MacKAY
(a) symbols generated as an incoming transform of incoming
signals, and
(b) symbols generated as the outward-directed (internal)
response of an imitative or adaptive system to the
incoming signals.
The distinction is not of course a merely verbal one. It determines in
practice the part of the mechanism to which dependent logical networks are
to be coupled.
C : Comparator.
E: Effector.
M: Control-signal
generator and
mixer.
SYSTEM 2
SYSTEM 1
SYSTEM 3
FIGURE 3
In type (a) connections have to be established between incoming
signal circuits and outgoing response-circuits. In type (b) connections
have to be established between one set of outgoing response-circuits and
another.
In metaphorical terms we might picture the distinction by saying
that for type (a) the act of knowing is an act of reception; for type (b)
it is an act of response.
Our example illustrates also one principle on which an automaton
could symbolize a hierarchy of abstract concepts, by the modes of response
of a hierarchically organized adaptive system.
To all these points we shall return in due course. We must first
digress slightly to clarify the principles on which error-sensitive systems
in general may be designed.
Having distinguished between systems which are "fully-informed"
and systems using trial and error, we shall see how the hierarchically-
organized adaptive system we have illustrated above can be generalized, on
a basis of statistically-controlled trial and error, so that abstract con
cepts of any order can be symbolized as required in the course of experi
ence, with the minimum of restriction (or instruction) by the designer.
THE EPISTEMOLOGICAL PROBLEM FOR AUTOMATA
k . Error-Sensitive Automata
The most familiar error-sensitive automata are so designed that
the nature and degree of error, as indicated by the error-signal, auto
matically prescribes the optimal corrective response. This requires that
all independent degrees of freedom of the response system be determined by
the error-signal, which must therefore have a corresponding information-
content. In the simplest type of servo, in fact, the error-signal must
have one logical degree of freedom for each that it determines in the
response.
A thermostat, for example, requires an error-signal of only one
dimension; a self-directed missile, in general, three; and so forth.
There are however two classes of error-sensitive automata in
which the logical dimensionality or "logon-content"^ [3] of the error-
signal can be less than the required number of degrees of freedom. In the
first class, the original error-indication is adequate (has the required
dimensionality) to prescribe the required response uniquely, but it is
transformed by a coding-process into an error-signal of lower dimension
ality (logon-content) for transmission to the response-mechanism. At the
receiving end it is then decoded into a form having the requisite number
of degrees of freedom.
This type of automaton differs only trivially from the first.
Although the logon-content of the error-signal may be less than that of
the response, the selective information-content must be at least equal to
that represented by the selection of the required response. In fact, if
an immediate and a correct response is required from one given mechanism
(leaving no time to take advantage of any redundancy, since this inher
ently entails delay [4 ])J+, then the error-signal must always comprise at
least m bits of selective-information, where m is the number of binary
decisions necessary in the response-mechanism to select the required
response. If there are n possible responses of equal complexity, then
m > log2n.
^The term "logon" was first used by D. Gabor (J. Inst. Elec. Engrs., 93,
III, 429-456, 194 6 ) in a more restricted sense.
k
In some cases delay can be reduced by having several response-mechanisms
operating in parallel and arranging that at least one produces the re
quired response (see below); but this does not get round the basic
theorems for one given mechanism.
MacKAY
We can thus describe both these automata as fully-informed.
Their corrective response can be immediate and can be deduced uniquely
from the error-signal.
There is however a second class of automaton in which the error-
signal is actually deficient in the selective information-content neces
sary to specify the required response uniquely.
This deficiency may arise through various physical limitations
even when the logon-content is adequate. The noise-level, for example,
or the resolving-power of the sensory system generating the error-signal,
may be such that the optimal response cannot be uniquely determined from
it. Indeed, in the strictest sense this is always the case, unless we
"quantize" the set of required responses.
It may however arise simply because it is desired to use an
error-signal of lower than adequate logon-content without the complexity
of a coding and decoding-process. In the extreme case we may wish to use
a simple one-dimensional two-valued ("go" - "no-go") error-signal to
control a response-mechanism of many degrees of freedom.
But whatever the reason for the inadequacy of selective
information-content, it is plain that the deficiency must be made good, if
adaptive success is to be ensured, by a process of trial and error on the
part of the response-mechanism. If only m bits are supplied by the error
signal, and one response must be selected out of n alternatives
(supposed equally likely meanwhile for simplicity), then after receiving
the error-signal the mechanism is still left with a choice from (n/2m )
possible responses.
If these can be tried in the most economical way possible and
full use is made of the information-capacity of the error signal at each
trial, the minimum average number of further trials necessary will be
[log2 (n/2m )]/m, or (log2n ) / m - 1 .
If, however, only a binary evaluation is possible at each later trial, then
on the average at least (log2 n - m) trials will be required in order to
find the required response (if all are equally likely).
We may of course interpret this as just another way of saying
that the number of bits of information by which the error-signal is
deficient must be gained from the environment in one way or another if the
final choice is to be adaptive to the required extent. Trial and error
(optimally designed) can enable the missing bits to be gained from suc
cessive error signals, provided that the environment is not changing too
rapidly.
This leads us to a further point. If the environment is changing
the adaptive response of the system must change to match it. The suc
cession of changes of response ideally required may be thought of as
THE EPISTEMOLOGICAL PROBLEM FOR AUTOMATA
2k 3
defining a certain information-rate: so many bits (on the-average) per
second.
It follows that if the error-signal carries m bits per trial,
and each trial requires At seconds, the system cannot respond completely
adaptively to an environment requiring an information-rate greater than
(m/At) bits/sec. from the response-mechanism.
In terms of discrete changes, the environment must alter at a
rate ((log2n)/m) times slower than the adaptive response mechanism if
the latter is to keep up with it.
Our chief point however is a positive one: it is possible for
a response-mechanism to be guided adaptively by an error-signal of any
selective-information-content, however small, to any required degree of
accuracy, given time. But in a fluctuating environment the average
selective information-content of successfully adaptive response cannot
exceed the product of the rate of trial and error with the selective
information-content of the error signal, or in general (and more obviously)
the error-signal channel-capacity.
As a final general point, leading to our next section, we may
note that where the probabilities of the adaptive responses required are
not equal--where there is redundancy (in the sense of communication theory)
in the response-pattern owing to the statistical structure of the pattern
of events to which response is required--then the foregoing statements
apply in the limit to the actual information-rates after allowing for
redundancy.
It follows that it should be possible in principle to devise an
automaton in which the adaptive response governed by an error-signal of
given selective-information-content may be much more complex than that
information-content might suggest, if advantage is taken of the redundancy
in the response-pattern required.
If the statistical structure were known to the designer, this
could be done by pre-setting the conditional probabilities of possible
trial-responses to match that structure, so that when the automaton is
confronted with a situation to which a certain response will be optimally
adaptive, the average time-lag before that particular response is tried
shall be the minimum made possible by fully exploiting redundancy.
From the discussion of Section 3 , however, there is evidently
another possibility. By generalizing the adaptive mechanism of Figure 3
to function on a basis of statistically-controlled trial and error, it
should be possible, even in the absence of prior knowledge of conditional
probabilities, to devise an automaton in which the optimal statistical
structure for its trial-procedure could be evolved automatically as the
result of experience.
2kk
MacKAY
Engineering details do not here concern us, but in the next
section we shall consider briefly how such an automaton could function.
5 . A Statistical Abstractive Mechanism
Automata have before been described, for example by Ashby [5],
in which the elements are interconnected by deterministic links, but the
error-signal functions by stimulating trial-and-error activity among relay-
switches altering the interconnections.
The mechanism we shall here consider is of a more general type.
We could describe it loosely as one in which the interconnections are
not strictly deterministic, but have a variable probability of functioning.
More precisely, it is a mechanism in which the probability of excitation
of each element (in a given time interval) can be made to depend on
continuously-variable physical factors as well as on the current states of
any number of other elements linked to it [2, 6, 7, and 8].
In the limiting case an element may be spontaneously active,
with a frequency depending in like manner on the current state of its
(topological) environment.
In an automaton of the general type of Figure 3 constructed of
such elements, adaptive activity can be guided by modifying the relative
probabilities of excitation. Trial-and-error normally takes place spon
taneously, and the error-signal functions by controlling the statistical
structure of the trial process.
The term "threshold” is used to denote, roughly speaking, the
resistance of an element to stimulation. In general each connecting link
to one element from another can have its own particular threshold, deter
mining the extent to which a signal in that link affects the probability
of excitation. The higher the threshold the lower the probability, or the
longer the interval of time that must elapse (with a given configuration
of stimuli) before excitation.
Error-stimulated adjustment of thresholds, then, or evaluatory
threshold-control for short, is the key notion on which we shall base our
statistical abstractive mechanism.
Assuming that we have elements in which the thresholds may be at least
semi-permanently modified according to the success or failure of current
trials, we can envisage a system initially devoid of instruction evolving
for itself a satisfactory pattern of adaptive activity.
To see how such statistically negative feedback can function, the
simple illustration provided by the model of Figure 4 may be helpful. It
was originally constructed to illustrate some points made in Reference [6],
and has of course no pretensions to animal status.
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