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

= R b m o ((P ' f m) T o Nf c (t+1)) c [ by theorem10.5]
= R b m o [(P 'f m) T o (P' f m o Tf (t+1)) c ] c . (10.22)
Expression (10.19) and (10.22), which are identities of N f and Tf
respectively, taken together is called a reciprocity relation. For testing
reciprocity conditions, one, however, has to use the results of Theorem 10.7.
Theorem 10.7: The condition of reciprocity in a FPN is given by
(Q ' f m ) T o Rf m o (Q ' f m o I c ) c = I (10.23 (a))
and R b m o [ (P ' f m ) T o (P ' f m) c ] c = I (10.23(b))
Proof: Proof is presented in Appendix C. €
Example 10.7: Consider the FPN given in fig. 10.13. Given R f m = I and
Rbm = I, we want to test the reciprocity property of the FPN.
φ φ I I φ φ
Here, P ' fm = I φ φ and Q ' fm = φ I φ
φ I φ φ φ I
p1
tr3
p3
tr1
Fig. 10.13: A FPN used to illustrate the reciprocity property.
tr2
p2