B1 ≡ bk+1, ..., bm.
Thus we permit a graded response, supposing that if
h(x) ∈ B0
the pattern is not recognized, and if
h(x) ∈ B1
the pattern is recognized and some action bj , k + 1 ≤ j ≤ m takes place.
We are interested in paths x which trigger pattern recognition-and-response
exactly once. That is, given a fixed initial state a0 , such that h(a0) ∈ B0 ,
we examine all possible subsequent paths x beginning with a0 and leading
exactly once to the event h(x) ∈ B1. Thus h(a0, ..., aj) ∈ B0 for all j < m,
but h(a0, ..., am) ∈ B1.
For each positive integer n let N (n) be the number of paths of length
n which begin with some particular a0 having h(a0) ∈ B0 and lead to the
condition h(x) ∈ B1. We shall call such paths ‘meaningful’ and assume N(n)
to be considerably less than the number of all possible paths of length n -
pattern recognition-and-response is comparatively rare. We further assume
that the finite limit
H ≡ lim log[N(n)∣
n→∞ n
both exists and is independent of the path x. We will - not surprisingly
- call such a pattern recognition-and-response cognitive process ergodic.
We may thus define an ergodic information source X associated with
stochastic variates Xj having joint and conditional probabilities P (a0, ..., an)
and P (an|a0, ..., an-1) such that appropriate joint and conditional Shannon
uncertainties may be defined which satisfy the relations
H[χ] = lim log∣N(n)∣ =
n→∞ n
lim H(Xn|X0,...,Xn-1)=
n→∞
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