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.
Possibilistic logic is a well-known logic for reasoning under uncertainty, which is based on the idea that the epistemic state of an agent can be modeled by assigning to each possible world a degree of possibility, taken from a totally ordered, but essentially qualitative scale. Recently, a generalization has been proposed that extends possibilistic logic to a meta-epistemic logic, endowing it with the capability of reasoning about epistemic states, rather than merely constraining them. In this paper, we further develop this generalized possibilistic logic (GPL). We introduce an axiomatization showing that GPL is a fragment of a graded version of the modal logic KD, and we prove soundness and completeness w.r.t. a semantics in terms of possibility distributions. Next, we reveal a close link between the well-known stable model semantics for logic programming and the notion of minimally specific models in GPL. More generally, we analyze the relationship between the equilibrium logic of Pearce and GPL, showing that GPL can essentially be seen as a generalization of equilibrium logic, although its notion of minimal specificity is slightly more demanding than the notion of minimality underlying equilibrium logic. Abstract This note points out the close relationships existing between recent proposals in the theory of belief revision made by Gardenfors based on the notion of epistemic entrenchment, and possibility theory applied to automated reasoning under uncertainty. It is claimed that the only numerical counterparts of epistemic entrenchment relations are so-called necessity measures that are dual to possibility measures, and are also mathematically equivalent to consonant belief functions in the sense of Shafer. Relationships between Spohn's ordinal conditional functions and possibility theory are also laid bare.