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

10.3.1 The Behavioral Model of FPN
Let us consider a transition tri, where I(tri)={pk ,pm} and O(tr1) = {pu ,pv}.
Assume that thi is the threshold vector, associated with the transition. The
transition tri is enabled if
Ri o ( nk Λ nm ) ≥ thi.
An enabled transition fires, resulting in a change in the FTT vectors at
its output arcs. It is to be noted that the FTT vectors at all its output arcs are
equal. In case the transition tri is not enabled, the FTT distribution at its output
arcs is set to null vector. The model of FPN, designed after Looney [24] and
based on the above considerations, is now formally presented.
ti (t+1)= ti (t) Λ [ Ri o (nk (t) Λ nm (t))] Λ
U [Ri o (nk (t) Λ nm (t))-thi ] (10.1)
In expression (10.1), U denotes a unit step vector, each component of
which becomes one when its corresponding argument ≥ 0 and becomes zero,
otherwise. In fact, the enabling condition of the transition tri is checked by
this vector. Moreover, the Λ operation between two vectors is done
component-wise like column vector addition in conventional matrix algebra.
It may be noted that if tri has m input places p1 ,p2 ,...pm and k output places
pm+1 ,pm+2 ,...pm+k (fig. 10.4) then expression (10.1) can be modified with the
replacement of
m
nk(t) Λ nm (t) by Λ nw (t).
w=1
After the FTT distribution at all the transitions in the FPN are updated
concurrently, the belief distribution at all places can be updated in parallel
following expression (10.2). Let us consider a place pj such that pj ∈ [O(tr1)
∩O(tr2)∩...∩O(trs)] (fig.10.5). The updating of belief distribution nj at place
pj is given by
nj (t+1)
= nj (t) V [t1 (t) V t2 (2)V... V ts (t)]
s
= nj (t) V ( V tr (t)). (10.2)
r=1