  Lehmann, Magidor, and Schlechta developed an approach to belief revision
based on distances between any two valuations. Suppose we are given such a
distance D. This defines an operator |D, called a distance operator, which
transforms any two sets of valuations V and W into the set V |D W of all
elements of W that are closest to V. This operator |D defines naturally the
revision of K by A as the set of all formulas satisfied in M(K) |D M(A) (i.e.
those models of A that are closest to the models of K). This constitutes a
distance-based revision operator. Lehmann et al. characterized families of them
using a loop condition of arbitrarily big size. An interesting question is
whether this loop condition can be replaced by a finite one. Extending the
results of Schlechta, we will provide elements of negative answer. In fact, we
will show that for families of distance operators, there is no "normal"
characterization. Approximatively, a normal characterization contains only
finite and universally quantified conditions. These results have an interest of
their own for they help to understand the limits of what is possible in this
area. Now, we are quite confident that this work can be continued to show
similar impossibility results for distance-based revision operators, which
suggests that the big loop condition cannot be simplified.
