(15)
where δRk is ‘small’ in some sense compared to R0 .
Note that in this analysis the operators Rk are, implic-
itly, determined by linear regression. We thus can invoke a
quasi-diagonalization in terms of R0 . Let Q be the matrix of
eigenvectors which Jordan-block-diagonalizes R0. Then
QΩk+ι = (QRoQ-1 + QδRk+1Q-1)QΩk.
(16)
If QΩk is an eigenvector of R0, say Yj with eigenvalue λj,
it is possible to rewrite this equation as a generalized spectral
expansion
Yk+1 = (J + δJk+1)Yj ≡ λjYj + δYk+1
n
= λjYj + aiYi.
i=1
(17)
J is a block-diagonal matrix, δJk+1 ≡ QRk+1Q-1, and
δYk+1 has been expanded in terms of a spectrum of the eigen-
vectors of R0, with
∣α ∣ ≪ ∣λj ∣, ∣αi+ι | ≪ ∣α |.
(18)
The point is that, provided R0 has been tuned so that this
condition is true, the first few terms in the spectrum of this
iteration of the eigenstate will contain most of the essential
information about δRk+1. This appears quite similar to the
detection of color in the retina, where three overlapping non-
orthogonal eigenmodes of response are sufficient to character-
ize a huge plethora of color sensation. Here, if such a tuned
spectral expansion is possible, a very small number of ob-
served eigenmodes would suffice to permit identification of a
vast range of changes, so that the rate-distortion constraints
become quite modest. That is, there will not be much dis-
tortion in the reduction from paths in R-space to paths in
Ω-space. Inappropriate tuning, however, can produce very
marked distortion, even institutional or machine inattentional
blindness, in spite of multitasking.
Note that higher order Rate Distortion Manifolds are likely
to give better approximations than lower ones, in the same
sense that second order tangent structures give better, if
more complicated, approximations in conventional differen-
tiable manifolds (e.g. Pohl, 1962).
Indeed, Rate Distortion Manifolds can be quite formally
described using standard techniques from topological mani-
fold theory (Glazebrook, 2006). The essential point is that
a rate distortion manifold is a topological structure which
constrains the ‘multifactorial stream of institutional or ma-
chine consciousness’ as well as the pattern of communication
between giant components, much the way a riverbank con-
strains the flow of the river it contains. This is a fundamental
insight, which we pursue further.
10. Directed homotopy of global workspace ma-
chines To reiterate, the groupoid treatment of modular cog-
nitive networks above defined equivalence classes of states
according to whether they could be linked by grammati-
cal/syntactical high probability ‘meaningful’ paths. The dy-
namical groupoid is based on indentification of equivalence
classes of languages. Next we ask the precisely complemen-
tary question regarding paths on dynamical manifolds: For
any two particular given states, is there some sense in which
we can define equivalence classes across the set of meaning-
ful paths linking them? The assumption is that the system
has been ‘tuned’ (i.e. programmed) using the rate distortion
manifold approach above, so that the problem to be solved is
more tractable, in a sense.
This is of particular interest to the second order hierar-
chical model which, in effect, describes a universality class
tuning of the renormalization parameters characterizing the
dancing, flowing, tunably punctuated accession to high order
congnitive function.
A closely similar question is central to recent algebraic ge-
ometry approaches to concurrent, i.e. highly parallel, comput-
ing (e.g. Pratt, 1991; Goubault and Raussen, 2002; Goubault,
2003), which we adapt.
For the moment we restrict the analysis to a giant compo-
nent system characterized by two renormalization parameters,
say ω1 and ω2 , and consider the set of meaningful paths con-
necting two particular points, say a and b, in the two dimen-
sional ω-space plane of figure 1. The arguments surround-
ing equations (6), (7) and (12) suggests that there may be
regions of fatal attraction and strong repulsion, Black holes
and White holes, which can either trap or deflect the path of
institutional or multitasking machine cognition.
Figures 1a and 1b show two possible configurations for a
Black and a White hole, diagonal and cross-diagonal. If one
requires path monotonicity - always increasing or remaining
the same - then, following, e.g. Goubault (2003, figs. 6,7),
there are, intuitively, two direct ways, without switchbacks,
that one can get from a to b in the diagonal geometry of
figure 1a, without crossing a Black or White hole, but there
are three in the cross-diagonal structure of figure 1b.
Elements of each ‘way’ can be transformed into each other
11