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We present an efficient algorithm [4]
to compute the membership function 
. Practically the generalized modus ponens is computed for each element
u of the universe U and for each element v of the
universe V that needs much computation time and memory space. In
effect, it needs a discretisation of fuzzy sets on the universe U
and V and computation of a Cartesian product and a projection.
So, this algorithm uses properties based on cores and supports of fuzzy sets involved in the generalized modus ponens to provide the final fuzzy set with a parametric computation that needs a couple of simple operations (figure 5). It uses only the five parameters of the representation of fuzzy sets (figure 3.1).
Assume the membership functions of the fuzzy sets A, 
, B and 
defined by:
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This algorithm is based on the following general properties of the implication of Brouwer-Gödel:
, it means that values of
are more specific than values of A, so 
.
, it means that values of
are less specific than values of A, so
is less specific than B. In this case:
) 
Supp(A) means that
contains uncertainty and
)
Core(A) means that the core of
is larger than the core of B. Assume 
and 
, two real numbers in the interval [0,1] characterizing the previous properties
defined with the 2 fuzzy sets A and
by:
|
and
are computed with intersections of segments of
the lines of cores and supports based on the five parameters of fuzzy sets
A and
. For example, in the figure 5,
and
are evaluated by:
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The fast algorithm called GMP is
defined by 
and this body is:
Begin ¯
If ¯ Else if
Else if
Else if
Else
End |
The computation of
and
can be integrated in the previous fast algorithm then denoted 
Figure 5: Parametric computation.
IRIT-UPS
Then 
(total uncertainty) 
Then 
Then 
