

![0_image_0.png](0_image_0.png)

where n is the cardinality of Θr and θi ∈ Θr. Recall that the DSm cardinality is simply given by the number of integers of the codification. The upper bound of the cardinality of S is given by:

$$\begin{array}{c}\mbox{\small$\cal S$}|<\sum\limits_{s=min_{S}}^{max_{S}}C_{n_{V}}^{s},\\ \mbox{\small$\cal S$}|<\sum\limits_{s=min_{S}}^{max_{S}}C_{n_{V}}^{s},\end{array}\tag{18}$$

where C
s nV

is the number of combinations of s elements among nV . Note that it also works if minS = 0 for the empty set.

## 3.6 Generation Of Dθ R

The generation of DΘ
rcould have the same complexity than the generation of DΘ if there is none constraint given by the user. Today, the complexity of the generation of DΘ is the complexity of the proposed code in [11]. Assume for example, the simple constrain θ1 ∩ θ2 ≡ ∅. First, the figures 7(a) and 7(b)
show the DSm cardinality for the elements of DΘ
r with |Θ| = 4 and the previous given constraint. On the left figure, the elements are ordered by increasing DSm cardinality and on the right figure with the same order than the figure 5. We can observe that the cardinality of the elements have naturally decreased and the number of non empty elements also. This is more interesting if the cardinality of Θ is higher. Figure 8 presents for a given positive DSm cardinality, the number of elements of DΘ
r for |Θ| = 5 and with the same constraint θ1 ∩θ2 ≡ ∅.

Compared to the figure 6, the total number of non empty elements (the integral of the curve) is considerably lower.

Thus, we have to generate DΘ
rand not DΘ. The generation of DΘ (see
[11] for more details) is based on the generation of monotone boolean functions.

A monotone boolean function fmb is a mapping of (x1, ..., xb) ∈ {0, 1}
bto a single binary output such as ∀x, x
′ ∈ {0, 1}
b, with x 4 x
′then fmb(x) ≤