to the total or relative ignorances associated with non existential constraints (if any, like in some dynamic problems); S3(A) transfers the sum of relatively empty sets directly onto the canonical disjunctive form of non-empty sets.

The hybrid DSm rule of combination generalizes the classic DSm rule of combination and is not equivalent to Dempter's rule. It works for any models (the free DSm model, Shafer's model or any other hybrid models)
when manipulating *precise* generalized (or eventually classical) basic belief functions. An extension of this rule for the combination of *imprecise* generalized (or eventually classical) basic belief functions is presented in next section. As already stated, in DSmT framework it is also possible to deal directly with complements if necessary depending on the problem under consideration and the information provided by the sources of evidence themselves.

The first and simplest way is to work with S
Θ on Shafer's model when a minimal refinement is possible and makes sense. The second way is to deal with partially known frame and introduce directly the complementary hypotheses into the frame itself. By example, if one knows only two hypotheses θ1, θ2 and their complements
¯θ1,
¯θ2, then we can choose switch from original frame Θ = {θ1, θ2} to the new frame Θ = {θ1, θ2,
¯θ1,
¯θ2}. In such case, we don't necessarily assume that ¯θ1 = θ2 and ¯θ2 = θ1 because ¯θ1 and ¯θ2 may include other unknown hypotheses we have no information about (case of partial known frame). More generally, in DSmT framework, it is not necessary that the frame is built on pure/simple (possibly vague) hypotheses θi as usually done in all theories managing uncertainty. The frame Θ can also contain directly as elements conjunctions and/or disjunctions (or mixed propositions) and negations/complements of pure hypotheses as well. The DSm rules also work in such non-classic frames because DSmT works on any distributive lattice built from Θ anywhere Θ is defined.

## 2.6 Examples Of Combination Rules

Here are some numerical examples on results obtained by DSm rules of combination. More examples can be found in [31].

2.6.1 Example with Θ = {θ1, θ2, θ3, θ4} Let's consider the frame of discernment Θ = {θ1, θ2, θ3, θ4}, two independent experts, and the two following bbas

$$m_{1}(\theta_{1})=0.6\quad m_{1}(\theta_{3})=0.4\quad m_{2}(\theta_{2})=0.2\quad m_{2}(\theta_{4})=0.8$$  of mass matrix  $${\bf M}=\begin{bmatrix}0.6&0&0.4&0\\ 0&0.2&0&0.8\end{bmatrix}$$

represented in terms of mass matrix

$${\mathrm{ise:}}\;\forall1\leq j\leq4,\,\mathrm{o}$$
$$\mathrm{ts}\;m(\theta_{j})=0/0$$

- Dempster's rule cannot be applied because: ∀1 ≤ j ≤ 4, one gets m(θj ) = 0/0 (undefined!).

- But the classic DSm rule works because one obtains: m(θ1) = m(θ2) = m(θ3) = m(θ4) = 0, and m(θ1 ∩ θ2) = 0.12, m(θ1 ∩ θ4) = 0.48, m(θ2 ∩ θ3) = 0.08, m(θ3 ∩ θ4) = 0.32 (partial paradoxes/conflicts).

- Suppose now one finds out that all intersections are empty (Shafer's model), then one applies the hybrid DSm rule and one gets (index h stands here for *hybrid* rule): mh(θ1 ∪ θ2) = 0.12, mh(θ1 ∪ θ4) = 0.48, mh(θ2 ∪ θ3) = 0.08 and mh(θ3 ∪ θ4) = 0.32.

2.6.2 Generalization of Zadeh's example with Θ = {θ1, θ2, θ3}
Let's consider 0 < ǫ1, ǫ2 < 1 be two very tiny positive numbers (close to zero), the frame of discernment be Θ = {θ1, θ2, θ3}, have two experts (independent sources of evidence s1 and s2) giving the belief masses

$$m_{1}(\theta_{1})=1-\epsilon_{1}\quad m_{1}(\theta_{2})=0\quad m_{1}(\theta_{3})=\epsilon_{1}$$
$$m_{2}(\theta_{1})=0\quad m_{2}(\theta_{2})=1-\epsilon_{2}\quad m_{2}(\theta_{3})=\epsilon_{2}$$

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