that one must work with this super-power set in general in real applications because most of the times the elements/granules of S
Θ have no clear physical meaning, not to mention the drastic increase of the complexity since one has 2 Θ ⊆ DΘ ⊆ S
Θ and

$$|2^{\Theta}|=2^{|\Theta|}<|D^{\Theta}|<|S^{\Theta}|=2^{|\Theta^{r e f}|}=2^{2^{|\Theta|}-1}$$
$$\left(1\right)$$
|Θ|−1(1)
Typically, $\epsilon$. 

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

In summary, DSmT offers truly the possibility to build and to work on refined frames and to deal with the complement whenever necessary, but in most of applications either the frame Θ is already built/chosen to satisfy Shafer's model or the refined granules have no clear physical meaning which finally prevent to be considered/assessed individually so that working on the hyper-power set is usually sufficient for dealing with uncertain imprecise (quantitative or qualitative) and highly conflicting sources of evidences. Working with S
Θ
is actually very similar to working with 2 Θ in the sense that in both cases we work with classical power sets; the only difference is that when working with S
Θ we have implicitly switched from the original frame Θ representation to a minimal refinement Θref representation. Therefore, in the sequel we focus our discussions based mainly on hyper-power set rather than (super-) power set which has already been the basis for the development of DST. But as already mentioned, DSmT can easily deal with belief functions defined on 2 Θ or S
Θ similarly as those defined on DΘ. Generic notation: In the sequel, we use the generic notation GΘ for denoting the sets (power set, hyper-power set and super-power set) on which the belief functions are defined.

Remark on the logical refinement: The refinement in logic theory presented recently by Cholvy in [2] was actually proposed in nineties by a Guan and Bell [14] and by Paris [20]. This refinement is isomorphic to the refinement in set theory done by many researchers. If Θ = {θ1, θ2, θ3} is a language where the propositional variables are θ1, θ2, θ3, Cholvy considers all 8 possible logical combinations of propositions θi's or negations of θi's (called interpretations), and defines the 8 = 23 disjoint parts/propositions of the Venn diagram in Fig. 2
[one also considers as a part the negation of the total ignorance] in the set theory, so that:

$$i_{1}=\theta_{1}\wedge\theta_{2}\wedge\theta_{3}$$
$$\begin{array}{l}{{i_{2}=g_{1}\wedge\neg\theta_{2}\wedge\neg\theta_{3}}}\\ {{i_{3}=\theta_{1}\wedge\neg\theta_{2}\wedge\theta_{3}}}\\ {{i_{4}=\theta_{1}\wedge\neg\theta_{2}\wedge\neg\theta_{3}}}\\ {{i_{5}=\neg\theta_{1}\wedge\theta_{2}\wedge\wedge\theta_{3}}}\\ {{i_{6}=\neg\theta_{1}\wedge\theta_{2}\wedge\neg\theta_{3}}}\\ {{i_{7}=\neg\theta_{1}\wedge\neg\theta_{2}\wedge\theta_{3}}}\\ {{i_{8}=\neg\theta_{1}\wedge\neg\theta_{2}\wedge\neg\theta_{3}}}\end{array}$$
$$i_{2}=\theta_{1}\wedge\theta_{2}\wedge\neg\theta_{3}$$
