[From Bill Powers (981201.1112 MST)]
Bruce Gregory (981201.1100 EDT)--
Rick Marken (981201.0745)
OK, so "react accordingly" means "have the appropriate effect".
Funny, I used exactly this example in just this way recently....This is
the >point I was trying to get at, knowing what "outputs to generate" (even
if >one could) is no help unless one can tell that they are reducing the
error.
There is a subtle and hard-to-grasp idea here, which is especially hard to
get across to most believers in conventional theories (S-R and cognitive).
I have often said that we learn control systems, not acts or responses.
What does this mean? It means that when we learn to control some variable,
we don't learn to produce any particular action, either as a reaction to
events in the world or as outputs that are planned and then executed.
To understand what we do learn, we have to see a control system as a
collection of variables which are functions of other variables. For
example, the output quantity is a (learned) function of the error signal.
To say this does not imply any particular state of the output quantity or
the error signal; it says only that whatever the value of the error signal
may be, the value of the output quantity will depend lawfully upon it. The
"law" is simply the form of the output function. It is that law that is
learned.
The transition from an event-based model to a system (PCT) model is just
like the transition from arithmetic to algebra. In arithmetic, we deal with
specific numbers. We are given a problem: 2 + 3 = ?. And we answer "5".
This is like stimulus-response theory, in which we are given a organism
plus a stimulus, and are asked to state what response will follow: boy +
girl + girl = ?, the answer being, possibly, "jealousy." Since the number
of combinations of organisms and stimuli is, for all practical purposes,
infinite, it would require an infinite number of such "arithmetic"
propositions to cover all possible kinds of behavior.
In algebra, we start dealing with symbols that stand for a _variable_
number -- that is, for something that can be represented by any number
within a range. When we say y = x + 1, we do not specify what the value of
x is. It could be any number between negative infinity and positive
infinity. We're just saying that whatever the value of x is, the value of y
is one greater. So we're describing a _relationship_ between two numbers,
without having to say what the values of the two numbers are. This is the
basic concept that we all had to learn in making the transition from
arithmetic to algebra.
This is what we're doing when we say that qo = f(e): the output quantity is
some function of the error signal. By specifying the form of the function
f, we can describe this relationship without having to refer to any
specific value of the error signal. We might say that qo = 10 * e, or if we
include the time dimension, qo = 0.1*INTEGRAL(e). Note that we can't say
what qo will be until we know what e will be. But the value of qo is
completely determined by e, for after we have observed the value of e, and
how it is changing, we can calculate the value of qo and its changes.
The equation qo = f(e) describes not a single response to a specific change
in an error signal, but ALL POSSIBLE RESPONSES TO ALL POSSIBLE ERROR SIGNALS.
It is the symbol f that stands for this universal relationship. Suppose
that f stands for multiplying e by 10 to produce qo. The specific form of
the function is qo = 10*e, and this remains the same for all values of e.
We can easily see that if the error signal is 1.5 units in magnitude, the
output quantity will be 15 units in magnitude. But it also follows that if
e is zero for five minutes, then jumps to a value of 2 and immediately goes
back to zero, qo will be zero for five minutes, then jump to a value of 20
and go immediately back to zero.
The output (qo) is a momentary event because the input (e) changes so as to
create a momentary event. BUT THIS DOES NOT MEAN THAT THE OUTPUT EVENT IS
"CAUSED" OR "TRIGGERED" BY THE INPUT EVENT. The correct explanation is
contained in the output equation: qo = 10*e. This relationship remains THE
SAME AT ALL TIMES during the changes in e. That is why a change in e is
accompanied by a corresponding change in qo. Even if the organism seems to
have learned to produce the output event when the input event "triggers"
it, what it has _really_ acquired is an output function such that qo =
10*e. Having acquired that _function_, the organism will produce whatever
output is implied by this equation, no matter how e changes -- even if it
simply remains zero or constant.
What's hard to communicate about this is that we can have a variety of
outputs in response to a variety of error signals without anything changing
in the system. When the output function is a simple proportionality, this
relationship is not hard to see. But if the output function is at all
complex, even if it's just an integration, there is no longer a one-to-one
correspondence between error signal and output. Then it may seem that each
different pattern of error signals causes a different "response". The fact
that the same relationship between error and output exists for all
apparently different responses is not so self-evident.
This same consideration holds for every relationship that makes one
variable dependent on another in the control system, or in the feedback
path. What is learned is a set of relationships between variables, not a
set of events caused by other events. Once the physical relationships have
been established, the whole control system creates specific and unchanging
relationships among its variables -- between reference signal and input, or
between disturbances and output. These relationships arise from the way
functions relate variables in the whole system. The relationship are
independent of any particular changes in the variables; no matter how the
reference signal and disturbance are changing (or staying the same), the
organization of the system remains exactly the same.
When you learn to see behavior in terms of relationships among variables
instead of causal connections between one event and another, you can see
invariance where formerly you could see only specific causal connections.
You can see that when someone builds a fire in the fireplace, two people
may show "the same behavior", even though one of them takes off a sweater
while another opens a window. Those are examples of _the same behavior_.
Best,
Bill P.