How to test PCT versus MCT: a proposal

[Hans Blom, 960803]

(Rick Marken (960801.1315))

A liberal theologean thinks all theories (religions) are equally
correct; that those who believe in one theory (religion) can profit
from the "wisdom" of the other theories; that value of a theory
(religion) is judged by believers in terms of how well the theory
"works" for them.

Thanks for the explanation. Let me break this long statement apart and
see what remains:

... thinks all theories are equally correct: This is setting up a
straw man, I think, if you interpret this too narrowly. Nobody would
deny that sometimes one theory is superior to another. But that
depends very much on context. When planning an interplanetary
mission we prefer Newton's laws to those of relativity theory,
because the former are so much simpler and the difference hardly
matters. But if we want to predict the path of a light beam being
deflected by the sun, we need Einstein's formulas in order to avoid
an error of 100%. If I am playing baseball, however, I need to rely on
my own "inner" laws, acquired by practice and listening to the coach,
rather than the laws of Newton or Einstein. Relying on a single
theory in all cases is limiting, to say the least. Although
physicists are searching for "grand unifying principles", these do
not (yet) exist. And if they existed, it is to be expected that they
will be so complex as to be useless in daily life. So we see a
proliferation of all kinds of "small", context-dependent theories in
science as well. Now the major task of a scientist is to know when to
apply which theory.

... profit from the "wisdom" of other theories: If that wisdom is
truly wisdom, who wouldn't want to do so? It might pay off to take
the trouble to find out whether a different theory has captured some
wisdom that yours misses.

... the value of a theory is judged by how well it works: I couldn't
have described the prime directive of science better! Note that you
can drop the "by them": who else can judge for me than I alone?

So all the above sound rather scientific to me, and I don't
understand your statement:

Liberal theology is the antithesis of science and (therefore) understanding.

Can you explain to me what leads you to think this way? You seem to
presuppose one objectively verifiable theory that is able to explain
everything.

You have never presented an example of a behavioral phenomenon (Y) that that
can be mimiced by an MCT model and _not_ by a PCT model. ...

Well, you must have missed something. I'll try again to show the
difference with a very simple example. I hope that you forgive me for
presenting an "unrealistic" example; I just want to clarify the
principle of what learning is about and why it might be important.

MCT, at least where it treats adaptive control, is explicitly about
learning and its laws, as well as about control. Assume that the
controller lives in a world where perception and action are related
according to

    p := x + y * a

where p is the perception, a is the action, and x and y are UNKNOWN
constants. The goals of the MCT controller are to control, that is to
maintain p at a certain reference value r, AND TO GET TO KNOW THE
VALUES OF THE UNKNOWNS x and y. Let's assume that r is constant as
well, to keep things simple, at least initially. And let's also assume
that there are no disturbances of any kind. These can be introduced
at a later time to see what their effect is.

Now, is it necessary to get to know the values of x and y? No. PCT
shows that one can bring p to r through a method that could be called
"climbing the gradient". For example, if r = 20, x = 10, and y = 2,
all one has to do is to somehow arrive at the value a = 5 to bring p
to the value of r.

Would it be nice to know the values of x and y? Yes. If we know them
and are presented with a new task (a different value of r), we will
immediately find the correct solution rather than having to grope
around. The above formula gives the solution:

ยทยทยท

a := (r - x) / y

So what is the advantage of "knowledge"? That we immediately arrive
at the correct solution rather than having to rely on a slow gradient
climb.

How do we find the values of x and y? Well, math says that two
unknowns can be solved if we have two independent equations. In this
case, two values of a will result in two values of p, from which x
and y can be solved. Now that is only theory. We still have to find a
practical method.

What is the disadvantage of "knowledge"? Let us (for now) define a
"theory" as our assumption of the values of x and y. The latter exist
in the world out there and are not immediately accessible. All we can
have is their assumed, estimated, or modeled values xm and ym, which
may be correct or incorrect. The "theory" is correct only if xm = x
and ym = y.

Can we still control correctly if we have an incorrect theory? Yes
and no. In the example above, r was constant (= 20). If x = 10 and y =
2 and we have somehow arrived at xm = 10 and ym = 2, our theory is
correct and the computation of the action a results in a = 5. If a
new value of r is presented, however, our first action will be
inaccurate and we will still have to do hill-climbing. If the theory
is almost correct, our first "impulse" will already take us close to
the solution, so that little hill-climbing is required. If the theory
is badly off, the first "impulse" might actually make things worse
compared to having no theory at all.

A bad theory does not necessarily lead to incorrect action. If we
somehow think that xm = 5 and ym = 3, we ALSO arrive at a = 5 if r =
20. In this strange case, control is still perfect although it is
based on an incorrect theory.

We can analyze when it is better to have a theory, even if it is
incorrect, than have no theory at all. In practice, having a theory
which is not too much off results in better control than having no
theory at all. This is simple to program. Set up the above equation

    p := x + y * a

as the true "world" equation, but use as the MCT solution

    a := (r - xm) / ym

to realize, IN ONE STEP, a perception

    p := x + y * (r - xm) / ym = r * y/ym + x - y * xm/ym

Now, if there were no feedback in an MCT controller, p would stay at
that value, which may or may not be equal to r. But an adaptive
controller will also have a built-in mechanism that adjusts the
values of xm and ym and, as a result, p will be brought to r.

You know how to set up a PCT controller for this case.

The questions are now: What is better, having a "theory" (internal
values of xm and ym) or no theory at all? When does an (incorrect)
theory result in worse behavior than having no theory at all? And the
final question: do humans have "theories"?

This leads to the following proposal: Implement a cursor tracking
task based on the above "world" equation. In each trial, r is kept
constant. In each new trial, r has a new (random) value. This can be
implemented by keeping r constant for e.g. 50 time intervals of 0.1
seconds; this is a trial. The action a will, I assume, stabilize
within that period. After 50 time intervals, a new r is presented
for, again, 50 time intervals.

In a first experiment, x and y are kept constant throughout all
trials. If learning is present, tracking will improve over time. If
not, then not.

A second experiment could be a sequence of the above multi-trial
experiments, with new values of x and y in each element of the
sequence. We expect control to deteriorate when new values of x and y
are installed, but only if a "theory" has been built.

[The second experiment would introduce the notion of "deception". If
in the previous trial r = 20, x = 10 and y = 2, and in the new trial
r = 20, x = 5 and y = 3, the appropriate action remains a = 5. Thus
it appears as if nothing has changed. If a successive trial, however,
has e.g. r = 30 but unchanged x = 5 and y = 3, the deception will be
uncovered.]

Would this be a set of experiments that could decide whether we build
internal models? If not, what would you suggest?

I expect one limitation to the algorithms: no GLOBAL computations,
i.e. no accumulation of values and "one shot" computations on large
arrays of data. The computations have to be identical after each new
iteration (e.g. 0.1 second).

Greetings,

Hans

PS: I'm relocating from Gainesville back to Eindhoven in the coming
days, so you have ample time to ponder this proposal and/or come up
with an alternative.