Introduction

Abstract

Reasoning under uncertainty is always based on a specified language or formalism, including its particular syntax and semantics, but also on its associated inference mechanism. In the present volume of the handbook the last aspect, the algorithmic aspects of uncertainty calculi are presented. Theory has sufficiently advanced to unfold some generally applicable fundamental structures and methods. On the other hand, particular features of specific formalisms and approaches to uncertainty of course still influence strongly the computational methods to be used. Both general as well as specific methods are included in this volume. Broadly speaking, symbolic or logical approaches to uncertainty and numerical approaches are often distinguished. Although this distinction is somewhat misleading, it is used as a means to structure the present volume. This is even to some degree reflected in the two first chapters, which treat fundamental, general methods of computation in systems designed to represent uncertainty. It has been noted early by Shenoy and Shafer, that computations in different domains have an underlying common structure. Essentially pieces of knowledge or information are to be combined together and then focused on some particular question or domain. This can be captured in an algebraic structure called valuation algebra which is described in the first chapter. Here the basic operations of combination and focusing (marginalization) of knowledge and information is modeled abstractly subject to simple axioms. These axioms allow for different schemes of so-called local computations. It is exactly this kind of local computation which makes inference feasible, which otherwise, in a naive approach would be computationally impossible. Although this algebraic structure and the associated computational architectures were motivated by numerical approaches to uncertainty, they apply also to symbolic structures. So valuation algebras present one unifying, generic approach to inference. In structures of logic designed for treating uncertainty, consequence finding has been identified as a basic problem. Consequence finding is therefore the subject of the second chapter. It places the problem in the context of propositional logic for the sake of simplicity. Consequence finding is concerned with the deduction of theorems in general. For practical purposes the search of theorems is limited to certain sublanguages. Or it is focused on the search of new theorems which can be deduced from additional knowledge. Various forms of common sense reasoning can be related to consequence finding. This includes abduction, default reasoning, assumption-based reasoning, circumscription and closed world reasoning. Although consequence finding is clearly a logical problem, it is a building block also for certain numerical approaches to uncertainty such as probabilistic argumentation systems (chapter 6) which illustrates once more the futility of separation of numerical from symbolic methods.