[From Bill Powers (940602.1520 MDT)]

Martin Taylor (940602.1400) --

It looks as though Mary stung you into showing that you really do

grasp PCT and HPCT. Good work, Mary. And Gary Cziko, mightn't it be

a good idea to post the part of Martin's summary on core,

classical, and conventional PCT, slightly edited for the audience,

in the monthly Intro to PCT? It's as good a summary of the basic

theory as I have seen.

You say

Anyway, the messages I hear Bill P sending are more along the

lines of "don't try applying to PCT principles that work

elsewhere; PCT is like nothing else in the world, so your

principles won't work here."

I am fortunately not roused to ire by this misconstruction of my

position, you slimy communist pervert, but it does offer an

opportunity to make my position clearer.

My position is that the first thing to do when trying to explain a

phenomenon not previously considered (like "alerting") is to see

whether an explanation for it already exists within PCT. If such an

explanation can be found, then there is no need to go further. If

something remains unexplained, and an alternative approach offers a

real or better explanation, then PCT will benefit from listening to

the other approach.

What bothers me is tacking an ad-hoc addition onto the model for

every new phenomenon, before trying to see whether the existing form

of the model already takes care of it. In the case of "alerting," I

think the phenomenon is adequately handled by a hierarchy of control

systems. In fact, the way in which alerting happens in examples I

can think of, the presence of a triggering perception is not

recognized just as an abstract fact, but results in an immediate

attempt to correct an error, which interferes with whatever control

process is going on. When the spider descended on Miss Muffet, what

did NOT happen was that an alerting system that monitors for spiders

advised the higher-level systems that an unwanted presence had

intruded, in case the higher systems wished, for cognitive reasons,

to terminate the present activity and reallocate resources to deal

with the spider. By the time that could have happened, Miss Muffet

would have been halfway to the next county, leaving her curds and

whey spilled on the ground.

It's true that there are fewer degrees of freedom of action than

there are of perception, so it's not physically possible to control

everything that can be perceived at the same time. It's true that if

an attempt were made to exceed the available degrees of freedom of

control, a conflict would result.

But as I've been hinting recently, the difference between input and

output degrees of freedom is probably not as large as it seems at

first glance. Large chunks of the perceived world covary because of

physical connections among them. Even though the receptor count

suggests a very large number of perceptual degrees of freedom, in

fact the connections in the environment vastly reduce the actual

degrees of freedom at the input. They also vastly reduce the

requirements on the output for effecting control. When you pull a

drawer out of a bureau, it is sufficient to pull only on one place

on a handle; the rest of the drawer, with its millions of

discriminable points, comes along without any need to control each

point. In fact, it would not be possible to control each point

individually; the points are attached together.

So your initial observations about input degrees of freedom

represent only a theoretical upper limit, which is probably many

orders of magnitude in excess of the actual degrees of freedom

available. This is only by way of introducing some doubt into the

argument by which you arrive at the concept of non-controlling

alerting systems.

Another important point is that given a large number of input

degrees of freedom, it is in principle possible to have a very large

number of perceptual variables under independent control at the same

time, much larger than the output systems could accomodate. So even

if there were control systems hooked up and active for many of these

currently uncontrolled systems, there is no mathematical prohibition

against their achieving control; only a practical one.

It is therefore possible that many control systems could be co-

active, but in such a way that no action is called for by many of

the excess control systems. In that way, the outputs of these

control systems could remain connected to the motor control system

reference inputs, without calling for any action that would disrupt

ongoing control processes. Their contributions to the net motor

control reference signal would normally be zero.

BUT -- and this is my main point -- if one of these control systems

were to find its perception departing from its reference level, it

would start sending signals to the motor control reference inputs

immediately. If those motor control systems were already in use, the

ongoing behavior would be disrupted, assuming there were no unused

degrees of freedom that would serve for control (waving the fly away

while you steer one-handed). Higher systems would experience

immediate error, as well as obtaining the information that set off

the normally-inactive control system. And that, I claim, is exactly

what happens. When the unwanted condition occurs, the reaction

against it is _immediate_, not waiting for a decision from higher-

level systems. The higher-level systems perceive what has happened

only after the action has already started.

This, it seems to me, completely explains the alerting phenomenon,

and in a way that fits observation better than your explanation

does, without requiring any specialized "alerting system."

Let's suppose that, contrary to the above, your proposal and mine

both explain the alerting phenomenon. The question then arises as to

which is the least complex explanation, requiring us to assume the

least amount of new machinery in the brain. My explanation requires

no additions at all. Yours requires a new system, which examines

error signals from the alerting systems, and makes a continual

judgment as to whether any of them has reached a magnitude that

calls for action. If any of them has, this monitoring system must

then decide which motor systems to turn off, to release the required

degrees of freedom, and then must connect the error signal it has

been monitoring to a pre-existing output function which is used only

when the alerting system in question is needed as a control system.

That choice depends, of course, on what the motor systems were

already being used for, which this monitoring system must also

understand.

That is a LOT of new machinery. And what you get from it is, at

best, only the ability to explain what my approach explains equally

well (actually, better).

My objection to the alerting system proposal is not that we should

not "try applying to PCT principles that work

elsewhere; PCT is like nothing else in the world, so your

principles won't work here." It is that you didn't try very hard to

see whether a better explanation could be found within PCT, and

seemed more concerned with preserving the value of an older

explanation that you happened to know about. Now you can see my

entire argument. Does it still seem so arbitrary of me to reject the

alerting system concept?

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RE: information theory

You say:

The objective of the second element is not to change PCT except

to provide other ways of looking at it, and thereby to show up

features that might be overlooked when we are restricted to a

single viewpoint.

The fact is, or seems to be, that information theory does not reveal

any new features of control processes; it merely derives well-known

features from a different computational stance. Consider the

relationship between disturbance bandwidth and magnitude of error.

When you first raised this issue, I was somewhat puzzled, because

you seemed to believe you were pointing out a new relationship. I

did not mention that interpretation because I could have been

mistaken.

In fact, from ordinary control theory it is quite possible to state

how _any_ signal in the control loop will behave, given the

disturbance and the reference signal (and a good model). In fact the

earliest approach to control theory was the frequency-domain

approach, in which such things as the spectral distribution of

disturbances was taken into account. With a good model of a control

system, and given any form of disturbance whatsoever, it is possible

to compute or find from simulation exactly the value of every signal

at every moment (save for system noise). From that result, of

course, it is then possible to characterize the signals in any terms

you like, RMS, averages, statistical distributions, information

rate, or what have you. Even system-generated noise can be taken

into account.

The information-theoretic approach starts with a different way of

characterizing signals, and different measures, but it ends up

predicting that there will be upper bandwidths of effective control,

or for given bandwidths, upper limits of usable gain (I am assuming

that some day you will complete this). So using information theory,

it is possible to derive some of the same general characteristics of

a control system that are normally derived, in control theory, in a

different way.

There is, however, one aspect of a control system that the

information-theoretic approach has not yet been shown to describe or

predict: the stability of the system. The reason is that the

information-theoretic calculations are not suited to dealing with a

closed-loop situation, except in terms of one function at a time.

Forms such as p(x|y) are modified in unknown ways when y is some

analytical function of x. The assumption in "x given y" is that y

can be "given" -- i.e., established independently of x. In a closed

loop it cannot. Setting y to a given value would break the loop.

No doubt an information-theoretic way to handle closed loops will be

found, perhaps along the lines your are now developing starting with

Laplace transforms. But that will mean only that information theory

will then be able to predict the same thing we can already predict

in another way -- judging from the computational examples I have

seen, a much simpler way.

So the question still remains open as to what IT can say about

control processes that can't be said using ordinary analytical

methods and simulations. So far I have seen nothing, either done or

proposed. And analytical methods and simulations give us one kind of

information that is forever beyond the scope of information theory:

a prediction of how the variables will change, in detail, through

time.

Over and again, I get from Bill a statement that I can

paraphrase as "I don't understand that in detail, and control

systems are simple, so it doesn't apply and I'm not

interested." "I'm not interested" is perfectly legitimate. "So

it doesn't apply" is not.

The problem that I keep running into is that in the first few pages

of your developments, I run into what seems a very poor analysis of

the physical situation to which the mathematics is supposed to

apply. Until that problem is cleared up, I can't raise any

enthusiasm for going on to study the ensuing complex (and for me,

very difficult) mathematical manipulations, nor, on the basis of

"garbage in, garbage out," can I accept that any outcome of the

mathematics is worth considering even if the mathematics is

perfectly correct. Mathematics works just like a computer program,

as I said earlier today. It stupidly does exactly what you tell it

to do, and never warns you when you have started off on the wrong

track.

You are correct that I trust simplicity much more than complexity.

At every step of a mathematical development, you have a choice of

ways to continue the development. You have to have in mind where you

want the development to go in order to choose the next steps. So

your own intentions strongly influence what the mathematics is going

to say and what is going to be ignored. Behind even the most complex

mathematical analyses, therefore, there is some simple intention,

some goal that one desires to reach. That, and not the mathematics,

is what ought to be discussed first, and what needs defending first.

Once the goal is selected, the only significant outcome would be

that the mathematics shows that the goal can't be reached. If the

goal is reached, all that is proven is that by choosing your path

carefully, you can lead the mathematics to do what you want it to

do.

This says that any long mathematical development needs to be checked

frequently against reality, if the development concerns a natural

system. Each time you take the next step in such a development,

you're making an assumption about what is relevant and important,

and often even about what _is_. Each such step should be checked

against nature, to see if the real system wants to go that way. If

the only constraint on the development is the next mathematical

result you want to demonstrate, the development as a whole can start

off on a random walk that leads only into imaginary universes.

So my preference for simplicity is at least principled. It's not

dictated entirely by my mental limitations. I learned these

attitudes at the feet of some lowly instructors, first in Navy

electronics and then in early courses in engineering physics. In

both venues, I was taught not to apply equations like cookbook

formulas, but to interpret them in terms of real phenomena. I was

taught that every way of writing the same equation can be seen to

have a specific meaning with respect to the real phenomenon. You can

write E = IR, which you interpret to mean that you're running a

known current through a resistor of known value, and computing the

voltage E. But you can also write I = E/R, meaning that you're

applying a known voltage E to a known resistance R, and measuring I.

You can do this with the control-system equations, too: however you

write them, the statement of equality has, if it's true, an

interpretation that you can apply to a real system, and if you like,

verify by measurement.

So I learned to take very small steps in analyzing any real system

mathematically. Of course like anyone else I became impatient with

this slow pace, and spent a good deal of time in mathematical

flights of fancy. I got to the point of convincing myself that there

was a way of trading angular momentum for linear momentum, and I had

pages and pages of mathematics to prove it, double and triple-

checked for errors. More than one conceptual mistake was involved,

of course, but the main ones were right at the beginning, where I

decided that this had to be possible and that I was going to prove

it possible and revolutionize space travel. Discarding this project

was excruciatingly painful, but probably less so at the age of 20

than it would have been at the age of 50, or now. After a few years,

perhaps five or six, I went back to taking small steps, and from

then on everything I tried, just about, worked. But the

disappointment took a long, long, time to wear off.

Perhaps you can conclude that I have a psychological problem about

complex mathematics, that I distrust it on sight. Perhaps so. But I

have seen more real results come out of simple treatments than

complex ones. Does this means that people who do complex mathematics

have a psychological problem with producing useful results? Or is it

that mathematics is only as reliable as its user makes it through

checking against nature?

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Best,

Bill P.