In this presentation we provide a personal perspective of the progress made in the field of approximate reasoning systems. Because of time and space limitations, we will limit the scope of our discussion to cover the most notable trends and efforts in reasoning with uncertainty and vagueness. The existing approaches to representing this type of information can be subdivided in two basic categories according to their qualitative or quantitative characterizations of uncertainty. Models based on qualitative approaches are usually designed to handle the aspect of uncertainty derived from the incompleteness of the information, such as Reasoned Assumptions (Doyle, 1983), and Default Reasoning (Reiter, 1980). With a few exceptions, they are generally inadequate to handle the case of imprecise information, as they lack any measure to quantify confidence levels (Doyle, 1983). A few approaches in this group have addressed the representation of uncertainty, using either a formal representation, such as Knowledge and Belief (Halpern and Moses, 1986), or a heuristic representation, such as the Theory of Endorsements (Cohen, 1985). We will further limit our presentation by focusing on the development the quantitative approaches. Over the past few years, quantitative uncertainty management has received a vast amount of attention from the researchers in the field, (Shachter et al., 1990, Henrion

The categorial approach to evidential reasoning can be seen as a combination of the probability kinematics approach of Richard Jeffrey (1965) and the maximum (cross-) entropy inference approach of E. T. Jaynes (1957). As a consequence of that viewpoint, it is well known that category theory provides natural definitions for logical connectives. In particular, disjunction and conjunction are modelled by general categorial constructions known as products and coproducts. In this paper, I focus mainly on Dempster-Shafer theory of belief functions for which I introduce a category I call Dempster?s category. I prove the existence of and give explicit formulas for conjunction and disjunction in the subcategory of separable belief functions. In Dempster?s category, the new defined conjunction can be seen as the most cautious conjunction of beliefs, and thus no assumption about distinctness (of the sources) of beliefs is needed as opposed to Dempster?s rule of combination, which calls for distinctness (of the sources) of beliefs.

We present examples where the use of belief functions provided sound and elegant solutions to real life problems. These are essentially characterized by ?missing' information. The examples deal with 1) discriminant analysis using a learning set where classes are only partially known; 2) an information retrieval systems handling inter-documents relationships; 3) the combination of data from sensors competent on partially overlapping frames; 4) the determination of the number of sources in a multi-sensor environment by studying the inter-sensors contradiction. The purpose of the paper is to report on such applications where the use of belief functions provides a convenient tool to handle ?messy' data problems.

Dempster-Shafer evidence theory is an efficient mathematical tool to deal with uncertain information. In that theory, basic probability assignment (BPA) is the basic element for the expression and inference of uncertainty. Decision-making based on BPA is still an open issue in Dempster-Shafer evidence theory. In this paper, a novel approach of transforming basic probability assignments to probabilities is proposed based on Deng entropy which is a new measure for the uncertainty of BPA. The principle of the proposed method is to minimize the difference of uncertainties involving in the given BPA and obtained probability distribution. Numerical examples are given to show the proposed approach.