Adaptive Fuzzy Petri Nets

An Adaptive Fuzzy Petri Nets is explained as a 9-tuple,

AFPN = {P, T , D, I , O, α, β, Th, W }

Here, Th: P → [0, 1] defines a function allocating a threshold value of λ_i from 0 to 1 to each place i. Hence we have,

Th = λ₁ , λ₂ , ... , λ_n

Now, for any type of transition t, the thresholds allocated to them then the transition is said to be enabled and can fire, if the certainty factors connected with the tokens of all its input places are greater.

Likewise, we have W explained as a set of W_I and W_O,

W_I: I → [- 1, 1]

And,

 W_O: O → [- 1, 1]

W_O and W_I are the set of output and input weights which allocates weights to all the arcs of the net W = W_I ∪ W_O . w_ij ∈ W_I  denoted how much a place impacts a following  transition connected by w_ij. Now a positive value of the factor means a positive impact and vice versa

Furthermore for any transition t, suppose I (t ) = { p₁ , p₂ , ... , p_n } .

The consequent input weights allocated to these places are as:

W_I1 , w_I2 , . . . , w_In

Now the sum of all the weights is unity such as:

w_I1  + w_I2  + .. . + w_In  = 1

All these situation result in one subsequent, the total of the impact is 1. In the similar way, w_Oj ∈ W_O denoted that how much a transition impacts an output places whether it fires.

Here compared to the structure of the Fuzzy Petri Nets, the Adaptive Fuzzy Petri Nets combine the advantages of Fuzzy Petri Nets via having the simpler structure better description ability. The advantages and the differences between Fuzzy Petri Nets and Adaptive Fuzzy Petri Nets are given as:

(a)   Th is explained as a mapping from each place to a threshold value in Adaptive Fuzzy Petri Nets quite than to a set of thresholds as in Fuzzy Petri Nets.

(b)   The set of weights W in the Adaptive Fuzzy Petri Nets are composed of two parts, name is: a set of output weights and a set of input weights. An input weight is allocated to an arc from a place to a transition and an output weight is allocated to an arc from a transition to a place. In Adaptive Fuzzy Petri Nets the weights are added to arcs upon the other hand they are added to places.

(c)    The unspecific reasoning process in Adaptive Fuzzy Petri Nets integrates the impact of each weighted branch. Conversely, the reasoning process is considers simply as the "more or less form" in the Fuzzy Petri Nets

(a)   In Adaptive Fuzzy Petri Nets we consider even the negative weights while the FPNs were merely considering the positive ones.