Alice and Kafka

[Martin Taylor 980228 22:40]

Bill Powers (980228.0308 MST)

I'm really beginning to understand Franz Kafka a lot better these last
couple of days, but now we seem to have gone through the looking glass
with him, to join Alice, as well.

Bill Powers (980227.0814 MST)--

I suppose that if I said "I was at the lake today" and you knew my wife
had been there with me, you would say "No you weren't; you were
not there alone."

I don't understand how this illustration applies.

I said "p is a function of d" and you said I was wrong, because p isn't
a function _only_ of d. I merely point out how nonsensical the same
transformation is in another, equivalent, circumstance. Perhaps I should
have been more direct, and pointed out that if y is y(x1, x2, x3, ...),
y is normally said to be a function of each of its arguments. In particular
y is a function of x1. Only one of Kafka's bureaucrats would find a way to
insert "only" into the statement so that the suspect could be shown thereby
to be a criminal.

In any case, I went further, and said that when the other arguments are
held constant, y is a pure function of x1 (as opposed to being simply
a function of x1).

Anyway, as Rick points out, your formula is not correct for the present
discussion. It should be written as

p = Fd(d)/(1+G) + Gr/(1+G)

That is _not_ what Rick pointed out. Rick suggested that my formula was
wrong. You show that you believe it to be right. In another message
directed to Rick, I have provided its derivation. For the "present
discussion," Rick uses "d" for the influence of the disturbance on
the CEV, as in the formula "p = o+d" which he often repeats, and which
you have not criticised.

In this Wander-land Through the Looking Glass, the validity of a mathematical
expression obviously depends on who says it.

nor is there any general basis for choosing one solution when the
input function is multiple-valued (as it most often is).

Are you saying (today) that p can take many values for a given condition
of the Controlled CEV (i.e. a particular set of sensory values, along with
their histories if the PIF is a time-function such as an integrator or
differentiator)? That's what a muliple-valued function is.

No, it's not.

I suggest you look in some kind of a mathematical dictionary, if you won't
take my word for the difference between a multi-valued function and a
function with multiple arguments. A multi-valued function has more than
one possible value for a particular set of values of its argument. A
square root is a trivially simple example. A function with multiple
arguments is like p(x1, x2) = x1 + x2, with its two arguments. A function
with multiple arguments may or may not be multiple-valued. A multi-valued
multi-argument function might be p(x1, x2) = sqrt(x1+x2).

Even if you won't take my word, I find it interesting to know that
(today) the CCEV is not a scalar quantity, as it was yesterday, and
probably will again be, tomorrow. And this isn't even a Leap Year.

Another simple function that has the property of multiple values is

p = x1 + x2.

OK, I'll believe you if you tell me what the multiple values are for, say,
x1 = 3 and x2 = 4.

We can have the SAME value of p for an infinity of pairs x1,x2. I think
you've got the problem reversed in your head.

I'm afraid I have. It's only through the Looking Glass that a function's
output determines its input; in the everyday world in which I usually live,
the input determines the output. In the world in which I live, any pair
of values of the two inputs results in only _one_ possible value of the
output.

It's not that the same
environment can produce two values of the controlled perception; it's that
the same value of the controlled perception can represent a multiplicity of
states of the environment.

But only one value of the CCEV, unless you are introducing some quite new
concept here, such as that the CCEV is multiple-valued, or perhaps that
the CCEV is multidimensional. Or both.

The controlled perception is the given; the
state of the environment needed to produce that given perception is what is
multiple-valued. Many different states of the environment can produce the
same perception; this is why a _single_ perception can't be used to deduce
the state of anything in the environment.

Except the state of the CCEV, perhaps?

Are you asserting that it most often is
true that perceptual functions are like that?

Yes, but not in the reversed sense you're talking about. In general, most
perceptions are functions of multiple input variables, and so the state of
a given perception can't be used to deduce the actual state of the input
variables.

In the world seen from the viewpoint of the control system, there is
only one input variable for which a state is accessible, and that is
the scalar variable called the Controlled Complex Environmental Variable.
The value of the perceptual signal is the internal representation of
the state of that variable.

An external analyst may observe that the CCEV is a complicated function
of many observables (available to _his/her_ perception), together with
their histories, but that complication doesn't affect the single control
system. (It can be important in analyzing the interactions among simple
control systems).

All that matters in analyzing the operation of an Elementary Control
loop is the set of relations among several scalar variables, namely the
two inputs to the loop (the reference signal and the disturbance signal),
the two outputs from the loop (the perceptual signal and the "output"
signal) and two internal variables (the error signal and the--usually
unnamed--influence of the output signal on the CCEV). The functions that
relate the variables each transform one scalar variable into another scalar
variable, except that the comparator and the CCEV both have two arguments,
the reference signal and the perceptual signal in the case of the comparator,
and the output signal and the disturbance signal in the case of the CCEV.

perceptual signal ^ | reference signal
                  > V
                  >____________-_____error signal___
                  > comparator |
perceptual signal | |
                  > output function
            input function |
                  > >output signal
             CCEV + -----environment function-------|
                  > >
disturbance signal| |output signal
                  ^ V

It's quite irrelevant to the control analysis whether the link between
one scalar variable in the loop and another goes by one wire or a million.
And if it goes by a million, the values in each micro-wire never
affect the operation of the loop independently of the value of the scalar
variable.

Apparently, the PCT world is one in which what matters is not what is
said but who said it, where functions go from output to input, where
functions of several arguments are automatically multiple-valued, and
where what is true on Tuesday is false on Thursday.

I do not care to continue to try to discuss matters using technical and
logical arguments in a world in which such arguments are invalid. And
I do not care to substitute personal for technical argument. When
you and Rick can come to an agreement on which terminology I should use,
when you can distinguish a function's arguments from its value, and
its output from its input, when you understand the difference between
"X is true" and "only X is true," then perhaps there might be merit in
continuing.

Martin

[From Bill Powers (980301.1123 MST)]

Martin Taylor 980228 22:40 --

In any case, I went further, and said that when the other arguments are
held constant, y is a pure function of x1 (as opposed to being simply
a function of x1).

This is a novel meaning of "function" to me, and I believe it would be even
to a real mathematician.

In this Wander-land Through the Looking Glass, the validity of a
mathematical expression obviously depends on who says it.

In mathematics there is no distinction between a "function" and a "pure
function." You're making up your own vocabulary.

I suggest you look in some kind of a mathematical dictionary, if you won't
take my word for the difference between a multi-valued function and a
function with multiple arguments.

I suggest you look up "function" and "pure function." And I suggest you
start thinking like a PCTer instead of an S-R psychologist. It is the
desired value of the perceptual signal (i. e., the reference signal) that
is the independent variable in a control system, because the inputs are
adjusted so as to produce a specified value of the perception. The control
system produces outputs that generate ONE of the sets of inputs that will
create the specified value of perceptual signal. It doesn't matter to the
control system which set of inputs is present.

The inputs to the PIF are not the givens from which the value of the
perceptual signal follows. That's the S-R analysis. Because of the control
loop, it is the other way around: the desired value of the perceptual
signal is what determines the values of the inputs to the perceptual input
function. The output action changes until those values are such as to
produce the desired perceptual signal. Since more than one set of values
will produce the same perceptual signal, the effective input relationship
is multiple-valued. Remember that the controlled input variable is the
_inverse input function_ of the reference signal, when control is working
properly. If the input function is a function of many variables, its
inverse is multiple-valued.

Even if you won't take my word, I find it interesting to know that
(today) the CCEV is not a scalar quantity, as it was yesterday, and
probably will again be, tomorrow. And this isn't even a Leap Year.

Martin, blame yourself if you're not getting the answers you expect. Yes,
the perceptual signal is a scalar, because it is only one signal and can
vary in only one dimension (as the theory now stands). But the signals of
which it is a function are generally multiple. If you specify a value of
the perceptual signal, there are multiple sets of inputs that can bring it
to that value, so the relationship of the perception to the input signals
is multiple-valued.

Another simple function that has the property of multiple values is

p = x1 + x2.

OK, I'll believe you if you tell me what the multiple values are for, say,
x1 = 3 and x2 = 4.

You're still in the S-R mode. Suppose x1 is fixed at 3 and the desired
perception is 6. We can deduce that the output will bring x2 to a value of
3. If x1 is arbitrarily changed to 1, x2 will change to 5, keeping the
perceptual signal at 6. So we have two different sets of x1,x3 leading to
the same perception, 6. Of course there are other scenarios that would
involve disturbances of both x1 and x2, and actions that affect both, but
this is one simple example of how this works.

Obviously, given that the perceptual signal is being maintained at 6, we
can't reason out what the two input variables will be unless we have more
information from the observer's point of view, particularly about the
disturbance.

We can have the SAME value of p for an infinity of pairs x1,x2. I think
you've got the problem reversed in your head.

I'm afraid I have. It's only through the Looking Glass that a function's
output determines its input; in the everyday world in which I usually live,
the input determines the output.

Then welcome to PCT World, where the output (via the control loop)
determines the input. Come on, Martin, stop trying to prove you're right
and think about this. If you double the sensitivity of the output function,
do you get twice as much action? No, you get half as much error. Causation
seems to work backwards in a control system, although of course it doesn't,
really.

In the world in which I live, any pair
of values of the two inputs results in only _one_ possible value of the
output.

That is true, if you reason forward (I presume you mean "output of the
PIF"). However, in a control loop, if you disturb x1, the resulting error
signal will cause x2 to be varied as well as x1, and you will get a
different pair of input values yielding the same perceptual signal. If you
like, I'll write a little model you can try to show how this works, but (a)
when you stop defending your conclusions I'm sure you will see how this
works, and (b) why can't you write your own damned programs?

It's not that the same
environment can produce two values of the controlled perception; it's that
the same value of the controlled perception can represent a multiplicity of
states of the environment.

But only one value of the CCEV, unless you are introducing some quite new
concept here, such as that the CCEV is multiple-valued, or perhaps that
the CCEV is multidimensional. Or both.

Martin, the CCEV is a figment of your imagination. There is no single
environmental variable that corresponds to the perceptual signal. There are
only the component input variables. I've never liked this idea of a CEV,
because it seems to sneak naive realism back into the theory even though it
is basically denied by the theory. It's like saying, "Sure, I know that the
taste of lemonade isn't really there in the mixture of acids, salts, and
oils -- but let's pretend it's really there anyway." That's all it is, a
pretense.

The perceptual signal is not "complex." It is a simple scalar. There is no
CEV and no CCEV, except in the eye of the external analyst.

The controlled perception is the given; the
state of the environment needed to produce that given perception is what is
multiple-valued. Many different states of the environment can produce the
same perception; this is why a _single_ perception can't be used to deduce
the state of anything in the environment.

Except the state of the CCEV, perhaps?

The CCEV is not different from the perceptual signal.

In the world seen from the viewpoint of the control system, there is
only one input variable for which a state is accessible, and that is
the scalar variable called the Controlled Complex Environmental Variable.

But it is not "complex;" it is a simple scalar. Its relation to the
multiple lower-level perceptions of which it is a function is not visible
except to the external analyst.

Apparently, the PCT world is one in which what matters is not what is
said but who said it, where functions go from output to input, where
functions of several arguments are automatically multiple-valued, and
where what is true on Tuesday is false on Thursday.

Get off your soapbox and try to understand what I'm saying. You are
complicating this argument beyond all reason. The fault is not in what I'm
saying, but in your perverse understanding of it.

I do not care to continue to try to discuss matters using technical and
logical arguments in a world in which such arguments are invalid.

How would you know? You use "technical and logical" arguments in a way
ideosyncratic to you, and always with a goal in mind: proving that you're
right, by hook or by crook. The old Oxbridge Syndrome.

Best,

Bill P.

[Martin Taylor 980303 11:20]

Bill Powers (980301.1123 MST)]

Martin Taylor 980228 22:40 --

In any case, I went further, and said that when the other arguments are
held constant, y is a pure function of x1 (as opposed to being simply
a function of x1).

This is a novel meaning of "function" to me, and I believe it would be even
to a real mathematician.

Ref: Mathematics Dictionary, James and James (2nd Edition).

Function: An association of a certain object (or objects) from one set
(the range) with each object from another set (the domain). ... A function
is said to be single valued if a unique value of the function is determined
by a choice of the independent variable (or variables). ... A function of
one variable is a function which has only one independent variable....A
function of several variables is a function that takes on a value or values
corresponding to every set of values of several variables (called the
independent variables).

In this Wander-land Through the Looking Glass, the validity of a
mathematical expression obviously depends on who says it.

In mathematics there is no distinction between a "function" and a "pure
function." You're making up your own vocabulary.

I'm not clear how this assertion relates to the fact that when I say that
the influence of the disturbing variable on the Controlled CEV is "d"
this is a big mistake, even though when Rick says the same thing it is
correct. That is the question I have asked several times in the last few
days, and to which the "Wander-land" comment applied. As I believe you
to know very well.

Could you explain the relationship between your comment and my question?

I suggest you look in some kind of a mathematical dictionary, if you won't
take my word for the difference between a multi-valued function and a
function with multiple arguments.

I suggest you look up "function" and "pure function."

I used "pure" so as to be explicit. I am sorry you don't like exlicitness.

And I suggest you

start thinking like a PCTer instead of an S-R psychologist.

Thank you for the suggestion. It is noted. But I'm afraid that I have the
contrary hope, that I will continue to think logically and, with luck,
precisely, using S-R ideas when dealing with input-output relations of
functions, and control loop ideas when dealing with the behaviour of
control loops.

It is the
desired value of the perceptual signal (i. e., the reference signal) that
is the independent variable in a control system, ...

But the desired value of the perceptual signal is _not_ the independent
variable of the perceptual input function. It is not _any_ variable of
the perceptual input function; it is neither an argument of the function
nor a value of the function.

...because the inputs are
adjusted so as to produce a specified value of the perception. The control
system produces outputs that generate ONE of the sets of inputs that will
create the specified value of perceptual signal. It doesn't matter to the
control system which set of inputs is present.

That's why the intermediate variables are irrelevant to an analysis of
the control system. There is _one_ variable in the environment that
corresponds to the perceptual signal. It has come to be given the label
"Controlled Complex Environmental Variable."

(I note later in your message that today you don't like the CCEV concept.
I can't help that, and I can't keep up with your rapid changes in what is
not acceptable PCT. I'll stick with slow changes, or changes that are
clearly demonstrable logically or technically. Whatever you call it, the
environmental state determined by the (complex) perceptual input function
has a value reflected in the perceptual signal. That environmental state
is the CCEV, or at least it has been over the last few years. And it
is _not_ the same as the perceptual signal, as you yourself pointed out
to Bruce Abbott when you introduced noise into the perceptual signal
to account for the decorrelation between perceptual and disturbance
signals.)

The inputs to the PIF are not the givens from which the value of the
perceptual signal follows.

Excuse me??? If the inputs of the function p = 3*x1 + 2*x2 are {1, 4},
the value of p cannot be computed? I'm afraid my poor S-R embogged brain
conceives the possibility that it can be, and will, when simulated, turn
out to be somewhere in the region of 11. More or less,though I admit
that Kafka wandering around through the Looking Glass might not be able
get this answer.

That's the S-R analysis. Because of the control
loop, it is the other way around: the desired value of the perceptual
signal is what determines the values of the inputs to the perceptual input
function.

Oh, come, now. The values of the inputs are _not_ determined by the
desired value of the perceptual signal. You've so often said so yourself.
Even as recently as the immediately preceding paragraph.

There is an infinity of sets of values that will produce the desired
value of the perceptual signal. In the external environment, only the
value of the CCEV is determined by the desired value of the perceptual
signal, and then only through the effective operation of a well structured
control loop that succeeds in bringing the perceptual signal to its
desired value.

···

----------------
Given what (unless your memory is really, really, going) you know about
what I know about control and PCT, could you explain what perception
you are trying to control to what reference value when you write stuff
like the following?

in a control loop, if you disturb x1, the resulting error
signal will cause x2 to be varied as well as x1, and you will get a
different pair of input values yielding the same perceptual signal. If you
like, I'll write a little model you can try to show how this works, but (a)
when you stop defending your conclusions I'm sure you will see how this
works, and (b) why can't you write your own damned programs?

And..

Martin, the CCEV is a figment of your imagination.

Oh, it is now, is it? I wonder what it will be in the next discussion?

There is no single
environmental variable that corresponds to the perceptual signal. There are
only the component input variables.

Which might they be? The individual impulses out of the individual rods and
cones in the eye, perhaps? A pulse out of an inner hair cell in the ear?

Let's not get _too_ ridiculous. Some others reading CSGnet might notice,
and might start wondering about the validity of some of the wonderful
and true things you so incisively write. That would be a shame. (No
smiley intended. I mean it).

The perceptual signal is not "complex." It is a simple scalar. There is no
CEV and no CCEV, except in the eye of the external analyst.

An interesting point. The subjective perception from inside the hierarchical
control system (me) that there are things "out there" is in the eye of
the external analyst. (Who? I'm not in therapy right now, in case you were
going to suggest my psychoanalyst:-).

Looked at from another point of view (the observer of the analyst), the
analysis of the control loop is the domain of the analyst's perceptions.
When we talk about _any_ entity in the control loop and discuss its
relation to the behaviour of the loop, we are taking the viewpoint of
the external analyst. From the viewpoint of the control loop (i.e.,
considering what happens at one point in the loop) there _is_ no
perceptual input function, no output function, and so forth. Those
are its own components. It doesn't perceive them. An external analyst
does. So if you want to say "there is no CCEV" except in the eye of
the external analyst, you should in fairness add "there is no CCEV, no
perceptual input function, no output function, no environmental feedback
function..." except in the eye of the external analyst.

In the world seen from the viewpoint of the control system, there is
only one input variable for which a state is accessible, and that is
the scalar variable called the Controlled Complex Environmental Variable.

But it is not "complex;" it is a simple scalar. Its relation to the
multiple lower-level perceptions of which it is a function is not visible
except to the external analyst.

Interesting how you can take my own statements and use them in a manner
implying criticism. I said almost precisely that, as a reason why one
should not be concerned with the multidimensional variations of the inputs
to the perceptual input function from lower-level perceptual signals. And
I went further, pointing out that they are of no concern _even_ to the
external analyst, unless the interactions among control systems is at
issue.

I do not care to continue to try to discuss matters using technical and
logical arguments in a world in which such arguments are invalid.

How would you know? You use "technical and logical" arguments in a way
ideosyncratic to you, and always with a goal in mind: proving that you're
right, by hook or by crook. The old Oxbridge Syndrome.

I leave others to form their own views on that. It is not my view.

I would note that your comment is itself neither a logical or
technical argument, nor a contribution to one.

But I suppose I do usually consider I am right when I have analyzed
a problem and satisfied myself with the answer, at least until somebody
shows, by logical or technical argument, that I am not. Attacks on my
motives don't really influence my understanding of the technology or
its underlying mathematics. But they do influence my willingness to
continue discussions on topics that might be controversial.

Martin

[From Bill Powers (980303.1327 MST)]

Martin Taylor 980303 11:20]--

Things have reached the shouting stage, so I guess I had better provide
that program and discuss its behavior:

The program, source code below, is called CTRLINPT.PAS or "control input."
We have here a perceptual signal p that is a function of two input
variables, x1 and x2. The input weights are ki1 = 1.0 and ki2 = 1.0.

The perceptual signal is therefore

p := ki1*x1 + ki2*x2

The two environmental variables are affected by the output o via two
environmental feedback functions ke1 and ke2, and are also affected by two
disturbing variables d1 and d2 through two disturbance functions kd1 and
kd2, so that

x1 := ke1*o + kd1*d1
x2 := ke2*o + kd2*d2.

Note that both input variables are jointly affected by the system's output
and by independent disturbances. For simplicity we use only one of the
disturbing variables here, although the other is in the program, set to
zero, for the sake of experimenting. The results will not be affected by
choosing different constants.

Since we're not concerned with dynamics, the control system is set up very
simply as an integrating-output system; it is run 1000 times for each value
of the disturbance and we record the final value each time a new
disturbance is applied.

In the forward direction, p is clearly a function (Fi) of x1 and x2.
However, x1 and x2 are both affected by the output of the system which is
in turn affected by p, so x1 and x2 are not independent variables.

The reference level of this system is set to 10. With an integrating output
function, the perceptual signal is brought very close to 10, regardless of
the size and direction of the disturbance d1. We observe that as d1
changes, both x1 and x2 change so that p remains close to 10.

Thus, as far as the input function is concerned, the values of x1 and x2
are the _inverse input function_ of the value of p or r (which are the
same); that inverse is necessarily multiple-valued, since p is a function
of two variables. X1 and x2, both dependent variables, assume values that
are determined by the inverse of the input function, the disturbance, and
the setting of the reference signal.

Here are the results when d1 is stepped from -5 to 5:

r= 10.000 p= 10.000 d1 = -5.000 x1 = 2.500 x2 = 7.500
r= 10.000 p= 10.000 d1 = -4.000 x1 = 3.000 x2 = 7.000
r= 10.000 p= 10.000 d1 = -3.000 x1 = 3.500 x2 = 6.500
r= 10.000 p= 10.000 d1 = -2.000 x1 = 4.000 x2 = 6.000
r= 10.000 p= 10.000 d1 = -1.000 x1 = 4.500 x2 = 5.500
r= 10.000 p= 10.000 d1 = 0.000 x1 = 5.000 x2 = 5.000
r= 10.000 p= 10.000 d1 = 1.000 x1 = 5.500 x2 = 4.500
r= 10.000 p= 10.000 d1 = 2.000 x1 = 6.000 x2 = 4.000
r= 10.000 p= 10.000 d1 = 3.000 x1 = 6.500 x2 = 3.500
r= 10.000 p= 10.000 d1 = 4.000 x1 = 7.000 x2 = 3.000
r= 10.000 p= 10.000 d1 = 5.000 x1 = 7.500 x2 = 2.500

Note that the perceptual signal remains the same, while x1 and x2 change
with the disturbance. Clearly, the _same_ perceptual signal is resulting
from _different_ values of x1 and x2, so there is no unique relationship
between the perceptual signal and the disturbance.

I may not have stated my meaning clearly in words, but I believe this
little program illustrates what I mean unambiguously.

Best,

Bill P.

P.S. RE the mathematics dictionary. The definition has changed since I
learned it. What your citation defines is what I learned to call a
_relation_; the word _function_ was reserved for relations involving only
one dependent variable.

···

--------------------------------------------------------------------------
program ctrlinpt;

uses dos, crt;

var x1,x2,ki1,ki2,ke1,ke2,kd1,kd2,d1,d2,p,r,e,o,gain,dt: real;
    i,j,k: integer;
    ch: char;
    outfile: text;

procedure initmodel;
begin
ki1 := 1.0;
ki2 := 1.0;
ke1 := 1.0;
ke2 := 1.0;
d1 := 1.0;
d2 := 0.0;
kd1 := 1.0;
kd2 := 1.0;
r := 10.0;
o := 0.0;
gain:= 1.0;
dt := 0.01;
end;

procedure controlsys;
begin
x1 := ke1*o + kd1*d1;
x2 := ke2*o + kd2*d2;

p := ki1*x1 + ki2*x2;
e := r - p;
o := o + gain*e*dt;
end;

begin
initmodel;
assign(outfile,'c:\junk.');
rewrite(outfile);
clrscr;
for j := 0 to 10 do
begin
  d1 := -5 + j;
  for i := 0 to 1000 do controlsys;
  writeln(outfile,'r= ',r:7:3,' p= ',p:7:3,' d1 = ',d1:7:3,
          ' x1 = ',x1:7:3, ' x2 = ',x2:7:3);
end;
close(outfile);
ch := readkey;
end.

The results will be found in the text file called "junk." in the root
directory of drive C.