By achieving their purposes through interactions with the physical world, Cyber Physical Systems (CPS) pose new challenges. Indeed, the evolution of the physical systems they control with transducers can be affected by surrounding physical processes over which they have no control and which may potentially hamper the achievement of their purposes. While it is illusory to hope for a comprehensive model of the physical environment at design time to anticipate and remove faults that may occur once these systems are deployed, it becomes necessary to evaluate their degree of effectiveness in vivo.In this paper, the degree of effectiveness is formally defined and generalized in the context of the measure theory and the mathematical properties it has to comply with are detailed. The measure is developed in the context of the Transferable Belief Model (TBM), an elaboration on the Dempster Shafer Theory (DST) of evidence so as to handle epistemic and aleatory uncertainties respectively pertaining the users expectations and the natural variability of the physical environment. This theoretical framework has several advantages over the probability and the possibility theories. (1) It is built on the Open World Assumption (OWA), (2) it allows to cope with dependent and possibly unreliable sources of information. The TBM is used in conjunction with the Input Output Hidden Markov Modeling framework (IOHMM) to specify the expected evolution of the physical system controlled by the CPS and the tolerances towards uncertainties. The measure of effectiveness is obtained from the forward algorithm, leveraging the conflict entailed by the successive combinations of the beliefs obtained from observations of the physical system and the beliefs corresponding to its expected evolution. The conflict, inherent to OWA, is meant to quantify the inability of the model at explaining observations.

Inappropriate use of Dempster's rule of combination has led some authors to reject the Dempster-Shafer model, arguing that it leads to supposedly unacceptable conclusions when defaults are involved. A most classic example is about the penguin Tweety. This paper will successively present: the origin of the miss-management of the Tweety example; two types of default; the correct solution for both types based on the transferable belief model (our interpretation of the Dempster-Shafer model (Shafer 1976, Smets 1988)); Except when explicitly stated, all belief functions used in this paper are simple support functions, i.e. belief functions for which only one proposition (the focus) of the frame of discernment receives a positive basic belief mass with the remaining mass being given to the tautology. Each belief function will be described by its focus and the weight of the focus (e.g. m(A)=.9). Computation of the basic belief masses are always performed by vacuously extending each belief function to the product space built from all variables involved, combining them on that space by Dempster's rule of combination, and projecting the result to the space corresponding to each individual variable.

Dempster-Shafer's model aims at quantifying degrees of belief But there are so many interpretations of Dempster-Shafer's theory in the literature that it seems useful to present the various contenders in order to clarify their respective positions. We shall successively consider the classical probability model, the upper and lower probabilities model, Dempster's model, the transferable belief model, the evidentiary value model, the provability or necessity model. None of these models has received the qualification of Dempster-Shafer. In fact the transferable belief model is our interpretation not of Dempster's work but of Shafer's work as presented in his book (Shafer 1976, Smets 1988). It is a ?purified' form of Dempster-Shafer's model in which any connection with probability concept has been deleted. Any model for belief has at least two components: one static that describes our state of belief, the other dynamic that explains how to update our belief given new pieces of information. We insist on the fact that both components must be considered in order to study these models. Too many authors restrict themselves to the static component and conclude that Dempster-Shafer theory is the same as some other theory. But once the dynamic component is considered, these conclusions break down. Any comparison based only on the static component is too restricted. The dynamic component must also be considered as the originality of the models based on belief functions lies in its dynamic component.

We derive axiomatically the probability function that should be used to make decisions given any form of underlying uncertainty.

Survey of several forms of updating, with a practical illustrative example. We study several updating (conditioning) schemes that emerge naturally from a common scenarion to provide some insights into their meaning. Updating is a subtle operation and there is no single method, no single 'good' rule. The choice of the appropriate rule must always be given due consideration. Planchet (1989) presents a mathematical survey of many rules. We focus on the practical meaning of these rules. After summarizing the several rules for conditioning, we present an illustrative example in which the various forms of conditioning can be explained.

The fundamental updating process in the transferable belief model is related to the concept of specialization and can be described by a specialization matrix. The degree of belief in the truth of a proposition is a degree of justified support. The Principle of Minimal Commitment implies that one should never give more support to the truth of a proposition than justified. We show that Dempster's rule of conditioning corresponds essentially to the least committed specialization, and that Dempster's rule of combination results essentially from commutativity requirements. The concept of generalization, dual to thc concept of specialization, is described.

Within the transferable belief model, positive basic belief masses can be allocated to the empty set, leading to unnormalized belief functions. The nature of these unnormalized beliefs is analyzed.

Jeffrey's rule of conditioning has been proposed in order to revise a probability measure by another probability function. We generalize it within the framework of the models based on belief functions. We show that several forms of Jeffrey's conditionings can be defined that correspond to the geometrical rule of conditioning and to Dempster's rule of conditioning, respectively.

In this paper, we present a decision support system based on belief functions and the pignistic transformation. The system is an integration of an evidential system for belief function propagation and a valuation-based system for Bayesian decision analysis. The two subsystems are connected through the pignistic transformation. The system takes as inputs the user's "gut feelings" about a situation and suggests what, if any, are to be tested and in what order, and it does so with a user friendly interface.

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.