'Classical' Negation in Nonmonotonic Reasoning and Logic Programming

Gelfond and Lifschitz were the first to point out the need for a symmetric negation in logic programming and they also proposed a specific semantics for such negation for logic programs with the stable semantics, which they called 'classical'. Subsequently, several researchers proposed different, often incompatible, forms of symmetric negation for various semantics of logic programs and deductive databases. To the best of our knowledge, however, no systematic study of symmetric negation in non-monotonic reasoning was ever attempted in the past. In this paper we conduct such a systematic study of symmetric negation. We introduce and discuss two natural, yet different, definitions of symmetric negation: one is called strong negation and the other is called explicit negation. For logic programs with the stable semantics, both symmetric negations coincide with Gelfond–Lifschitz' 'classical negation'. We study properties of strong and explicit negation and their mutual relationship as well as their relationship to default negation 'not', and classical negation '¬'. We show how one can use symmetric negation to provide natural solutions to various knowledge representation problems, such as theory and interpretation update, and belief revision. Rather than to limit our discussion to some narrow class of nonmonotonic theories, such as the class of logic programs with some specific semantics, we conduct our study so that it is applicable to a broad class of non-monotonic formalisms. In order to achieve the desired level of generality, we define the notion of symmetric negation in the knowledge representation framework of AutoEpistemic logic of Beliefs, introduced by Przymusinski.