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

Theorem 16.1: With initial values of ni(o) ≤1 and Wij ≤ 1, the ni remains
bounded in the interval (0,1).
Proof: Proof of the theorem directly follows from the recursive definition of
expression (16.7).
Theorem 16.2: The recall model described by expression (16.7) is
unconditionally stable.
Proof: For oscillation, the value of a function f should increase as well as
decrease with time. Since ni(t+1) can only increase or remain constant, but
cannot decrease (vide expression (16.7)), therefore ni(t+1) cannot exihibit
oscillations. Further, since ni(t+1) is bounded and is not oscillatory, thus it
must be stable.
It may be added here that the recall model of Kosko, which excludes ni(t) from
the right hand side of expression (16.7), is oscillatory for a cyclic cognitive
net.
Theorem 16.3: The encoding model represented by expression (16.6) is
stable, when 0 ≤ α ≤ 2
unstable, when α ≥ 2
oscillatory, when α =2.
Proof : Replacing ∆ by E - 1, we have
(E - 1 + α ) Wij = f(ni) f (nj) (16.8)
Since at steady-state f (ni), f (nj) become constant, then let f(ni) f(nj) at
steady-state be denoted by f (ni)* and f(nj)* respectively.
Thus, (E- 1+α ) Wij (t) = f (ni)* f (nj)*. (16.9)
The complementary function for the above equation is
(E- 1 +α) Wij =0
which yields Wij (t) = C (1-α) t . (16.10)
The particular integral for equation (16.9) is given by
Wij(t ) = (1 / α ) f(ni)* f (nj)*. (16.11)
Combining expression (16.10) and (16.11), we find: