This paper introduces four alternative representations of a set function: the Möbius transformation, the co-Möbius transformation, and the interactions between elements of any subset of a given set as extensions of Shapley and Banzhaf values. The links between the five equivalent representations of a set function are emphasized in this presentation.

There is introduced the notion of σ-null-additive set function as a generalization of the classical measure. There are proved the relations to disjoint and chain variations. The general Lebesgue decomposition theorem is obtained. AMS Mathematics Subject Classification (2000): 28A25

We describe a combinatorial algorithm which, given a monotone and consistent symmetric set function d on a finite set V in the sense of Rizzi [Riz00], constructs a non trivial set S minimizing d(S,V \ S). This includes the possibility for the minimization of symmetric submodular functions. The presented algorithm requires at most as much time as the one in [Riz00], but depending on the function...

Can we learn a sparse graph from observing the value of a few random cuts? This and more general problems can be reduced to the challenge of learning set functions known to have sparse Fourier support contained in some collection P. We prove that if we choose O(k log |P|) sets uniformly at random, then with high probability, observing any k-sparse function on those sets is sufficient to recover...

To compute a function such as a mapping from vertices to colors in the graph coloring problem, current practice in Answer Set Programming is to represent the function as a relation. Among other things, this often makes the resulting program unnecessarily large when instantiated on a large domain. The extra constraints needed to enforce the relation as a function also make the logic program less...

The idea of probabilistic metric space was introduced by Menger and he showed that probabilistic metric spaces are generalizations of metric spaces. Thus, in this paper, we prove some of the important features and theorems and conclusions that are found in metric spaces. At the beginning of this paper, the distance distribution functions are proposed. These functions are essential in defining p...