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

Here W T is a transposed weight matrix, whose (i, j)th element denotes the
weight from transition tri to place pj. ‘Th’ is the threshold vector, the i-th
component of which is a scaler denoting the threshold of transition tri. ‘U’ is a
unit step vector, which applies step function to all its arguments. ‘N’ is the
belief vector, whose i-th component is a scaler, representing the fuzzy belief of
proposition di. ‘T’ is the fuzzy truth token (FTT) vector, whose i-th component
is a scaler denoting the FTT of transition tri. ‘Q’ is a binary place to transition
connectivity matrix, whose (i, j)-th element denotes the existence (or no
existence) of connectivity from the place pj to transition tri.
A new weight adaptation rule, presented below, is used for the encoding of the
weights:
dWij
----- = - α Wij + ti (t) . nj (t) (16.16)
dt
where ti(t) and nj (t) denote the FTT and the belief of the transition tri and the
place pj respectively. The above equation can be discretized into the following
form
Wij (t+1) = (1- α) Wij + ti(t) . nj(t). (16.17)
The vector-matrix form of the above equation is given by
W (t+1) = (1 - α) W (t) + [ ( N (t) . T T (t)) ^ P ] (16.18)
where P is a binary connectivity matrix from transition to places in a FPN.
The recall model, represented by expression 16.14 and 16.15, is stable, when
N(t+1) = N (t) = N* at t = t* , the equilibrium time.
Now, the above relation holds good for ( Q o N* C ) C > Th if
N* ≥ [ W T o ( Q o N* C ) C ]. (16.19 )
The steady-state values of the components of N* thus can be easily
evaluated, with the known values of the Q and W matrix.
The conditional convergence: 0 < α