This article discusses several implementation aspects for Dempster-Shafer belief functions. The main objective is to propose an appropriate representation of mass functions and efficient data structures and algorithms for the two basic operations of combination and marginalization. © 2003 Wiley Periodicals, Inc. 1.

The paper presents a generic approach of approximating inference. The method is based on the concept of valuation algebras with its wide range of possible applications in many different domains. We present convenient resource-bounded anytime algorithms, where the maximal time of computation is determined by the user.

Probability is usually closely related to Boolean structures, i.e. Boolean algebras or propositional logic. Here we show, how probability can be combined with Non-Boolean structures, and in particular Non-Boolean logics. The basic idea is to describe uncertainty by (Boolean) assumptions, which may or may not be valid. The uncertain information depends then on these uncertain assumptions, scenarios or interpretations. We propose to describe information in information systems, as introduced by Scott into domain theory. This captures a wide range of systems of practical importance such as many propositional logics, first order logic, systems of linear equations, inequalities, etc. It covers thus both symbolic as well as numerical systems. Assumption-based reasoning allows then to deduce supporting arguments for hypotheses. A probability structure imposed on the assumptions permits to quantify the reliability of these supporting arguments and thus to introduce degrees of support for hypotheses. Information systems and related information algebras are formally introduced and studied in this paper as the basic structures for assumption-based reasoning. The probability structure is then formally represented by random variables with values in information algebras. Since these are in general Non-Boolean structures some care must be exercised in order to introduce these random variables. It is shown that this theory leads to an extension of Dempster-Shafer theory of evidence and that information algebras provide in fact a natural frame for this theory.

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Technical systems are in general not guaranteed to work correctly. They are more or less reliable. One main problem for technical systems is the computation of the reliability of a system. A second main problem is the problem of diagnostic. In fact, these problems are in some sense dual to each other.

This papers proposes a new approximation method for Dempster-Shafer belief functions. The method is based on a new concept of incomplete belief potentials. It allows to compute simultaneously lower and upper bounds for belief and plausibility. Furthermore, it can be used for a resource-bounded propagation scheme, in which the user determines in advance the maximal time available for the computation. This leads then to convenient, interruptible anytime algorithms giving progressively better solutions as execution time goes on, thus offering to trade the quality of results against the costs of computation. The paper demonstrates the usefulness of these new methods and shows its advantages and drawbacks compared to existing techniques. Key words: belief functions, Dempster-Shafer theory, incompleteness, lower and upper approximation, join tree propagation, fusion algorithm, interruptible anytime algorithm.

We focus on a well-known classification task with expert systems based on Bayesian networks: predicting the state of a target variable given an incomplete observation of the other variables in the network, i.e., an observation of a subset of all the possible variables. To provide conclusions robust to nearignorance about the process that prevents some of the variables from being observed, it has recently been derived a new rule, called conservative updating. With this paper we address the problem to efficiently compute the conservative updating rule for robust classification with Bayesian networks. We show first that the general problem is NP-hard, thus establishing a fundamental limit to the possibility to do robust classification efficiently. Then we define a wide subclass of Bayesian networks that does admit efficient computation. We show this by developing a new classification algorithm for such a class, which extends substantially the limits of efficient computation with respect to the previously existing algorithm. 1

Abstract. Representing and reasoning with an agent’s preferences is important in many applications of constraints formalisms. Such preferences are often only partially ordered. One class of soft constraints formalisms, semiring-based CSPs, allows a partially ordered set of preference degrees, but this set must form a distributive lattice; whilst this is convenient computationally, it considerably restricts the representational power. This paper constructs a logic of soft constraints where it is only assumed that the set of preference degrees is a partially ordered set, with a maximum element 1 and a minimum element 0. When the partially ordered set is a distributive lattice, this reduces to the idempotent semiring-based CSP approach, and the lattice operations can be used to define a sound and complete proof theory. A generalised possibilistic logic, based on partially ordered values of possibility, is also constructed, and shown to be formally very strongly related to the logic of soft constraints. It is shown how the machinery that exists for the distributive lattice case can be used to perform sound and complete deduction, using a compact embedding of the partially ordered set in a distributive lattice. 1

In the latest issue of this journal, Lefevre et al.’s paper [1] raises the important question of how to combine belief functions from different sources and how to manage conflicts. They propose a parametrized combination rule that includes several existing combination rules as special cases. Their method is based on so-called weighting factors that determine how conflicting masses are to be distributed among the subsets of the corresponding frames of discernment. Since my understanding of Dempster-Shafer theory and my practical experience with belief function modelisation and corresponding computations is quite different, I do not support several points in their approach. From a practical point of view, I have several major concerns. First of all, their method requires to specify actual weighting factors for every combination operator. This may be feasible for simple problems where the corresponding model consists of a few belief functions only. However, in cases where many belief functions are to be combined (several hundreds or thousands are common in many practical cases), the problem of specifying weighting factors becomes

Subsets of the Negation Normal Form formulas (NNFs) of propositional logic have received much attention in AI and proved as valuable representation languages for Boolean functions. In this paper, we present a new framework, called VNNF, for the representation of a much more general class of functions than just Boolean ones. This framework supports a larger family of queries and transformations than in the NNF case, including optimization ones. As such, it encompasses a number of existing settings, e.g. NNFs, semiring CSPs, mixed CSPs, SLDDs, ADD, AADDs. We show how the properties imposed on NNFs to define more “tractable ” fragments (decomposability, determinism, decision, read-once) can be extended to VNNFs, giving rise to subsets for which a number of queries and transformations can be achieved in polynomial time. 1