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

When the belief distribution of the axiom predicates are given, one has
to use the forward reasoning model. On the other hand, when the belief
distribution of the predicates for the concluding places is known, one should
use the back-directed reasoning model of the IFF relation. Moreover, when
the belief distributions of the predicates at the non-terminal places are
available, one has to use both forward and back-directed reasoning models to
estimate the belief distribution of the predicates corresponding to respective
predecessors and successors of the given non-terminal places. However, under
this case, the estimated beliefs of the predicates may not be consistent. In
other words, after obtaining steady-state beliefs at all places, if one recomputes
beliefs of the non-axiom predicates with the known beliefs of the
axiom predicates, the computed beliefs may not tally with their initial values.
In order to overcome this problem, one requires a special relationship, called
reciprocity [26]. It may be noted that in a FPN that holds (perfect) reciprocity
property, n successive steps of forward (backward) reasoning followed by n
successive steps of backward (forward) reasoning restores the value of the
belief vector N(t).
Definition 10.12: A FPN is said to hold reciprocity property if updating
FTT (belief) vector in the forward direction followed by updating of FTT
(belief) vector in the backward direction restores the value of the FTT (belief)
vector.
Formally, we estimate Tf (t+1) from given Nf (t) and Nf (t) from Tf (t+1) in
succession,
i.e., Tf (t+1) = Rf m o (Q ' f m o Nf c (t)) c (10.17)
and Nf (t) = P ' b m o Tf (t+1). (10.18 )
Combining equations (10.17) and (10.18), we have
Nf (t) = P ' b m o Rf m o (Q ' f m o Nf c (t)) c
= (Q ' f m) T o Rf m o (Q ' f m o Nf c (t)) c . [ by theorem 10.5] (10.19)
Further, from the definition 10.14, one may first estimate Nf (t+1) from Tf
(t+1) and then Tf (t+1) from Nf (t+1). Formally
N f (t+1) = P ' f m o Tf (t+1) (10.20)
and Tf (t+1) = Rb m o (Q ' b m o Nf c (t+1)). (10.21)
Combining (10.20) and (10.21) we have
Tf (t+1) = R b m o (Q ' b m o Nf c (t+1)) c