Artificial Intelligence and Soft Computing: Behavioral ... - Arteimi.info

Since such choice of Rfm and Rbm satisfy the reciprocity condition, it is
expected that the belief distribution at a given place of the FPN would retrieve
its original value after n-forward steps followed by n-backward steps of
reasoning in the network. Consequently the steady-state belief distribution at
all places in the FPN will be consistent independent of the order of forward
and backward computation. This, in fact, is useful when the initial belief
distribution of the intermediate [12] places only in the FPN is known.
Example 10.8: Consider the diagnosis problem, cited in example 10.7. We
assume that the bi-directional IFF relationship exists between the predicates
corresponding to input-output place pairs of the transitions in the network of
fig. 10.14. We also assume that the belief distribution at places p4 and p5 only
is known and one has to estimate the consistent beliefs at all places in the
network. In the present context, we first estimate Rf m and Rbm by using
expressions (10.24) and (10.25) and then carry out one step forward reasoning
followed by two steps back-directed reasoning using expression (10.13)
through (10.16). It has been checked that the steady-state belief vector, thus
obtained, is unique and remains unaltered if one carries out one step backdirected
reasoning followed by two steps forward and two steps back-directed
reasoning.
10.7 Fuzzy Modus Tollens and Duality
In classical modus tollens [15], for predicates A and B, given the rule A→B
and the observed evidence ¬ B, then the derived inference is ¬A. Thus the
contrapositive rule: (A→B)⇔ (¬ B→ ¬ A) follows. It is known that in fuzzy
logic the sum of the belief of an evidence and its contradiction is greater than
or equal to one [22]. So, if the belief of an evidence is known, the belief of its
contradiction cannot be easily ascertained. However, in many real world
problems, the belief of non-occurrence of an evidence is to be estimated,
when the belief of non-occurrence of its causal evidences is known. To tackle
such problems, the concept of classical modus tollens of Predicate logic is
extended here to Fuzzy logic for applications in FPN.
Before formulation of the problem, let us first show that implication
relations (A→B) and (¬B→ ¬A) are identical in the fuzzy domain, under the
closure of Lukasiewciz implication function. Formally let ai , 1≤ i ≤n and bj ,
1≤ j ≤m be the belief distribution of predicates A and B respectively. Then
the (i, j)th element of the relational matrix R 1 for the rule A→B by
Lukasiewciz implication function is given by
R1 ( i , j ) = Min { 1, ( 1-ai+ bj ) } (10.26)