that is known.
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