Motivation
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 will succeed, because the program includes as a fact; the query will fail, because it does not occur in the head of any of the rules. The query will fail also, because the only rule with in the head contains the subgoal in its body; as we have seen, that subgoal fails. Finally, the query succeeds, because each of the subgoals, succeeds. (The latter succeeds because the corresponding positive goal fails.) To sum up, the behavior of SLDNF resolution on the given program can be represented by the following truth assignment:
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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
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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.
Read more about this topic: Stable Model Semantics
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