Plausibility Measures and Default Reasoning: An Overview

We introduce a new approach to modeling uncertainty based on plausibility measures. This approach is easily seen to generalize other approaches to modeling uncertainty, such as probability measures, belief functions, and possibility measures. We then consider one application of plausibility measures: default reasoning. In recent years, a number of different semantics for defaults have been proposed, such as preferential structures, -semantics, possibilistic structures, and -rankings, that have been shown to be characterized by the same set of axioms, known as the KLM properties. While this was viewed as a surprise, we show here that it is almost inevitable. In the framework of plausibility measures, we can give a necessary condition for the KLM axioms to be sound, and an additional condition necessary and sufficient to ensure that the KLM axioms are complete. This additional condition is so weak that it is almost always met whenever the axioms are sound. In particular, it is easily seen to hold for all the proposals made in the literature. Finally, we show that plausibility measures provide an appropriate basis for examining first-order default logics. 1 Plausibility Measures As the title suggests, this overview considers two (apparently unrelated) notions: plausibility measures, which provide a general framework for modeling uncertainty, and default reasoning, which involves making sense of stateSome of this work was done while both authors were at the IBM Almaden Research Center. The first author was also at Stanford while much of the work was done. IBM and Stanford’s support are gratefully acknowledged. The work was also supported in part by the Air Force Office of Scientific Research (AFSC), under Contract F49620-91-C-0080 and grant F94620-96-1-0323 and by NSF under grants IRI-95-03109 and IRI-96-25901. The first author was also supported in part by an IBM Graduate Fellowship and by the Rockwell Science Center. ments such as “birds typically fly”. We start by discussing plausibility measures. The standard approach to modeling uncertainty is probability theory. In recent years, researchers, motivated by varying concerns including a dissatisfaction with some of the axioms of probability and a desire to represent information more qualitatively, have introduced a number of generalizations and alternatives to probability, such as Dempster-Shafer belief functions [28] and possibility theory [5]. Rather than investigating each of these approaches separately, we focus on one measure of belief that generalizes them all, and lets us understand their commonalities and differences. A plausibility measure associates with a set a plausibility, which is just an element in a partially ordered space. Formally, a plausibility space is a tuple (W;F ; Pl), where W is a set of worlds, F is an algebra of measurable subsets of W (that is, a set of subsets closed under union and complementation to which we assign plausibility) and Pl is a probability measure, that is, a function mapping each set in F to an element of some partially-ordered set D. We use D to represent the partial order on D. We read Pl(A) as “the plausibility of set A”. If Pl(A) D Pl(B), then B is at least as plausible as A. Since D is a partial order, there may be sets in F which are incomparable in plausibility. We assume that D is pointed: that is, it contains two special elements >D and ?D such that ?D D d D >D for all d 2 D; we further assume that Pl(W ) = >D and Pl(;) =?D. The only other assumption we make is A1. If A B, then Pl(A) D Pl(B). Thus, a set must be at least as plausible as any of its subsets. Probability measures are clearly a subset of plausibility measures, in which the plausibilities lie in [0,1]. Indeed, every systematic approach for dealing with uncertainty of which we are aware can be viewed as a plausibility measure. We provide a few examples here. A belief functionB onW is a functionB : 2W ! [0; 1] satisfying certain axioms [28]. These axioms certainly imply property A1, so a belief function is a plausibility measure. A fuzzy measure (or a Sugeno measure) f on W [31] is a function f : 2W 7! [0; 1], that satisfies A1 and some continuity constraints. A possibility measure [5] Poss is a fuzzy measure with the additional property that Poss(A) = supw2A Poss(fwg). An ordinal ranking (or -ranking) onW (as defined by Goldszmidt and Pearl [19], based on ideas that go back to Spohn [30]) is a function : 2W ! IN , where IN = IN [ f1g, such that (W ) = 0, (;) = 1, and (A) = mina2A (fag) if A 6= ;. Intuitively, an ordinal ranking assigns a degree of surprise to each subset of worlds in W , where 0 means unsurprising and higher numbers denote greater surprise. Again, it is easy to see that a -ranking is a plausibility measure. Given how little structure we have required of a plausibility measure, it is perhaps not surprising that plausibility measures generalize so many other notions. However, this very lack of structure turns out to be a significant advantage of plausibility measures. By adding structure on an “as needed” basis, we are able to understand what is required to ensure that a plausibility measure has certain properties of interest. This gives us insight into the essential features of the properties in question while allowing us to prove general results that apply to many approaches to reasoning about uncertainty. In previous work, we provided three examples of this phenomenon. One of them—default reasoning—will be the focus of this overview; we discuss default reasoning in more detail in Section 2. The other two involve showing how plausibility can give useful insights into notions normally associated with probability, such as conditioning and independence, and using plausibility as a basis for a model of belief change. We briefly discuss these points in Section 3.