Sequency vs. Simultaneity

[From Bill Powers (940413.0900 MDT)]

Martin Taylor (940412.1800) --

If the past values of p are (as they in fact are) still
reflected in o, how can you say the function does not give
previous values of p an effect on present values of o?

I'm trying to emphasize that the workings of a control system (or
any physical system) are best expressed in terms of present-time
operations. Even in a transport lag, the values of the lagged signal
that produce effects are those existing in the mechanism right now,
with past values having disappeared forever and being incapable of
having any effects in the present.

Consider a transport lag physically implemented as an analog shift
register like a charge-coupled device or an axon. In the register at
a given time is a set of stored values v of the input:

input value ----> v5 v4 v3 v2 v1 v0 ----> output value

The input value right now is v5, the output value right now is v0.
After the next shift we have

input value ----> v6 v5 v4 v3 v2 v1 ----> output value

Now the input value is v6 and the output value is v1. The value v5
still exists, having been moved into the second slot from the left.
The value v0 no longer exists; it can have no further effects. The
value v7 does not yet exist.

This is not, by the way, a convolution; it is only a transport lag.
A convolution is implemented this way:

input value ----> v6 v5 v4 v3 v2 v1 ---> shifts
> > > > > >
weightings--> w1 w2 w3 w4 w5 w6
> > > > > >
SIMULTANEOUS SUMMATION
>
output value

Again, the only effects are present-time effects. The past never
affects the present.

In an integration, there is a cumulative summation of inputs:

Input Sum
0 (initial state)
1 1
3 4
7 11
2 13
-1 12
----> -5 7
-3 4
-4 0
... ...

At the time shown by the arrow, the value of the sum before the new
input arrives is 12, and afterward is 7. From the previous value of
12, there is no way to tell what the input was at any still earlier
time; that information is lost. The same sum of 12 could have been
produced by any number of different past values of the input. So
even though the 12 does represent effects left over from past values
of the input, it is not a unique indicator any one past value. All
that is carried forward through time is the current sum. The only
value that can have any effect on a subsequent process is the final
sum of 0, which will change simultaneously with the next input. As
it happens, in the example that final value is the same as the
initial state of the integrator; the situation as far as the
integrator is concerned is just as if none of the intervening values

This is what is meant in physics by saying that the history of a
system is of no importance in determining its future behavior. If
you know all the values of variables at a given time, and all the
derivatives, it makes no difference from then on how those variables
and their derivatives got into those states. It is only the
_present_ state of the system that determines what will happen next.

Simulations necessarily have this character, because the only values
of variables that figure into the next iteration are those that
exist, somewhere, right now. The next state of a simulated system is
completely determined by its present state, and only by its present
state.

This principle is not adhered to in some approaches. A predictor-
corrector integrator, for example, uses both past and future values
of a variable as part of computing its integral at a given time.
Analytical approaches in general are free to take information from
any place in a data record, because they treat the entire data set
as if it is all equally available at all times. This is how
statistical approaches work; they treat the entire data set as a
static entity, so that, for example, in computing conditional
probabilities the bins can be filled in any sequence. While I'm not
sufficiently adept mathematically to justify my feeling, I feel that
this can lead to spurious results, if only because averages or sums
computed over a whole data set can't possibly have been correct for
a point in that data set in the middle, where later values had not
yet occured.

ยทยทยท

-----------------------------------------
The way you talk about control systems involves ranging freely
backward and forward through time, and seems to me undisciplined.
The idea that a control system somehow uses perceptual information
in a way conditional on its probable future effects on the
controlled quantity is precisely the kind of proposal that made
scientists justifiably reject the concept of purpose. PCT explains
purpose strictly in terms of a real physical system that operates
only with present-time information. In analyzing a control system
there is NEVER a need to speak of predicting the future effects of a
variable; there is NEVER a need to explain behavior in terms of a
past value of a variable that no longer exists.

There have been numerous abortive attempts to work control theory
into mainstream ideas about behavior. Most of those have failed
because, in one way or another, they insisted on preserving a
sequential view of behavior. This sequential view fails precisely
because it fails to show that all parts of a control loop are in
operation at the same instant. In large part, this failure comes
from the tradition of trying to understand behavior as a sequence of
events, a succession of states that proceeds in jumps from one
frozen condition to another discretely-different frozen condition.
Only from this point of view can you speak of a "change" in a
variable like the CEV as if it could occur independently of the
behavior of the other variables in the system that are always
determining the state of the CEV. It leads to thinking of "typical"
inputs as impulses or step-functions -- functions that are designed
to cause changes abruptly at a given instant, before anything else
can interfere.
---------------------------------------

The p that you substitute from p = o + d is identically the p
in do/dt = k*(r - p), save for whatever transport lags are
present in the input function and comparator.

It is that, but the "o" in the differential equation doesn't >plug

into the other equation. It isn't a straight

substitution.

That's true only if you think that there is no output until an
effect of the input gets to it. This is the sequential view of the
system. In fact, it is the _current_ value of output that adds to
the _current_ value of disturbance to create the _current_ input to
the perceptual function. It is the _current_ value of perceptual
signal, now being generated by that function, that enters the
comparator. The CURRENT value of error is entering the output
function at the same time that the CURRENT value of output variable
is coming out of it. ALL the values of variables everywhere in the
loop are the CURRENT values.

What the functions do is to explain how those current values are
being generated out of inputs to the functions. If the system
equations don't give the right answers when you plug in all the
current values at the same time, then the functions are wrong and
need to be changed. When you have the right functions, a
simultaneous plot of all the variables in the model will match a
simultaneous plot of all the accessible variables in the real
system.

If I think of three more ways of saying this, I don't suppose you
will agree any more readily than you will now. The whole point of
speaking this way is to avoid the trap of trying to trace individual
events around and around the loop. The whole loop is always in some
state, and that state is changing with time. Transport lags and
integral lags are irrelevant; they do not alter the fact that the
whole loop operates exactly simultaneously. At a given instant, you
may find the effect of a previous input existing half-way through
the physical process that constitutes a transport lag -- but at that
same instant the input to the lag has one value and the output from
it has another. The system is not behaving according to the value
that is halfway through the transport lag, but only according to the
value presently at its output.
----------------------------------------------
Ursula Le Guin wrote what is to my mind one of the greatest science-
fiction stories ever written: The Dispossessed. Embedded in the
story was a conflict between two major world-views, which were
called Sequency (the official view) and Simultaneity, the view of
the principal character, Shevek. I think Le Guin was reading my
mind. I see the struggle between the modeling view exemplified by my
approach to PCT and certain traditional approaches as being over
exactly this point: whether we are to understand physical systems as
having overall shapes that evolve through time in an eternal now, or
as consisting of punctuate events which cause future punctuate
events along a line in which both past and future have real
existence.
---------------------------------------------------------------
Best,

Bill P.

<Martin Taylor 940413 13:50>

Bill Powers (940413.0900 MDT)

Ursula Le Guin wrote what is to my mind one of the greatest science-
fiction stories ever written: The Dispossessed. Embedded in the
story was a conflict between two major world-views, which were
called Sequency (the official view) and Simultaneity, the view of
the principal character, Shevek. I think Le Guin was reading my
mind. I see the struggle between the modeling view exemplified by my
approach to PCT and certain traditional approaches as being over
exactly this point: whether we are to understand physical systems as
having overall shapes that evolve through time in an eternal now, or
as consisting of punctuate events which cause future punctuate
events along a line in which both past and future have real
existence.

Your differences with me are not on this point. We have exactly the same
view on it, so far as I can see. Only the present values exist now, and
only the present values are used now. I love that phrase "overall shapes
that evolve through time in an eternal now." Exactly the way I see it,
but had never expressed it so felicitously.

That doesn't mean that the values at one point in a system NOW can be
determined from the values at any other point in the system NOW. In fact,
in a physically realizable system, the current values at any point
are physically INDEPENDENT of the current values at any other point. The
value at a point NOW can relate only to what an observer would see as past
values at other points in the loop.

If you try to analyze the behaviour of any interconnected system and deal
physically, though it might by chance be close numerically, if the variables
change slowly enough. It will be wrong physically because none of the
values you have measured will have had any effect on any of the other values,
and the system you will have analyzed would be physically a disconnected
one, not the interconnected one you thought you were analyzing.

Only an outside observer can see all the points in an interconnected system
at once. It takes some time for events at point A to affect an observer
at point O. You can talk about the NOW value of a signal at point A only
in the observer's NOW, at O. If the observer is careful, it can be arranged
that there is the same time lag for observations of events at points X, Y,
and Z. Then something NOW at O would have happened at the same clock time
at X, Y, and Z. This cannot be so if the observer is at any point within
the system observed. There would have been no effect at Y of what happened
at X, even though the observer at O outside the observed system saw them
both as NOW. At Y, what is seen as NOW at X is something that the observer
at O would have seen as happening in X's past.

O
^^^
/ | \
signals / | \
/ | \
/ | \
X-->--Y-->--Z

This is what is meant in physics by saying that the history of a
system is of no importance in determining its future behavior. If
you know all the values of variables at a given time, and all the
derivatives, it makes no difference from then on how those variables
and their derivatives got into those states. It is only the
_present_ state of the system that determines what will happen next.

Right. But you have to remember that effects take time to propagate,
as we discussed in a private interchange a couple of weeks ago on
relativity theory. What is happening NOW depends on where you are
looking from. And what an observer sees as happening NOW all around
a loop is indeed what determines the future of the signals in the loop,
though nothing that is seen as happening NOW at any point has yet had
any effect at any other point.

The way you talk about control systems involves ranging freely
backward and forward through time, and seems to me undisciplined.

I see no ranging backward and forward through time in what I have been
trying to say. And I think your way of dealing with present values as
if they could influence each other may not be undisciplined, but is
definitely unphysical.

PCT explains
purpose strictly in terms of a real physical system that operates
only with present-time information.

That's a general physical or engineering approach. I think we all
would go along with it, unless we are practicing mystics.

In analyzing a control system
there is NEVER a need to speak of predicting the future effects of a
variable; there is NEVER a need to explain behavior in terms of a
past value of a variable that no longer exists.

But this, on the other hand, is theology, not physics. You may easily
assert that YOU have never found a need, but these statements do not
themselves follow from the previous arguments on using present values.
Present values were generated from past values, which may be retained in
memory uniquely, may be part of the value of an integral, may remain
inside a delay line, and so forth. As such, they either are, or are
components of, present values. It may well be useful to take advantage
of them in explaining behaviour. Likewise with the predicted future.
The future isn't in the present, but the predicted value is, and can
affect behaviour in just the same way as any other present value can.
A predicted probability likewise is a present value, and only a present
value, since the future to which it refers can, by its nature, never happen.
So to say that _there is_ never a need is a theological statement, an
absolutist regulation, not a statement about the physics or mathematics
of control. Say "I see" no need, and I will have no quarrel with you.

===============start digression=======
(Incidentally, I'm sure you were well aware that I would disagree with
you about probability, and I presume you wrote the following in order to
provoke a reaction. I note it here, to show that your attempt did not go
unnoticed, but I don't think it valuable to reprise the discussion).

This is how
statistical approaches work; they treat the entire data set as a
static entity, so that, for example, in computing conditional
probabilities the bins can be filled in any sequence.

================end digression==========

This sequential view fails precisely
because it fails to show that all parts of a control loop are in
operation at the same instant. In large part, this failure comes
from the tradition of trying to understand behavior as a sequence of
events, a succession of states that proceeds in jumps from one
frozen condition to another discretely-different frozen condition.

Right. I've always been puzzled by people who have to think this way,
since decades before I ever heard of PCT.

Only from this point of view can you speak of a "change" in a
variable like the CEV as if it could occur independently of the
behavior of the other variables in the system that are always
determining the state of the CEV. It leads to thinking of "typical"
inputs as impulses or step-functions -- functions that are designed
to cause changes abruptly at a given instant, before anything else
can interfere.

Wrong. All events NOW do in fact occur independently of anything else that
is NOW elsewhere. In the future, those things that are NOW elsewhere will be
NOW here, and will have whatever effect they may have. But not now.

A change in the CEV, for example, occurs because NOW there is an effect
of some prior output (possibly extended over long prior time) at the CEV,
together with an effect NOW of possibly many disturbing variables that
did something at a variety of prior times. All of these are NOW at
the CEV, but not elsewhere in the loop. Later, they will be NOW at other
points in the loop, but not in a stepwise manner. They will be NOW
elsewhere, as the effects propagate, and not sooner.

As for "typical" inputs being step functions, step functions and the like
are useful in discussion and in analysis, because they are readily reduced
to infinitesimal dimensions (in linear systems) and are the way elementary
calculus concepts are usually introduced. By considering the integrated
effect of an infinite number of infinitesimal steps, you get any other
waveform at all, continuous or discontinuous. You can do this with other
basic functions, but steps are very convenient and simple for most purposes,

Notice that your own wording "before anything else can interfere" is a
somewhat undisciplined mixture of present and future in the analysis.

The p that you substitute from p = o + d is identically the p
in do/dt = k*(r - p), save for whatever transport lags are
present in the input function and comparator.

It is that, but the "o" in the differential equation doesn't plug
into the other equation. It isn't a straight
substitution.

That's true only if you think that there is no output until an
effect of the input gets to it. This is the sequential view of the
system.

How can you say that with (I assume) a straight face? The problem has
nothing to do with whether there is output due to prior effects. It has
to do with a simple mathematical fact:

If do/dt = x(t), then

o = integral(x(t) dt) + constant.

That constant is what makes the two "o"s different. In the differential
equation, any constant whatever will give the same result, whereas in
p = o+d, the constant is a critical part of the equation. Change it,
and you change p. I believe I did mention that this was the reason why
you couldn't plug one into the other, but if I forgot, it was not hard
to deduce.

In the integral form of the equations, this problem does not occur, but
then the two p values are different, as I have tried to show in a
straightforward way.

It has nothing to do with when effects occur.

If I think of three more ways of saying this, I don't suppose you
will agree any more readily than you will now. The whole point of
speaking this way is to avoid the trap of trying to trace individual
events around and around the loop. The whole loop is always in some
state, and that state is changing with time.

The trouble is that I DO agree, and never have disagreed with this point.
A lot of our difficulty in communication is that on several issues you
complain I take no notice of what you try to convince me of, when all along
I have taken as true the thing you go on about, and use it as a basis for
further discussion.

The further trouble is that you don't take the implications of seeing the
loop as a whole using only present values, seriously, but slough them off
as irrelevant trivia. The system of algebraic analyses works exactly at zero
frequency, in the stable limit, if everything is slow enough... Sure it
does. And under those conditions, transport lags and the like truly don't
matter. How could they, if nothing is allowed to change?

If one treats the effects ONLY in the present, as effective at each point in
the loop rather than at the external observer's viewpoint, one will get
it right. This has the result of following events around the loop, but
they are not _individual_ events, since all points of the loop are equally
important at all times.

The system is not behaving according to the value
that is halfway through the transport lag, but only according to the
value presently at its output.

Precisely. Deal with the implications of this, and I think we will find
ourselves in agreement.

Martin