Robustness (was What are we arguing about?)

CSGnet Archive CSG1999
1

[From Rick Marken (991228.0800)]

Me:

With this level 2 reference fixed, the spreadsheet was unable
to bring the level 3 perceptions into a match with 4 of the 64
possible combinations of level 3 references.

Bill Powers (991228.0726 MDT)

Excellent. However, let's give credit where it's due. It seems
to me that Martin predicted that with a single fixed second-
level reference signal, the third-level systems would be able
to continue controlling in 15/16 of the cases, or in other words
all but 4 of 64.

Yes. Nice going Martin!

Me:

So with respect to the issue we have been debating, which, as
you say [Bill Powers (991225.0221 MDT)], is "whether requiring
(or forbidding) a specific set of overt actions from a child is
likely to create internal conflict in the child, and therefore
likely to result in some form of resistance" the answer from the
hierarchcal spreadsheet model is (as I sai when this discussion began)
"yes".

Bill:

No, it is "no" if you mean by the word "likely" what other people
mean. Chances are 15 out of 16 that no internal conflict would be
created, which makes _that_ the most "likely" case.

Yes. Good point. I agree.

I think we all agree that the spreadsheet program is only a first
approximation to the real case. However, support for it will quite
likely increase now that it has proven someone _else_ right.

Yes. Excellent point!

Martin conjectured that the more systems that are involved at a given
level, the more robust the hierarchy will be against loss of a single
intermediate-level control system. I think your test vindicates his claim.
Try it with 12 systems at each level and see what happens!

Of course! Will do ASAP! Maybe there's a paper on the robustness of
hierarchical control lurking in this debate.

I would not yet conclude from these simulation-experiments that
causing children to adopt arbitrary reference levels (assuming that
is possible) would be harmless to the child. But we have to conclude
that in the spreadsheet world, it is not expecially harmful unless
the minority of cases in which a problem does occur are important
to the child.

Yes. But it should also be noted that the spreadsheet simulation
provides no evidence that causing children to adopt arbitrary
reference levels (assuming that is possible) would be _good for
the child_. I think some people were arguing that forcing kids to
adopt particular references was good for the kid (it made them
"responsible"). I think the spreadsheet simulation suggests that
this conceit (that one can improve kids by forcing them to adopt
particular goals) may, indeed, result in less harm to a kid (assuming
that a kid is a hierarchy of many control systems) than I had thought.

Best

Rick

···

--
Richard S. Marken Phone or Fax: 310 474-0313
Life Learning Associates mailto: rmarken@earthlink.net
http://home.earthlink.net/~rmarken

2

[From Rick Marken (991228.0930)]

Bill Powers (991228.0909 MDT)--

I'm not very concerned about forcing kids to adopt particular
references...

What I do object to is forcing kids to do things and then telling
them, or strongly implying, that you didn't force them to do anything.
This is especially ridiculous when you go around telling everyone that
you're teaching responsible thinking...

Yes. I agree.

What do you think of idea of testing the robustness of a hierarchical
model? I think the most general test would involve the 2 systems;
the level 2 perceptions are functions of _all_ the level 1 inputs.
The effect of "freezing" one level one output (comparable to a higher
level reference) could be measured in terms of expected versus actual
error in all second level systems. Then I could test the effect of
the number of level two systems on this ratio.

Best

Rick

···

--
Richard S. Marken Phone or Fax: 310 474-0313
Life Learning Associates mailto: rmarken@earthlink.net
http://home.earthlink.net/~rmarken

3

[From Bill Powers (991228.0909 MDT)]

Rick Marken (991228.0800)--

Yes. But it should also be noted that the spreadsheet simulation
provides no evidence that causing children to adopt arbitrary
reference levels (assuming that is possible) would be _good for
the child_.

That's a quibble; whether it's good for the child depends on exactly what
intervention is used. Causing a child to adopt an arbitrary (as far as the
child is concerned) reference condition of not playing in traffic could
reasonably be assumed to be good for the child, or less bad than the
alternative. And as always, the condition: assuming it is possible for an
external agency to alter a reference condition in the child. That is by no
means established as a fact, and is actually in doubt. You can tell a child
not to do something, meaning not to produce certain actions that have
effects to which you object. But what stopping those actions entails in the
way of altered internal reference conditions is unknown to the external
observer without a great deal of testing (which has never, so far, been done).

I think some people were arguing that forcing kids to
adopt particular references was good for the kid (it made them
"responsible"). I think the spreadsheet simulation suggests that
this conceit (that one can improve kids by forcing them to adopt
particular goals) may, indeed, result in less harm to a kid (assuming
that a kid is a hierarchy of many control systems) than I had thought.

I'm not very concerned about forcing kids to adopt particular references.
If you think you need to do that (and that you can), and if nobody has
demonstrated a better way to achieve the higher goal, then do what you
think you need to do. PCT doesn't say you shouldn't. It just tells you not
to be surprised if there's opposition.

What I do object to is forcing kids to do things and then telling them, or
strongly implying, that you didn't force them to do anything. This is
especially ridiculous when you go around telling everyone that you're
teaching responsible thinking. If the teacher won't take responsibility for
what he or she does, how does he or she expect to get the child to take on
similar responsibilities? If an adult says one thing but does another, the
child is going to learn the most from what the adult does, and develop
considerable cynicism about what the adult says. That's not PCT talking;
it's me, talking from experience.

Best,

Bill P.

4

[From Bill Powers (991228.1651 MDT)]

Rick Marken (991228.0930)--

What do you think of idea of testing the robustness of a hierarchical
model? I think the most general test would involve the 2 systems;
the level 2 perceptions are functions of _all_ the level 1 inputs.
The effect of "freezing" one level one output (comparable to a higher
level reference) could be measured in terms of expected versus actual
error in all second level systems. Then I could test the effect of
the number of level two systems on this ratio.

Good, I like the continuous measure. Total error squared would probably be
a good indicator, and maximum error in any system would be another.

Best,

Bill P.

5

[Martin Taylor 991228 20:03]

[From Bill Powers (991228.1651 MDT)]

Rick Marken (991228.0930)--

>What do you think of idea of testing the robustness of a hierarchical
>model? I think the most general test would involve the 2 systems;
>the level 2 perceptions are functions of _all_ the level 1 inputs.
>The effect of "freezing" one level one output (comparable to a higher
>level reference) could be measured in terms of expected versus actual
>error in all second level systems. Then I could test the effect of
>the number of level two systems on this ratio.

Good, I like the continuous measure. Total error squared would probably be
a good indicator, and maximum error in any system would be another.

If you would like another prediction, here's one. If you fix a single
level-one reference value (or cut off one level-one output) you will
get level-2 conflict. The same kind of thing will happen as with a
canonical conflict. The output of one or more (probably just two, but
maybe more) level-2 control units will increase without limit (these
being linear integrators). How this will affect the error values
depends on the balances among the weights in the system. If they are
finely balanced, it will be the same as in a canonical conflict--
essentially nothing will happen. But if they are not, the error in
at least one level-2 control system will increase until the errors in
the others approach zero.

Sorry, I haven't tried this or analyzed it properly. But intuitively
that's what I feel should happen. Could be way wrong; control systems
don't always work as one expects.

Martin

6

[From Bill Powers (991229.0235 MDT)]

Martin Taylor 991228 20:03--

[From Bill Powers (991228.1651 MDT)]

Rick Marken (991228.0930)--

>What do you think of idea of testing the robustness of a hierarchical
>model?

Martin:

If you would like another prediction, here's one.

I'm not sure of the value of making intuitive predictions once we are about
to perform an actual test. "Intuition" is not much of a basis, although
prior to actually testing models it can certainly help to suggest ways of
_eliminating_ hypotheses. I remember that when the first Ranger probes went
to the moon, there was a flurry of predictions about what they would find
just before the first flight, and I wondered what the point of them was,
since we were about to find out the truth. I finally concluded that this
was an academic game that people like to play. If your guess turns out
wrong you just quietly retire into the background and hope nobody brings it
up again (other players of the game will not do so, by tacit consent). On
the other hand, if it turns out to have been right (and with enough
different guesses being made, _somebody_ is bound to be right, or close
enough), the winner can modestly say it was just a guess, while broadly
hinting at some mysterious ability to divine the truth without the normal
need for observations, quantitative theories, and so on. I'm sure this is
not _your_ modus operandi, but it does sound like something we could be
aware of and avoid.

If you fix a single
level-one reference value (or cut off one level-one output) you will
get level-2 conflict.

Cutting off one level-2 output, since it goes to _all_ level-1 systems,
would instantly remove control of the variable that system controls; it
would not produce conflict because that system would be unable to disturb
any other systems at the same level. That is my prediction, although it's
not based on intuition. Perhaps you meant cutting off the connection
between one level-2 control system and a single level-1 reference input.

What would be your prediction concerning a "rotated" logic-level control
system in which every perceptual signal was a function of all lower
variables (each treated as being above or below some fixed threshold)? Or
shall we simply wait and see? I'm not as confident today, by the way, that
I know what I mean by "rotation" in the logical case.

I should point out that pure analog neural integrators do not exist in a
real brain; they are all leaky, if only because of "coincidence losses".
This means that gain is finite, and so conflict is not inevitable unless
contradictions between control systems are fairly direct. This means, as
usual, that we do not have an either-or case; conflict is not something
that either exists or does not exist. Instead, there are degrees of
conflict, with the total amount of system error increasing as the
conditions of orthogonality are violated more and more. In control systems
with a loop gain of 200, those conditions don't have to be violated very
much to result in saturated output, which destroys control. But for systems
with lower gains, there is more tolerance of conflict (and of course errors
are larger even when control continues).

I await the results of Rick's experiments with interest, and make no
predictions.

Best,

Bill P.

7

[Martin Taylor 991219 15:43]

[From Bill Powers (991229.0235 MDT)]

I'm not sure of the value of making intuitive predictions once we are about
to perform an actual test.

if it turns out to have been right (and with enough
different guesses being made, _somebody_ is bound to be right, or close
enough), the winner can modestly say it was just a guess, while broadly
hinting at some mysterious ability to divine the truth without the normal
need for observations, quantitative theories, and so on. I'm sure this is
not _your_ modus operandi, but it does sound like something we could be
aware of and avoid.

The point is well taken.

>If you fix a single
>level-one reference value (or cut off one level-one output) you will
>get level-2 conflict.

Cutting off one level-2 output, since it goes to _all_ level-1 systems,
would instantly remove control of the variable that system controls; it
would not produce conflict because that system would be unable to disturb
any other systems at the same level. That is my prediction, although it's
not based on intuition. Perhaps you meant cutting off the connection
between one level-2 control system and a single level-1 reference input.

Actually, the parenthesis was a bit stupid, wasn't it? To cut off an
output means, as you said, that the higher (level-2, now) system won't
influence the level 1 systems at all, and therefore won't disturb the other
5 systems. The "cut-off" output will increase without limit, since it's
doing the same as trying to push down a brick wall, while the othe 5
level-2 systems have 6 degrees of freedom to help them control.

Fixing one level-1 reference does the opposite. It makes the 6 level-2
systems try to work through 5 level-1 degrees of freedom. What was
"intuitive" about my guess was whether it would be one, two, or all of
the level-2 systems that would have ever-increasing output and sustained
error, because I think what happens probably depends on the weightings
in the linkages up and down and through the environment, and perhaps on
the order in which the spreadsheet computes. I wouldn't call it intuitive
that at least one of the level 2 systems will experience increasing error.

What would be your prediction concerning a "rotated" logic-level control
system in which every perceptual signal was a function of all lower
variables (each treated as being above or below some fixed threshold)? Or
shall we simply wait and see? I'm not as confident today, by the way, that
I know what I mean by "rotation" in the logical case.

I started to answer the message in which you brought up this issue, and
I'm still working on it. I thought I had an answer, but when I thought
more about it, it was a wrong answer. So I think I will post the first
half, which deals with the degrees of freedom issue, and not deal with
the "logical rotation" qustion until I can say something useful about it.

I should point out that pure analog neural integrators do not exist in a
real brain; they are all leaky, if only because of "coincidence losses".
This means that gain is finite, and so conflict is not inevitable unless
contradictions between control systems are fairly direct.

I think this needs a little expansion. At this moment, I don't understand
what the value of the gain has to do with whether systems trying to
control through fewer degrees of freedom than are available has to do
with whether they conflict.

In control systems
with a loop gain of 200, those conditions don't have to be violated very
much to result in saturated output, which destroys control.

I do understand the implications of non-linearity of this kind, though.
What it means is that the saturated one is no longer taking up one of
the lower-level degrees of freedom, leaving that df free for non-saturated
ones. It's the same thing as happens if there is a "dead-zone" of zero
gain near zero error (i.e. if there is a non-linear comparator that
outputs "zero-error" when r-p is less than some threshold). That's one
way of getting moderately good control at higher levels when there are
too few lower-level degrees of freedom.

8

[From Bill Powers (991229.1456 MDT)]

Martin Taylor 991219 15:43--

I think this needs a little expansion. At this moment, I don't understand
what the value of the gain has to do with whether systems trying to
control through fewer degrees of freedom than are available has to do
with whether they conflict.

For general interest, Kent McClelland's paper on conflict (including
simulations) is at

http://www.grinnell.edu/sociology/ccp.html

Conflict does not disable control systems unless it demands from them more
output than they can produce. That is more likely to happen with high-gain
control systems than with low-gain systems. Even if the "axes" along which
two high-gain (or integrating) control systems operate are 89 degrees away
from orthogonal, that remaining 1 degree allows the two systems to control
independently -- albeit by using HUGE and almost-cancelling outputs. When
realistic limits are put on the outputs, the huge outputs do not occur, and
one or both control systems loses control. Kent explores some of these
properties of conflict in his paper.

Interestingly, Kent found that conflicted control systems can resist
disturbances and control quite well around a virtual reference level. This
is easily understood after the fact, of course -- but it wasn't obvious
until the simulations were done.

Best,

Bill P.

9

[Martin Taylor 9912 29 17:20]

[From Bill Powers (991229.1456 MDT)]

Martin Taylor 991219 15:43--

>I think this needs a little expansion. At this moment, I don't understand
>what the value of the gain has to do with whether systems trying to
>control through fewer degrees of freedom than are available has to do
>with whether they conflict.

Conflict does not disable control systems unless it demands from them more
output than they can produce.

The issue isn't whether the control systems are disabled, but whether they
can bring their perceptions to their individual reference levels.

That is more likely to happen with high-gain
control systems than with low-gain systems. Even if the "axes" along which
two high-gain (or integrating) control systems operate are 89 degrees away
from orthogonal, that remaining 1 degree allows the two systems to control
independently -- albeit by using HUGE and almost-cancelling outputs.

In that case, there are still 2 degreed of freedom for control. There is
mutual disturbance, but no conflict--at least not in the sense that the
number of available degrees of freedom is too small. The gain as such
only determines how quickly the integration builds the output, assuming that
the output functions are pure integrators. It doesn't determine whether
there is conflict, or even whether there is mutual disturbance.

Interestingly, Kent found that conflicted control systems can resist
disturbances and control quite well around a virtual reference level.

But not bring all their perceptions to their individual reference levels.

This
is easily understood after the fact, of course -- but it wasn't obvious
until the simulations were done.

Simulations are getting to be a major source of insight in many sciences.
The problem is of knowing what about the simulation results can be
generalized, and to what that aspect can be generalized. You've pointed
out several issues of that nature during the discussions about the
spreadsheet.

Talking of that, I was surprised that Rick (991227.1610) found 15/16 of
the sets of reference values could be achieved by the perceptions when he
fixed one of the level 2 references, since my (corrected) analysis
suggested that 15/16 would be true only for one specific fixed level 2
reference (F7 in the spreadsheet). If I7 were fixed, that analysis
suggested that half the sets should be achievable. If it were H7, 3/4;
G7, 7/8; E7, 31/32, and D7, 63/64.

Unless F7 was the one Rick fixed, my "correction" was apparently wrong.

Martin

10

[Martin Taylor 991230 01:44]

[Martin Taylor 9912 29 17:20]

Talking of that, I was surprised that Rick (991227.1610) found 15/16 of
the sets of reference values could be achieved by the perceptions when he
fixed one of the level 2 references, since my (corrected) analysis
suggested that 15/16 would be true only for one specific fixed level 2
reference (F7 in the spreadsheet). If I7 were fixed, that analysis
suggested that half the sets should be achievable. If it were H7, 3/4;
G7, 7/8; E7, 31/32, and D7, 63/64.

Unless F7 was the one Rick fixed, my "correction" was apparently wrong.

Since I wrote that, Rick sent me his "testing" spreadsheet with the macro
that tests all different possible combinations of level 3 reference values.
I used it to try what happens when you fix each of the level 2 references
one at a time. Here are the results--number of unachievable sets when the
level 2 reference value in question is fixed

fixed unachievable
I7 -- 32
H7 -- 16
G7 -- 8
F7 -- 4
E7 -- 2
D7 (3 runs) -- 2, 1, 2.

Apart from D7, these numbers are as the analysis of 991222 23:30
indicated they should be.

What happened when D7 was fixed? I was a bit puzzled with the first run
that had 2 unachievable sets rather than the expected 1. Then I realized
that in the case of fixing I7 something similar can happen, and probably
it can happen as well for all the others. If I7 is fixed at 5, the
corresponding perceptual signal I8 fluctuates above and below 5 as it is
disturbed. The third-level perceptual function is (I8 > 5). If I8 is
fluctuating randomly above and below 5, the third-level perceptual signal
fluctuates equally randomly between false and true, regardless of what
its reference value is. The criterion Rick has used to test whether
third-level control is good is that the perception should match its
reference value for a reasonably long time. If I8 is set to a number
far enough from 5, it can do this (or conversely it can consistently
fail to match its reference). So the number of failures detected by the
spreadsheet macro will fluctuate, usually being above 32, when I8 is
fixed at 5.

The same thing can happen for all the others, since they are linked
to I8 by a chain of >, >...>5. If the gap between a fixed reference
level and 5 is inadequate, any or all of the intermediate perceptions
may be able to match their references most of the time, but be driven
away by occasional large disturbances. And this is what I observed for
the third of the D7-fixed runs when I watched the "all true" set of
reference values. Most of the time, all the perceptions were "true",
but every now and then one or other would momentarily go "false" and
this was enough for the spreadsheet to assert failure of control.

What I thought I might do next is in the direction of what Bill P has
been suggesting--making the six level 3 perceptions depend on all the
level 2 perceptions. I might try a set of six different weighting
functions that would be orthogonal if we were dealing with analogue
perceptions,
and see what happens. With such a set of functions, there wouldn't be
any chains of linkages among the functions, and the effects of fixing
any one level 2 reference ought to be symmetric. Who knows if it will
work? Will all 64 different reference sets be achievable when all level
2 references are fixed?

I don't know how much more relevant this may be to living control systems,
but it's interesting to try, if only as a _slightly_ mor generalizable
condition.

Martin

11

[From Rick Marken (991230.0800)]

Martin Taylor (9912 29 17:20)

Talking of that, I was surprised that Rick (991227.1610) found
15/16 of the sets of reference values could be achieved by the
perceptions when he fixed one of the level 2 references, since
my (corrected) analysis suggested that 15/16 would be true only
for one specific fixed level 2 reference (F7 in the spreadsheet).

Bill Powers (991230.0758 MDT)

Rick, how about trying some other level-2 references to fix? The
simulation may have to run all night, but another macro ought to do
the trick. I trust that it's possible to print out or otherwise save
automatically the results.

I'm working on something else rather intently at the moment. I should
have it finished up this evening so I'll do the tests tomorrow. It only
takes a few minutes of run time on the machine here at work. I did
quickly test a couple of Martin's predictions and he seems to be spot
on; my finding of 15/16 apparently did occur when I fixed F7. When I
fixed
I7, half the sets were achieved (as Martin predicted) and when I fixed
D7 only one out of 64 references was achieved (again as Martin
predicted).
I think I see a trend;-)

Best

Rick

···

---
Richard S. Marken Phone or Fax: 310 474-0313
Life Learning Associates mailto: rmarken@earthlink.net
http://home.earthlink.net/~rmarken

12

[From Bill Powers (991230.0758 MDT)]

Martin Taylor 9912 29 17:20 --

Bill:

Conflict does not disable control systems unless it demands from them more
output than they can produce.

Martin:

The issue isn't whether the control systems are disabled, but whether they
can bring their perceptions to their individual reference levels.

Bill:
When I say a control system is disabled, I mean that it is unable to brings
its perception to its reference level.

Bill:

Even if the "axes" along which
two high-gain (or integrating) control systems operate are 89 degrees away
from orthogonal, that remaining 1 degree allows the two systems to control
independently -- albeit by using HUGE and almost-cancelling outputs.

Martin:

In that case, there are still 2 degreed of freedom for control. There is
mutual disturbance, but no conflict--at least not in the sense that the
number of available degrees of freedom is too small. The gain as such
only determines how quickly the integration builds the output, assuming that
the output functions are pure integrators. It doesn't determine whether
there is conflict, or even whether there is mutual disturbance.

Bill:
Right. But how quickly the integration builds the output determines how
quickly the output will reach its limit. When a disturbance (from any
source including another control system) demands the maximum possible
output, any further increase in the disturbance will simply produce more
error, and will be unopposed.

Bill:

Interestingly, Kent found that conflicted control systems can resist
disturbances and control quite well around a virtual reference level.

Martin:

But not bring all their perceptions to their individual reference levels.

Bill:
Yes, that's right.

Martin:

Talking of that, I was surprised that Rick (991227.1610) found 15/16 of
the sets of reference values could be achieved by the perceptions when he
fixed one of the level 2 references, since my (corrected) analysis
suggested that 15/16 would be true only for one specific fixed level 2
reference (F7 in the spreadsheet).

Bill:
Rick, how about trying some other level-2 references to fix? The simulation
may have to run all night, but another macro ought to do the trick. I trust
that it's possible to print out or otherwise save automatically the results.

Best,

Bill P.

13

[From Bill Powers (991230.0808 MDT)]

Martin Taylor 991230 01:44--

What happened when D7 was fixed? I was a bit puzzled with the first run
that had 2 unachievable sets rather than the expected 1. Then I realized
that in the case of fixing I7 something similar can happen, and probably
it can happen as well for all the others. If I7 is fixed at 5, the
corresponding perceptual signal I8 fluctuates above and below 5 as it is
disturbed. The third-level perceptual function is (I8 > 5). If I8 is
fluctuating randomly above and below 5, the third-level perceptual signal
fluctuates equally randomly between false and true, regardless of what
its reference value is. The criterion Rick has used to test whether
third-level control is good is that the perception should match its
reference value for a reasonably long time. If I8 is set to a number
far enough from 5, it can do this (or conversely it can consistently
fail to match its reference). So the number of failures detected by the
spreadsheet macro will fluctuate, usually being above 32, when I8 is
fixed at 5.

Bill:
The question here is whether these effects reflect transient effects of
changing disturbances, or instabilities. If the effects are transient,
applying (and holding) or removing (and leaving removed) a disturbance
might cause some temporary errors, but if you wait long enough the system
should settle to a constant state in all systems at all levels, with all
errors small. If, however, we are seeing instability, the system will never
settle down -- values will keep changing forever. Errors might temporarily
go to zero (or small values), but they won't stay there. In that case I
would say that the system has lost control. At the logic level, all the
errors must be stable at zero after a disturbance has been applied and held
for a sufficient time, since logical errors can be only 1 or 0. If any
perceptual signal continually oscillates between 1 and 0, the corresponding
system is not controlling.
....
Martin:

What I thought I might do next is in the direction of what Bill P has
been suggesting--making the six level 3 perceptions depend on all the
level 2 perceptions. I might try a set of six different weighting
functions that would be orthogonal if we were dealing with analogue
perceptions,
and see what happens. With such a set of functions, there wouldn't be
any chains of linkages among the functions, and the effects of fixing
any one level 2 reference ought to be symmetric. Who knows if it will
work? Will all 64 different reference sets be achievable when all level
2 references are fixed?

Bill:
Bravo. Way to go. I await the next installment with interest.

Best,

Bill P.

14

[Martin Taylor 991230 13:15]

[From Bill Powers (991230.0808 MDT)]

>What I thought I might do next is in the direction of what Bill P has
>been suggesting--making the six level 3 perceptions depend on all the
>level 2 perceptions. I might try a set of six different weighting
>functions that would be orthogonal if we were dealing with analogue
>perceptions,
>and see what happens.

Bravo. Way to go. I await the next installment with interest.

I was not able to find a set of six orthogonal functions each of which
involved all of the second-level perceptions in a single inequality
while giving them all the same weight in each function. I don't think
such a set of functions exists, any more than there are orthogonal
diagonals in an ordinary 3-D cube. But I've done the next best thing,
and generated a set of nearly orthogonal functions, as follows:

+ + + + + + (meaning A + B + C + D + E + F > 0)
- - - + + +
+ - - - + +
- + + - + +
+ - + + - +
- + - + - + (meaning -A + B - C + D - E + F > 0)

Taking + to mean +1 and - to mean -1, the rows are the weights of the six
level 2 preceptions in the six level 3 perceptions. The columns are the
weights of the six level 3 outputs in the six level 2 references (i.e.
the fourth level 2 reference (G7 in the spreadsheet) is the sum of
the first two and last two level 2 outputs minus the sum of the third
and fourth level 2 outputs). That pattern ensures each control unit at
level 3 individually has negative feedback.

I've sent a copy of this revised spreadsheet to Rick so that he can do
his own tests with it if he wants. What I've done so far is show that
all 64 sets of level 3 references are achievable, but if you fix reference
D7 to be zero, there are 8 failures in the 64. I'm going to try fixing
each of the level 3 references singly, in turn, but not until tonight or
tomorrow, as I'm out for the rest of the day.

Martin

15

[From Bill Powers (991231.0152 MDT)]

Note that tomorrow, the year will be 7D0 hexadecimal, or 3720 octal. Big deal.

from Rick Marken (991230.0800)--

I did
quickly test a couple of Martin's predictions and he seems to be spot
on; my finding of 15/16 apparently did occur when I fixed F7. When I
fixed I7, half the sets were achieved (as Martin predicted) and when I fixed
D7 only one out of 64 references was achieved (again as Martin
predicted).

All is well with the world again. Different observers independently
recreating and observing the same phenomenon report the same results,
regardless of disparities in what they initially believed. Surely this
argues that there are observer-independent truths, that this is one of
them, and that there are ways to discover them.

Best,

Bill P.