Abstract 

We illustrate an approach to uncertain knowledge based on lower conditional probability bounds.  Our
results and algorithms exploit a concept of generalized coherence (g-coherence), which is a
generalization of de Finetti's coherence principle and is equivalent to the avoiding uniform loss
property for lower and upper probabilities(a la Walley).  By our algorithms, given a g-coherent
assessment, we can also correct it obtaining the associated coherent assessment (in the sense of
Walley and Williams).  Our algorithms work with a reduced set of variables and a reduced set of
constraints.  Such reduced sets are computed by suitably exploiting the additive structure of the random
gains.  In this paper, we study in detail imprecise assessments defined on families of three
conditional events.  We give some necessary and sufficient conditions and, then, we generalize some of
the theoretical results obtained.  We also exploit such results by proposing two algorithms which
provide new strategies for reducing the number of constraints and for deciding g-coherence.  Finally,
we illustrate our approach by giving some examples.  

Keywords:  uncertain knowledge, probabilistic reasoning un-der coherence, lower conditional probability bounds, g-coherence checking, not relevant
gains, basic sets, algorithms, computational aspects, reduced sets of variables, reduced sets of
linear constraints.