KR Conference 2012 Conference Paper
- Didier Dubois
- Henri Prade
- Steven Schockaert
An important aspect of possibilistic logic is that models correspond to epistemic states, rather than to propositional interpretations, which forms a natural basis for epistemic reasoning. However, possibilistic logic only takes sets of formulas of the form (α, λ) into account, which we could interpret as conjunctions of assertions of the form N (α) ≥ λ. In some applications, on the other hand, we may want to link such assertions using different propositional connectives. In logic programming, for instance, a (negation-free) rule such as α → β intuitively means that whenever α is known to be true, we should accept β to be true as well. This could be expressed using necessity measures and material implication as N (α) ≥ 1 ⇒ N (β) ≥ 1. This implication, however, cannot be expressed in possibilistic logic, an observation which stands in stark contrast to the expressivity of modal logics for epistemic reasoning. In (Banerjee and Dubois 2009), a so-called Meta-Epistemic Logic (MEL) was introduced as a first step to bridge this gap, in the form of a simple modal logic with a semantics in terms of Boolean possibility distributions (i. e. possibility distributions π such that π(ω) ∈ {0, 1} for every ω ∈ Ω). Essentially, MEL is a fragment of the modal logic KD, in which neither the nesting of modalities nor the occurrence of nonmodal propositional formulas is allowed. Recently, generalized possibilistic logic (GPL) was introduced as a graded version of MEL (Dubois, Prade, and Schockaert 2011; Dubois and Prade 2011), developing an original proposal of (Dubois and Prade 2007). 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 wellknown 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.