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

a finite set of places, D= {d1, d2 ..., dn } is a finite set of predicates, each di
having a correspondence to each pi for 1≤ i ≤ n , N = {n1,, n2 ,...,nn } is a
finite set of discrete fuzzy membership distributions, called belief
distributions, each distribution ni having correspondence to each predicate di.
Tr = {tr1 ,, tr2 , ..., trm } is a finite set of transitions; P ∩ Tr ∩ D = ∅. ' t' and
'th' represent respectively sets of fuzzy truth token (FTT) distribution and
thresholds, associated with each transition. I and O: Tr → P represent
mapping from transitions tri to their input and output places. Ri , associated
with each transition tri, represents the certainty factor (CF) of a rule: I(tri )
→O (tri ) and is represented by a fuzzy relational matrix.
Example 10.1: To bring out the implications of the above definitions, let us
consider the following production rules and database.
Production Rules (PR)
PR1: Tall(x), Stout (x) →Fast-runner (x)
PR2: Fast-runner (x) →Has-nominal-pulse-rate (x), Stout (x).
Database: Tall (ram ), Stout (ram).
In the PR above, Tall (x), Stout (x), etc. denote predicates and the
comma in the left and right hand sides of the implication sign (→) denote
AND and OR operations respectively. Given the measured membership
distribution of Tall(x), Stout (x) and Fast-runner (x) in PR1, one can easily
construct a relational matrix, representing the CF of the rule for each possible
membership values of the antecedent and consequent predicates under the
rule. For example, let us consider the membership distribution of Tall (x),
Stout (x) and Fast-runner (x) as shown in fig. 10.1. The relational matrix for
the rule can be constructed first by ANDing the distribution of Tall (X) and
Stout (x) and then by using an implication function [30-32] over the derived
distribution and the distribution of Fast-runner (x).
µ tall(x) Λ µ stout (x)
= [0.2 0.4 0.6 0.8] T Λ [0.1 0.2 0.9 0.2 ] T = [ 0.1 0.2 0.6 0.2] T .
Here T denotes the transposition operator and the ' Λ' operation between
two vectors has been computed by taking component-wise minimum of the
two vectors.
R1 = [ µµ tall (x) Λ µµ stout(x)] o [µ fast-runner (x)] T
= [0.1 0.2 0.6 0.2 ] T o [0.1 0.2 0.6 0.9]