What Is Memory that It May Have Hindsight and Foresight as well? - Understanding Understanding: Essays on Cybernetics and Cognition - page 115

Of course, I could have used other examples, as, for instance, the gener-
ation of a “color space” by the resolution of a triple-paradox which is pro-
duced by the divergent reports of the three types of cones with different
pigmentation regarding the appearance of one and the same spot in the
external world. However, this case and other cases are not minimal.
I shall now enlarge on my earlier brief comment regarding the use of
recursive functions as a more powerful tool in accounting for past experi-
ence than simple storage of the outcomes of individual acts. This comment
was prompted by Young's observation of recursive loops that report back
to a central station via some synaptic delays. Turning again to Figure 3 and
following the arrows leading from (A) to (A + ) back to (A), we realize that
an action A that took place in the past, say, one cumulative synaptic delay
D ago, i.e., A(t -D), is evaluated by (A + ) at time t, i.e., A + (t), which in turn
modifies the cellular aggregate in (A) that will at best respond with new
action after a cumulative synaptic delay D, i.e., with A(t +D).
I propose now to make changes neither in the structure nor in the func-
tion of this subsystem, but only in the interpretation of the modifications
that are supposed to take place. Instead of interpreting the synaptic
modifications in the cellular complex A as stores of the outcomes of various
individual actions, I propose that these modifications should be interpreted
as a modification of the transfer function of the whole subsystem (A,A + ).
Let me demonstrate this idea again with a minimal example, this time of
recursive functions.
First, I have to point out that the term “recursive function” is a misnomer,
for these functions are like any other function, and it is only that they are
not as usual defined explicitly but are defined recursively. By this is meant
that a function which relates a dependent variable y to an independent vari-
able, say time t, is not explicitly given in terms of this independent variable,
say y = t 2 , y-sinwt, or in general y = f(t), but is given in terms of its own
values at earlier instances y(t) = F(y{t -D}), where D expresses the interval
between the earlier instance and the instance of reference t. A typical
example of a recursive definition of a function is, for instance, the descrip-
tion of growth of a bacterial colony:

“The number of bacteria in a bacterial colony at any time is twice the number it
was one generation ago.”

If it takes on the average the time D for a bacterium to divide—i.e., one
generation extends over a time interval D—then the recursive description
of the size y of this colony is

() =◊ - ( )

yt

2 yt D

I shall not discuss the mathematical machinery that “solves” these expres-
sions, i.e., transforms them into explicit statements with respect to the
independent variable t only. For instance, in the above case the “solution”
is, of course,