Fuzzy Sets and Possibility Theory - Information Fusion in Signal and Image Processing - page 174

Another distinctive element, the zero element, means that a source yielding this
value has complete determination over the result of the fusion. Such elements also
exist for t-norms and t-conorms.
The property of increasingness is usually imposed on operators and matches what
our intuition tells us.
Boundary conditions, which define the behavior of the operators when the infor-
mation to combine has extreme values, guarantee compatibility with the binary case,
where all of the propositions are simply either true or false (this corresponds to the
constraint of complying with deductive logic imposed by Cox for defining an induc-
tive logic [COX 46]).
The continuity property satisfied by most operators guarantees the robustness of
the fusion. However, this property is not always necessary, since natural phenomena
(particularly time phenomena) are not always continuous.
Idempotence means that providing information that is already available will not
change the fusion result. This property is not systematically imposed. It is verified by
means, the t-norm min and the t-conorm max (and those are the only ones). We might
want, on the contrary, to have the combination of two identical values reinforce or
weaken the overall result. Let us consider the example of identical simultaneous testi-
monies. If the witnesses are plotting together, it is not surprising to see them saying the
same thing and the associated degrees of confidence will therefore be combined in an
idempotent way. Whereas if they are independent, the credibility of what they are say-
ing will be reinforced if they are trusted, or weakened if they are not. Let us note that
the combination rules modeling these behaviors have been known since Bernoulli.
Generally speaking, it is considered that if sources are dependent (in the cognitive
sense), idempotence can be imposed, whereas if they are independent, reinforcement
effects can be needed.
Along the same lines, the nilpotence property will be imposed, for example, to
combine consecutive testimonies, in order to model the deterioration of information
along a chain of witnesses that are not completely reliable. For example, for certain
t-conorms, satisfying this property will help achieve a result equal to 1 by combining
a certain number of measures, which are not all equal to zero. This type of behavior
may be useful when the information is the result of a long processing chain.
The excluded middle and non-contradiction properties, satisfied only for certain
operators, have an accepted interpretation in reasoning terms, in the field of artificial
intelligence and fuzzy reasoning. There are examples in image processing where the
excluded middle is not advisable, whenever there is a need for introducing absence of
knowledge regarding an event and its complements and therefore to relax the compre-
hensiveness constraint applied, for example, in probabilities.