ROSS ASHBY - The Cybernetic Brain - page 162

array of binary elements that switch each other on and off according to some
rule—as did Alexander with his imaginary lightbulbs. The important point to
stress, however, is that even such simple models are impossible to solve ana-
lytically. One cannot calculate in advance how they will behave; one simply
has to run through a series of time steps, updating the binary variables at each
step according to the chosen transformation rules, and see what the system
will in fact do. This is the cybernetic discovery of complexity transcribed from
the field of mechanisms to that of mathematical formalisms. Idealized binary
arrays can remain Black Boxes as far as their aggregate behavior is concerned,
even when the atomic rules that give rise to their behavior are known.
The only way to proceed in such a situation (apart from Alexander's trick
of simply assuming that the array breaks up into almost disconnected pieces)
is brute force. Hand calculation for a network of any size would be immensely
tedious and time consuming, but at the University of Illinois Crayton Walker's
1965 PhD dissertation in psychology reported on his exploration of the time
evolution of one-hundred-element binary arrays under a variety of simple
transformation rules using the university's IBM 7094-1401 computer. Walker
and Ashby (1966) wrote these findings up for publication, discussing how
many steps different rule systems took to come to equilibrium, whether the
equilibrium state was a fixed point or a cycle, how big the limit cycles were,
and so on. 64 But it was Kauffman, rather than Walker and Ashby, who obtained
the most important early results in this area, and at the same time Kauffman
switched the focus from the brain to another very complex biological system,
the cell.
Beginning in 1967, Kauffman published a series of papers grounded in
computer simulations of randomly connected networks of binary elements,
which he took to model the action of idealized genes, switching one another
on and off (like lightbulbs, which indeed feature in At Home in the Universe ).
We could call what he had found a discovery of simplicity within complexity. A
network of N binary elements has 2 N possible states, so that a one-thousand-
element network can be in 2 1000 distinct states, which is about 10 300 —another
one of those hyperastronomical numbers. But Kauffman established two fun-
damental findings, one concerning the inner, endogenous, dynamics of such
nets, the other concerning exogenous perturbations. 65
On the first, Kauffman's simulations suggested that if each gene has exactly
two inputs from other genes, then a randomly assembled network of one thou-
sand genes would typically cycle among just twelve states—an astonishingly
small number compared with 10 300 (Kauffman 1969b, 444). Furthermore the
lengths of these cycles—the number of states a network would pass through