Answer Set Programming (ASP) is a popular framework for modelling combinatorial problems. However, ASP cannot be used easily for reasoning about uncertain information. Possibilistic ASP (PASP) is an extension of ASP that combines possibilistic logic and ASP. In PASP a weight is associated with each rule, whereas this weight is interpreted as the certainty with which the conclusion can be established when the body is known to hold. As such, it allows us to model and reason about uncertain information in an intuitive way. In this paper we present new semantics for PASP in which rules are interpreted as constraints on possibility distributions. Special models of these constraints are then identified as possibilistic answer sets. In addition, since ASP is a special case of PASP in which all the rules are entirely certain, we obtain a new characterization of ASP in terms of constraints on possibility distributions. This allows us to uncover a new form of disjunction, called weak disjunction, that has not been previously considered in the literature. In addition to introducing and motivating the semantics of weak disjunction, we also pinpoint its computational complexity. In particular, while the complexity of most reasoning tasks coincides with standard disjunctive ASP, we find that brave reasoning for programs with weak disjunctions is easier.
Not all existing extensions of disjunction allow non-minimal models. For example, in @cite_32 an extension of disjunctive logic programs is presented which adds the idea of inheritance. Conflicts between rules are then resolved in favor of more specific rules. Such an approach allows for an intuitive way to deal with default reasoning and exceptions. In particular, the semantics allow for rules to be marked as being defeasible and allows to specify an order or inheritance tree among (sets of) rules. Interestingly, the complexity of the resulting system is not affected and coincides with the complexity of ordinary disjunctive programs.
@cite_32
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The paper proposes a new knowledge representation language, called DLP<, which extends disjunctive logic programming (with strong negation) by inheritance. The addition of inheritance enhances the knowledge modeling features of the language providing a natural representation of default reasoning with exceptions. A declarative model-theoretic semantics of DLP< is provided, which is shown to generalize the Answer Set Semantics of disjunctive logic programs. The knowledge modeling features of the language are illustrated by encoding classical nonmonotonic problems in DLP<. The complexity of DLP< is analyzed, proving that inheritance does not cause any computational overhead, as reasoning in DLP< has exactly the same complexity as reasoning in disjunctive logic programming. This is confirmed by the existence of an efficient translation from DLP< to plain disjunctive logic programming. Using this translation, an advanced KR system supporting the DLP< language has been implemented on top of the DLV system and has subsequently been integrated into DLV.