by continuous deformation without crossing either the Black
or White hole. Figure 1a has two additional possible mono-
tonic ways, involving over/under switchbacks, which are not
drawn. Relaxing the monotonicity requirement generates a
plethora of other possibilities, e.g. loopings and backwards
switchbacks, whose consideration is left as an exercise. It is
not clear under what circumstances such complex paths can
be meaningful, a matter for further study.
These ways are the equivalence classes defining the topo-
logical structure of the two different ω-spaces, analogs to the
fundamental homotopy groups in spaces which admit of loops
(e.g. Lee, 2000). The closed loops needed for classical homo-
topy theory are impossible for this kind of system because
of the ‘flow of time’ defining the output of an information
source - one goes from a to b, although, for nonmonotonic
paths, intermediate looping would seem possible. The theory
is thus one of directed homotopy, dihomotopy, and the cen-
tral question revolves around the continuous deformation of
paths in ω-space into one another, without crossing Black or
White holes. Goubault and Rausssen (2002) provide another
introduction to the formalism.
These ideas can, of course, be applied to lower level cogni-
tive modules as well as to the second order hierarchical cog-
nitive model of institutional or machine cognition where they
are, perhaps, of more central interest.
We propose that empirical study will show how the in-
fluence of cultural heritage or developmental history defines
quite different dihomotopies of attentional focus in human
organizations. That is, the topology of blind spots and their
associated patterns of perceptual completion in human orga-
nizations will be culturally or developmentally modulated. It
is this developmental cultural topology of multitasking orga-
nization attention which, acting in concert with the inher-
ent limitations of the rate distortion manifold, generates the
pattern of organizational inattentional blindness. Analogous
developmental arguments should apply to highly parallel dis-
tributed machine cognition.
Such considerations, and indeed the Black Hole develop-
ment of equation (12), suggest that a multitasking system
which becomes trapped in a particular pattern of behavior
cannot, in general, expect to emerge from it in the absence of
external forcing mechanisms.
A second perspective, perhaps more relevant to this work,
is that the solution to a highly parallel computing problem
might well be cast in terms of ‘finding the right Black Hole’.
That is, the machine begins at some starting point, and, if
the problem has been used to properly constrain the shape of
the underlying dynamic manifold, i.e. to act as a tuning ‘goal
context’ in Baars’ terminology, solution involves identifying
the various topological modes possible to that complicated
geometric object. Among those modes will be the absorbing
singularites which constitute problem solution. Evolutionary
algorithms rear their heads as obvious potential tools, but
some preliminary simplying analytic factoring, via topological
analysis, might well be possible before they need be applied.
The essential insight is that the equivalence class structure
to highly parallel problems which Asanovic et al. (2006) rec-
ognized can be used to define a groupoid symmetry which can
then be imposed on the ‘natural’ groupoids underlying mas-
sivly parallel computation, as we have characterized it here.
This sort of behavior is central to ecosystem resilience the-
ory (Gunderson, 2000; Holling, 1973), a matter which Wallace
et al. (2007) explore in more detail. The essential idea is that
equivalence classes of dynamic manifolds, and the directed
homotopy classes within those manifolds, each and together
create domains of quasi-stability requiring action of some
external executive - programmer, operating system, higher
level adaptive cognitive module - for change. A self-dynamic,
adaptive machine which enters a pathological quasi-stable re-
silience mode will stay there until moved out of it by executive
act.
The multi-level topology of institutional cognition provides
a tool for study of resilience in human organizations or social
systems, and, according to our perspective, probably for ma-
chines as well. Apparently the set of dynamic manifolds, and
its subsets of directed homotopy equivalence classes, formally
classifies quasi-equilibrium states, and thus characterizes the
different possible resilience modes. Some of these may be
highly pathological. Others will represent solutions to the
computing problem, according to this scheme.
DISCUSSION AND CONCLUSIONS
The simple groupoid defined by a distributed cognition ma-
chine’s basic cognitive modular structure can be broken by
intrusion of (rapid) crosstalk within it, and by the imposi-
tion of (slower) crosstalk from without. The former, if strong
enough, can initiate a set of topologically-determined giant
component global workspaces, in a punctuated manner, while
the latter deform the underlying topology of the entire system.
Broken symmetry creates richer structure in systems charac-
terized by groupoids, just as it does for those characterized
by groups. Equivalence classes of ‘cognitive languages’, in a
certain sense, generate the dynamical groupoid of the system,
and the associated dihomotopy groupoids the topologies of
those individual manifolds, containing ‘solutions’ to the com-
puting problem.
Multitasking machine attention - the program - acts
through a Rate Distortion manifold, a kind of retina-like
filter for grammatical and syntactical meaningful paths.
Signals outside the topologically constrained tunable syn-
tax/grammar bandpass of this manifold are sub ject to less-
ened probability of punctuated conscious detection: inat-
tentional blindness. Path-dependent machine developmental
history will, according to this model, profoundly affect the
phenomenon by imposing additional topological constraints
defining the ‘surface’ along which this second order behavior
can (and cannot) glide.
The key trick is to use the problem itself as a ‘goal context’
to define that ‘surface’, whose singularities or absorbing states
constitute the solution to the problem. Characterizing the
topology of the problem-constrained dynamic manifold would
probably greatly simplify the use of evolutionary algorithms
or other techniques to actually identify the singularities, i.e.
to ‘find the solution’.
12