[From Bill Powers (960101.1115)]

Yes, Happy New Year to all. When I was a kid, science fiction stories

about the far future were set in 1980. Here we are in the future beyond

the future. It's not as different as I once hoped it would be, but we're

working on it.

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Rick Marken (951231.1900) --

... anyone who understood what was going on in the basic

compensatory tracking task could have come up with these analyses.

But you do have be able see the generality of the principle

illustrated by these experiments; in particular you have to be able

to see that the perception of distance between cursor and target in

the compensatory tracking task is the same, in principle, as the

perception of the grammatical structure or phonemic content of an

utterance.

Exactly so. The person who understands these basic relationships tries

them out FIRST as an explanation of behavior. A lot of our discussions

arise from those who try some other explanation first, and then try to

fit PCT into the left-over spaces.

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Shannon Williams (960101) --

We are not visualizing this "pattern generator" the same way. I

visualize something that changes its patterns every time that it is

used. The parts of the pattern that correctly generates an output

which will achieve control of the input should remain steady. But

it will not be invariant at all.

An invariant is not a constant, as I have come to understand it. I think

it's worthwhile discussing invariants, to see the difference in how we

would handle patterns in a control model, as opposed to an open-loop

pattern generator model.

Consider the equation

y = a*x1 + b*x2

This could represent a simple perceptual function with two inputs

(weighted according to a and b) and one output, y. There are ways in

which x1 and x2 can vary which result in y varying, and other ways they

can vary that result in y remaining constant.

If y is constant, the points defined by x1 and x2 will lie on a straight

line with a slope defined by a and b (a straight line because I chose a

linear function of two variables). All pairs of values of x1 and x2 that

lie on this line produce the same output y (because we have said that y

is constant), so we can say that the value of y is _invariant_ with

respect to this particular way of changing x1 and x2. The perceptual

function will treat all pairs of inputs that lie on this line as the

same perception.

We can also change x1 and x2 in a way that moves the point x1,x2 _at

right angles_ to the line of invariance. This will produce different

values of y. Or, approaching the same result from another direction, if

we change the value of y to a new constant value, we will have a new

straight line parallel to the original line, but different from it. So

if the point x1,x2 moves from anywhere on the first line to anywhere on

the second line, y will change from its first invariant value to a new

invariant value. The kind of invariance remains the same: y is still

invariant with respect to combined changes in x1 and x2 that move the

point parallel to the line of invariance -- in the same spatial

direction.

> y = 6 *

> *

> * *

> # * *

x2 | # * y = 1 *

> * * *

> * * *

* # * y = -5 * #

> * * #

> * # * #

> * # * #

> * * # #

* *

> *

----------------------------------------------------

x1

Fig. 1: Lines of invariance.

The starred lines in Fig. 1 show the ways x1 and x2 can change together

that leave y constant (I haven't specified a and b; they're whatever

produces these lines). The #'s show a general point moving in x1-x2

space as the values of x1 and x2 change independently. Starting with the

upper-left # and tracing the movement down , to the right, and then

finally up and to the right, you can see that the value of y will start

at something greater than 6, decine to 1 and then to -5, and finally

become constant with y a little less than -5 as the #'s begin moving up

and to the right parallel to the line at y = -5. Notice that the value

of y does not reflect any movements of the # points parallel to the

lines of invariance. It reflects only changes in directions at right

angles to those lines.

Note that the directions of the lines of invariance are determined by

the constants a and b. Those constants are properties of the input

function and determine which lines of invariance will exist.

Suppose we're controlling the perceptual signal y at a value y0. The

error, y0 - y, is amplified and send to two output functions, which

affect the values x1 and x2 in appropriate directions. The result will

not be to bring x1 and x2 to any particular values, but to bring y to

the value y0. This means that disturbances which act in the x1-x2 plane

parallel to the lines of invariance in the figure above will NOT be

resisted. Why not? Because they will leave y invariant: the control

system will detect no change in the error. If the reference value is y0

= 1 unit, disturbances could move the # marks parallel to the line y = 1

above without causing any change in the system's action.

So this control system does not control x1 and x2 in the general x1-x2

plane. It controls only in a direction at right angles to the lines of

invariance.

Suppose that x1 is the magnitude of one formant and x2 is the magnitude

of another. We can assume that the same phoneme will be heard if some

function of x1 and x2 remains constant. This will most probably involve

curves, not lines, of invariance. The perceptual function is probably

nonlinear in x1 and x2. But it is still true that any combination of

formant magnitudes that leaves the output of the function anywhere on

one curve of invariance will lead to the same perception, the same

phoneme. The value of the perceptual signal will be the same anywhere

along the curve.

By changing the reference level for the perceptual signal of the

phoneme-control system, a higher system can change the phoneme that is

heard. That is, it can move the two formant magnitudes x1 and x2 so the

point they define lies near a new curve of invariance. A listener who is

perceiving through a similar input function will hear the phoneme

change, too. However, a sound spectrometer will not necessarily show the

same combination of formant magnitudes each time the same phoneme is

generated. The reason is that there is no control parallel to the curves

of invariance -- nor does there have to be any control. The listener and

the speaker will hear the same new phoneme, even though the formants may

change in combinations that also have an irrelevant component. Note that

you can have a change from one phoneme to another over a wide range of

changes in the two formants. One of the formants might increase,

decrease, or remain the same in magnitude, depending on what the other

one does. Just look at Fig. 1, and pick any two points on different

lines of invariance, with each line representing a different phoneme

(along a continuum, as from eeee-ih-eh ... oh-ooo). Think of a trombone,

not a trumpet.

This principle can be extended to input functions having any number of

inputs, and for any kinds of nonlinear functions (it even works with

nultiple-valued functions). The principle will remain the same. There

will be lines, or curves, or surfaces, or hypersurfaces of invariance,

parallel to which disturbances will not be resisted. Changing the

reference signal will result in moving the output of the perceptual

function to a new surface, etc.. An observer using the same perceptual

function will see a change of the controlled variable from one value to

another, even though an objective measure of the actual inputs to the

perceptual function shows that the values do not repeat from one

instance of the same perception to the next.

A control system, therefore, is not like an open-loop pattern generator.

It does not simply convert inputs to outputs. It varies its outputs to

make the value of some function of its inputs match a reference signal,

and in doing so it may actually produce many different values of the

inputs to the perceptual function while matching the same reference

condition. The perception it controls can, in general, be recognized by

an observer only if the observer passes the same physical inputs through

a highly similar input function. A sound spectrometer can't reveal the

invariances.

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This reference *must* evolve. If it does not evolve then it would

mean that our perception of the physical world is not subject to

our goals. In other words, the sounds we hear as infants would

sound the same to us as adults.

The perceptual functions and output functions must evolve; a reference

signal is just a signal with a particular magnitude. I haven't dealt

with change of organization. I think it's best to start by defining what

has to evolve; then we can see what aspects of the system need to be

modified during evolution.

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Best to all,

Bill P.