Evaluating the Uncertainty of a Boolean Formula with Belief Functions

Abstract

In fault-tree analysis, probabilities of failure of components are often assumed to be precise and the events are assumed to be independent, but this is not always verified in practice. By giving up some of these assumptions, results can still be computed, even though it may require more expensive algorithms, or provide more imprecise results. Once compared to those obtained with the simplified model, the impact of these assumptions can be evaluated. This paper investigates the case when probability intervals of atomic propositions come from independent sources of information. In this case, the problem is solved by means of belief functions. We provide the general framework, discuss computation methods, and compare this setting with other approaches to evaluating the uncertainty of formulas.

Inclusion–exclusion principle for belief functions

“…We can mention several lines of research that would complement the present results: (1) find necessary and sufficient conditions for the inclusion/exclusion principle to hold for plausibilities in the general case (a counterpart to Proposition 5); (2) investigate the relation between the assumption of random set independence (made in this paper) and other types of independence [12]; (3) investigate how to decompose an event / a formula into a set of event satisfying the inclusion/exclusion principle (e.g., classical BDDs do not always provide adequate solutions).…”