toire of possible responses.
Cognitive pattern recognition-and-response proceeds by an
algorithmic combination of an incoming external sensory sig-
nal with an internal ongoing activity - incorporating the
learned picture of the world - and triggering an appropriate
action based on a decision that the pattern of sensory activity
requires a response.
More formally, a pattern of sensory input is mixed in an un-
specified but systematic algorithmic manner with a pattern of
internal ongoing activity to create a path of combined signals
x = (a0, a1 , ..., an, ...). Each ak thus represents some func-
tional composition of internal and external signals. Wallace
(2005a) provides neural network examples.
This path is fed into a highly nonlinear, but otherwise sim-
ilarly unspecified, decision oscillator, h, which generates an
output h(x) that is an element of one of two disjoint sets B0
and B1 of possible system responses. Let
B0 ≡ b0, ..., bk,
B1 ≡ bk+1,..., bm.
Assume 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.
The principal objects of formal interest are paths x which
trigger pattern recognition-and-response. That is, given a
fixed initial state a0 , we examine all possible subsequent paths
x beginning with a0 and leading to the event h(x) ∈ B1 . Thus
h(a0, ..., aj) ∈ B0 for all 0 < j < m, but h(a0, ..., am) ∈ B1 .
For each positive integer n, let N (n) be the number of
high probability grammatical and syntactical paths of length
n which begin with some particular a0 and lead to the condi-
tion h(x) ∈ B1. Call such paths ‘meaningful’, assuming, not
unreasonably, that N (n) will be considerably less than the
number of all possible paths of length n leading from a0 to
the condition h(x) ∈ B1.
While combining algorithm, the form of the nonlinear os-
cillator, and the details of grammar and syntax, are all un-
specified in this model, the critical assumption which permits
inference on necessary conditions constrained by the asymp-
totic limit theorems of information theory is that the finite
limit
H ≡ ∣im l°g[N(„)|
n→∞ n
(1)
both exists and is independent of the path x.
We call such a pattern recognition-and-response cognitive
process ergodic. Not all cognitive processes are likely to be
ergodic, implying that H, if it indeed exists at all, is path
dependent, although extension to nearly ergodic processes, in
a certain sense, seems possible (Wallace, 2005a).
Invoking the spirit of the Shannon-McMillan Theorem, it
is possible to define an adiabatically, piecewise stationary, er-
godic 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 satisfy the classic relations
H[X] = lim log[N(n)| =
n→∞ n
lim H(Xn|X0, ..., Xn-1 ) =
n→∞
lim H(Xo,...,Xn)
n→∞ n
This information source is defined as dual to the underlying
ergodic cognitive process (Wallace, 2005a).
Recall that the Shannon uncertainties H (...) are
cross-sectional law-of-large-numbers sums of the form
- k Pk log[Pk |, where the Pk constitute a probability
distribution. See Khinchin (1957), Ash (1990), or Cover and
Thomas (1991) for the standard details.
3. The cognitive modular network symmetry
groupoid
A formal equivalence class algebra can be constructed by
choosing different origin points a0 and defining equivalence
of two states by the existence of a high probability mean-
ingful path connecting them with the same origin. Disjoint
partition by equivalence class, analogous to orbit equivalence
classes for dynamical systems, defines the vertices of the pro-
posed network of cognitive dual languages. Each vertex then
represents a different information source dual to a cognitive
process. This is not a representation of a neural network as
such, or of some circuit in silicon. It is, rather, an abstract
set of ‘languages’ dual to the cognitive processes instantiated
by biological structures, social process, machines, or their hy-
brids.
This structure generates a groupoid, in the sense of We-
instein (1996). States aj, ak in a set A are related by the
groupoid morphism if and only if there exists a high prob-
ability grammatical path connecting them to the same base
point, and tuning across the various possible ways in which
that can happen - the different cognitive languages - parame-
tizes the set of equivalence relations and creates the groupoid.
This assertion requires some development.
Note that not all possible pairs of states (aj, ak ) can be
connected by such a morphism, i.e. by a high probability,
grammatical and syntactical path linking them with some