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 form ulation of the problem , let us first show that im plication 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 Rj for the rule A^ B by Lukasiewciz implication function is given by
Again, the (i, j) th element of the relational matrix R2 for the rule —B — —A using Lukasiewciz implication function is given by
R2 ( i , j ) = Min [ 1, {1- (1-bj ) + (1-ai )}] = Min {1, ( 1-a + bj ) } (10.27)
Thus it is clear from expressions (10.26) and (10.27) that the two relational matrices R1 and R2 are equal. So, classical modus tollens can be extended to fuzzy logic under Lukasiewciz implication relation. Let us consider a FPN (vide fig. 10.15(a)), referred to as the primal net, that is framed with the following knowledge base rule 1: di ,d2 — d3 rule 2: d2 — d4 rule 3: d3 ,d4 — di
The dual of this net can be constructed by reformulating the above knowledge base using the contrapositive rules as follows:
rule 1: —d3 — —di, —d2 rule 2: —d4 — —d2 rule 3: —di — —d3, —d4
Here the com ma in the R.H.S. of the if-then operator in the above rules represent OR operation. It is evident from the reformulated knowledge base that the dual FPN can be easily constructed by replacing each predicate's di by its negation and reversing the directivity in the network. The dual FPN of fig. 10.15(a) is given in fig. 10.15(b).
Reasoning in the primal model of FPN may be carried out by invoking the procedure: forward reasoning. Let R= Rp , P' =Pp and Q' = Q p denote the matrices for the primal model in expression (10.8). Then for forward reasoning in the dual FPN, one should initiate P' = (Qp)T and Q' = (Pp)T (vide theorem 10.5), and R = Rp prior to invoking the procedure forward reasoning. If the belief distributions at the concluding places are available, the belief distribution at other places of the dual FPN may be estimated by invoking the procedure backward-reasoning with prior assignment of Q' f m = (Pp )T ,P' f m = (Qp )T and Rf m = Rp.
Example 10.9: Consider the FPN of fig. 10.15(a),where di = Loves (ram, sita), d2 = Girl-friend (sita, ram), d3 = Marries (ram, sita) and ^ = Loves(sita, ram). Suppose that the belief distribution of — loves (ram, sita) and —Loves(sita, ram ) are given as in fig. 10.16(a) and (b) respectively. We are interested to estimate the belief distribution of —girl-friend (sita, ram). For the sake of simplicity in calculation, let us assume that Th = 0 and R = I and estimate the steady-state belief distribution of the predicates in the network by using forward reasoning. The steady-state belief vector is obtained only after one step of belief revision in the network with the steady-state value of the predicates -d equals to [0.85 0.9 0.95] T.
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