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

formulation of the state space model, we first construct a belief vector N (t) of
dimension (z. n) × 1, such that
N(t) = [ n1 (t) n2 (t) ....nn (t) ] T ,
where each belief vector ni (t) has z components. Analogously, we construct a
FTT vector T and threshold vector Th of dimension (z. m) ×1, such that
T(t) =[ t1 (t) t2 (t) ...tm (t) ] and Th = [ th1 th2, . . . ,thm ]
where each t j (t) and thi have z components. We also form a relational matrix
R, given by
R1 φ φ -- φ
φ R2 φ -- φ
φ φ R3 . φ
R = - - - --
φ φ φ -- Rm
where Ri for 1 ≤ i ≤ m is the relational matrix associated with transition tri and
Φ denotes a null matrix of dimension equal to that of Ri's, with all elements =
0. It may be noted that position of a given Ri on the diagonal in matrix R is
fixed. Moreover, extended P and Q matrices denoted by P ' and Q'
respectively may be formed by replacing each unity and zero element in P and
Q by square identity and null matrices respectively of dimensions equal to the
number of components of ti and nj respectively. Now consider a FPN with n
places and m transitions. Omitting the U vector for brevity, the FTT updating
equation at a given transition tri may now be described by expression (10.3)
n
ti (t+1)= ti (t) Λ[ Ri o ( Λ nw (t) ) ] (10.3)
∃ w =1
n
= ti (t) Λ[ Ri o { V n c w (t)} c ]
∃ w =1
n
= ti (t) Λ [ Ri o { V qw / { V n c w (t)} c } ]
∀ w =1