On the cardinality index of fuzzy measures and the signatures of coherent systems

Boolean and pseudo-Boolean functions play a central role in various areas of applied mathematics. We will focus here on their use in decision making, cooperative game theory, and engineering reliability theory. A discrete fuzzy measure on the finite set X = {1, . . . ,n} is a nondecreasing set function μ : 2 → [0,1] satisfying the boundary conditions μ(∅) = 0 and μ(X) = 1. For any subset S ⊆ X , the number μ(S) can be interpreted as the certitude that we have that a variable will take on its value in the set S ⊆ X . A cooperative game on a finite set of players N = {1, . . . ,n} is a set function v : 2 →R which assigns to each coalition S of players a real number v(S). This number represents the worth of S. (Even though the condition v(∅) = 0 is often required for v to define a game, here we do not need this restriction.) A system is defined by a finite set of components C = {1, . . . ,n} that are interconnected according to a certain structure. The components are either in function or in a failed state, and the same holds for the whole system. It is common to associate the Boolean value 0 with a failed state and the value 1 with a component that is in function. Therefore the structure function of a system is the function φ from 2 to B = {0,1} which associates with any set A of components that are in function the corresponding state of the system. The system is semicoherent if the structure function is nondecreasing and satisfies the conditions φ(∅) = 0 and φ(C) = 1. It is coherent if in addition all the components are essential. We identify any subset S of {1, . . . ,n}with its characteristic vector 1S ∈ {0,1} n (defined by (1S)k = 1 if and only if k is in S). This identification allows us to identify set functions and pseudo-Boolean functions, i.e., functions from B to R. Therefore discrete fuzzy measures, cooperative games, and structure functions of coherent systems are all described by pseudo-Boolean functions. The use of discrete fuzzy measures allows us to model real situations where additivity is not suitable since the set of such measures is richer than the set of classical additive measures. In the same way, cooperative games allow us to take into account possible interactions between the players and need not be additive. Finally, the set of all increasing pseudo-Boolean functions is necessary to describe all the possible semicoherent systems. However, the variousness of this set of functions also has the drawback