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Assume the inference of two parallel rules represented in a schematic form as:
| Rule 1 | : | IF | X is 
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THEN | Y is 
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| Rule 2 | : | IF | X is 
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THEN | Y is 
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| Fact | : | X is 
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| Conclusion | : | Y is 
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The basic idea of this algorithm consists in generating
with a core 
obtained by a numerical method, and 
approximated in a good way to be represented with the parameterized function.
The parameterized computation is based on the inclusion of 
in 
and/or in 
characterized by two parameters 
and 
. This algorithm uses the simple generalized modus ponens 
for four major cases characterized by four possible binary values of
and
:
( 
)
( 
)
( 
)
( 
) However
does not always respect the previous properties, so
and
are not always binary values and we have defined four new major cases:
close to
( 
)
close to
( 
)
close to 
( 
)
close to 
( 
) The fast parallel generalized modus ponens
algorithm called 
is defined by 
and his body is:
Begin ¯
If ¯ If ( ¯
If (
Else if
If (
If (
Else if
If (
If (
Else if
If (
If (
End |
The principle of this fast parallel algorithm is to select a case with
respect to
and
and compute
with
as if
and
are binary values.
If
and
are not binary value, the algorithm modifies the core and the support of
with respect to 
and 
which characterize the inclusion of 
in 
and/or in 
. A parameter 
is equally introduced to compute the final uncertainty of
.
Figure 7: Core computation for 2 parallel rules.
In case (a),
is computed only from
parameters, and there is a possible displacement of
if 
and 
. This displacement is obtained by projecting onto V, the intersection
of the core of
and the core of 
to find the core of
as shown in the figure 7.
The core of is
represented in grey.
Moreover supports are approximated to eliminate scale effect in
as shown in figure 8. Supports
are cut from the core along the support to eliminate the part of the support
under 
if 
. Equally supports are averaged to take into account the part of the support
under
if 
.
This solution is applied for all the cases (a), (b), (c), (d) and tries
to respect the shape of
as possible.
Figure 8: Support approximation.
The case (b) is a similar to case (a).
The case (c) modify cores and supports with an extension if 
.
The case (d) modifies cores and supports as does case (c), with a reduction
if 
.
But if 
,
the fuzzy set
cannot be evaluated by 
( 
) because the extension of the core is not equal for each side of
and
. So we have introduced two parameters 
and 
to take into account this particularity.
IRIT-UPS
\ 
and 
Then 
and 
Then 
) Then 
) Then 
and 
Then 
and 
Then 
