θ1 ∧θ2 = p12 ∪p123 (i.e. all Venn diagram disjoint parts that contain the index "12" in their indexes) equals to (θ1 ∧ θ2 ∧ ¬θ3) ∨ (θ1 ∧ θ2 ∧ θ3).

The refinement based on Venn Diagram, becomes very hard and almost impossible when the cardinal of Θ,
n, is large and all intersections are non-empty (the free model). Suppose n = 20, or even bigger, and we have the free model. How can we construct a Venn Diagram where to show all possible intersections of 20 sets? Its geometrical figure would be very hard to design and very hard to read (you don't identify well each disjoint part of a such Venn Diagram to what intersection of sets it belongs to). The larger is n, the more difficult is the refinement. Fortunately, based on Smarandache's codification, we can algebraically design in an easy way for all such intersections (for example, if n is very big, we can use computer programs to make combinations of indexes {1, 2**, ..., n**} taken in groups or 1, of 2, ..., or of n elements each), so the refinement should not be a big problem from the programming point of view, but we must always keep in mind if such refinement is really necessary and if it has (or not) a deep physical interpretation and justification for the problem under consideration.

The assertion in [2], upon Milan Daniel's, that hybrid DSm rule is equivalent to Dubois-Prade rule is untrue, since in dynamic fusion they give different results. Such example has been already given in [7] and is reported in section 2.6.3 for the sake of clarification for the readers. The assertion in [2] that "from an expressivity point of view DSmT is equivalent to DST" is partially true since this idea is true when the refinement is possible (not always it is practically/physically possible), and even when the spaces we work on, S
Θ = 2Θref , where the hypotheses are exclusive, DSmT offers the advantage that the refinement is already done (it is not necessary for the user to do (or implicitly presuppose) it as in DST). Also, DSmT accepts from the very beginning the possibility to deal with non-exclusive hypotheses and of course it can a fortiori deal with sets of exclusive hypothesis and work either on 2 Θ or 2 Θref whenever necessary, while DST first requires implicitly to work with exclusive hypotheses only.

The main distinctions between DSmT and DST are summarized by the following points:
1. The refinement is not always (physically) possible, especially for elements from the frame of discernment whose frontiers are not clear, such as: colors, vague sets, unclear hypotheses, etc. in the frame of discernment; DST does not fit well for working in such cases, while DSmT does; 2. Even in the case when the frame of discernment can be refined (i.e. the *atomic* elements of the frame have all a distinct physical meaning), it is still easier to use DSmT than DST since in DSmT framework the refinement is done automatically by the mathematical construction of the super-power set; 3. DSmT offers better fusion rules, for example Proportional Conflict redistribution Rule \# 5 (PCR5) -
presented in the sequel - is better than Dempster's rule; hybrid DSm rule (DSmH) works for the dynamic fusion, while Dubois-Prade fusion rule does not (DSmH is an extension of Dubois-Prade rule);
4. DSmT offers the best qualitative operators (when working with labels) giving the most accurate and coherent results; 5. DSmT offers new interesting quantitative conditioning rules (BCRs) and qualitative conditioning rules
(QBCRs), different from Shafer's conditioning rule (SCR). SCR can be seen simply as a combination of a prior mass of belief with the mass m(A) = 1 whenever A is the conditioning event; 6. DSmT proposes a new approach for working with imprecise quantitative or qualitative information and not limited to interval-valued belief structures as proposed generally in the literature [5, 6, 47].

## 2.2 Notion Of Free And Hybrid Dsm Models 10

Free DSm model: The elements θi, i = 1**, . . . , n** of Θ constitute the finite set of hypotheses/concepts characterizing the fusion problem under consideration. When there is no constraint on the elements of the frame, we call this model the *free DSm model*, written Mf(Θ). This free DSm model allows to deal directly with fuzzy