Bomb in the Hierarchy Simulation (was Back in control)

Re: Bomb in the Hierarchy Simulation (was Back in
control)
[Martin Taylor 2005.02.22.0929]

[From Bill Powers (2005.02.19.1604
MST)]

Martin Taylor 2005.01.19.15.59–

They are only numbers, whatever the
external analyst might think they represent. The one level doesn’t
know it controls the amount of sensation. It only controls the value
of a perceptual variable. Likewise for the level that the external
observer thinks is controlling the shape of a configuration. It’s just
another value of a scalar variable. The control system doesn’t
“know” the meaning of what it controls.

No, but the computations that take place determine that meaning. Each
level of perception computes invariants from the level below, so if a
particular perception at the level below somehow goes out of control,
there is no reason to suppose that anything at the next level will
also go out of control. Perceptions at a given level are functions of
many at a lower level, not just one.

That last was exactly the point I was making, wasn’t it? That if
one of the inputs ran away, there was a possibility that the
higher-level system might not be able to sustain control by means of
its other inputs?

You say “there is no reason to suppose that anything at the
next level will also go out of control.” Of course there isn’t.
But there’s also no reason to suppose that nothing at the next level
will go out of control.

The Bomb is (necessarily in a functioning hierarchy) something
that happens only rarely, and that rarity is increased the larger the
explosion under consideration. Trivially, assuming equal probabilities
everywhere, if the probability of it propagating through one level is
p, then the probability of it propagating throung n levels is
p^n.

(Of course that probability will be different for different
control units, whether they are at the same level or at different
levels, but the same principle applies; it would be interesting,
though, to determine whether in a heterogeneous hierarchy, the Bombs
follow paths of least resistance in the way landslides and snow
avalanches do).

Furthermore, each level introduces new
information. The shape of a cube is computed from where
sensations occur in the visual map, not on the magnitudes of the
sensations, and the whereness is not indicated in the magnitude of any
sensation signal. Sensations must be controlled to change whereness,
but the magnitude of a given sensation is not the critical variable.
The spatial relationship between two objects is left unchanged if both
objects move in the same way, or if they change brightness or
orientation or color. A runaway magnitude at one level does not
necessarily imply a runaway magnitude at the next level. There can,
perhaps, be special cases where that link between levels might exist,
but as a general rule I don’t think it does.

I illustrated the Bomb algebraically in a linear system (as has
[Erling Jorgensen (2005.02.22 0100 EST)] numerically). That is the
kind of system usually used in simulations to demonstrate the
viability of PCT (e.g. Rick’s spreadsheet). We all know that
mathematically a hierarchy of linear control systems is exactly
equivalent to a one-level control system, so demonstrating the Bomb in
a linear hierarchy really is no demonstration at all.

In general, P(n) = p(P(n-1,1)…P(n-1,k)) where P(n) is the
perceptual signal of some level n control unit, p is an arbitrary
function, and P(n-1,m) is the contribution of the m’th level n-1
perceptual signal to P(n).

The Bomb can explode if for a particular function p, the level n
control unit has a finite probability of being unable to compensate
for a runaway in any of the P(n-1,m). Only if there is NO
combination of input values for which the control unit cannot
compensate will the Bomb explosion definitively stop if an explosion
reaches that control unit.

Well, I still believe that this sort of
verbal argumentation doesn’t get us anywhere. Better to produce a
mathematical demonstration; then the outcome won’t depend on who finds
the cleverest argument.

There are two kinds of simulation that I can imagine being
useful. One demands that a heterogeneous hierarchy (one not involving
linear systems at more than one consecutive level) be natively
reorganized in a sufficiently complex varying environment, in which
feedback strands turn positive from time to time. That would be the
ideal case, but a difficult one to set up and run, and even then,
whatever the result, the verbal discussion would continue. If no Bombs
showed up, I could argue both that it’s just a matter of time, and
that the PIFs had been inappropriately chosen. If the Bomb did show
up, you could argue that a better choice of PIFs would have eliminated
the Bomb.

The second kind of simulation would replace actual control
systems by a network of nodes and links in which the propoagation of a
signal (the Bomb explosion front) out of a node was a probabilistic
function of the inputs. Such a netwrok could be tested analytically or
by Monte Carlo simulations, and parametric variation of the different
node probabilities could show how the Bomb propagation and
level-by-level damping were affected by probability variation. That
experiment would also be subject to verbal argument, both as to its
applicability to the real world, and as to the ways of choosing the
probabilistic functions at issue.

“Cleverest argument” determines only who of an audience
comes to believe something. It doesn’t determine the reality behind
the argument.

Martin

[From Rick Marken (2005.02.22.0920)]

Martin Taylor (2005.02.22.0929) --

I illustrated the Bomb algebraically in a linear system (as has [Erling
Jorgensen (2005.02.22 0100 EST)] numerically). That is the kind of system
usually used in simulations to demonstrate the viability of PCT (e.g. Rick's
spreadsheet).

The spreadsheet hierarchy is linear at levels 1 and 2 only. Level 3 is non-
linear (It is controlling for logical relationships). The linear definitions
of perceptions at levels 1 and 2 could by made non-linear pretty easily and
I believe the whole hierarchy would still work just the same. That might be
a good exercise for someone to do with the hierarchy.

Best

Rick

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[Martin Taylor 2005.02.22.16.22]

[From Rick Marken (2005.02.22.0920)]

Martin Taylor (2005.02.22.0929) --

I illustrated the Bomb algebraically in a linear system (as has [Erling
Jorgensen (2005.02.22 0100 EST)] numerically). That is the kind of system
usually used in simulations to demonstrate the viability of PCT (e.g. Rick's
spreadsheet).

The spreadsheet hierarchy is linear at levels 1 and 2 only. Level 3 is non-
linear (It is controlling for logical relationships).

Sorry. I knew that!

The linear definitions
of perceptions at levels 1 and 2 could by made non-linear pretty easily and
I believe the whole hierarchy would still work just the same. That might be
a good exercise for someone to do with the hierarchy.

Yes, it would be. One easy and useful trick would be to make each analogue perceptual signal the logarithm of the function it now uses (or rather, of the absolute value, multiplied by the sign of the output). I suggest that as a very crude analogy to what a lot of physiological perceptual functions do.

Martin

[From Rick Marken (2005.02.22.1355)]

Martin Taylor (2005.02.22.16.22) --

Rick Marken (2005.02.22.0920)]

The linear definitions
of perceptions at levels 1 and 2 could by made non-linear pretty easily and
I believe the whole hierarchy would still work just the same. That might be
a good exercise for someone to do with the hierarchy.

Yes, it would be. One easy and useful trick would be to make each
analogue perceptual signal the logarithm of the function it now uses
(or rather, of the absolute value, multiplied by the sign of the
output). I suggest that as a very crude analogy to what a lot of
physiological perceptual functions do.

Yes. That is a great idea. The only problem is that all variables in the
spreadsheet range from minus to plus infinity. And you know how upset
mathematicians get when you try to take the log of a negative number. Maybe
a cubic function would do it. I'll try.

Best

Rick

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[Martin Taylor 2005.02.22.17.32]

[From Rick Marken (2005.02.22.1355)]

Martin Taylor (2005.02.22.16.22) --

Rick Marken (2005.02.22.0920)]

The linear definitions
of perceptions at levels 1 and 2 could by made non-linear pretty easily and
I believe the whole hierarchy would still work just the same. That might be
a good exercise for someone to do with the hierarchy.

Yes, it would be. One easy and useful trick would be to make each
analogue perceptual signal the logarithm of the function it now uses

>. I suggest that as a very crude analogy to what a lot of

physiological perceptual functions do.

Yes. That is a great idea. The only problem is that all variables in the
spreadsheet range from minus to plus infinity. And you know how upset
mathematicians get when you try to take the log of a negative number. Maybe
a cubic function would do it. I'll try.

You missed "(or rather, of the absolute value, multiplied by the sign of the output)" And then the kettle boiled for tea before I re-read my posting, or I would have noted the infinity at log(zero). Actually, what I think you want is sgn(f)log(abs(f) + k) where f is the existing function, k a small positive constant (acting like a noise floor), and sgn(f) = -1 if f is negative, +1 if f is positive. Physiologically, you wouldn't have negative inputs, anyway, but this suggestion allows you to compress into one control unit what might well otherwise exist in two complementary control units.

So, if you can, why not try sgn(f)*log(abs(f) + k)?

Martin

[From Rick Marken (2005.02.22.1540)]

Martin Taylor (2005.02.22.17.32) --

So, if you can, why not try sgn(f)*log(abs(f) + k)?

Actually, I had already tried this! It doesn't work, though, because some of
the higher level perceptions require that some of the lower level inputs go
negative.

I just tried making the level one perceptions a cubic function of the
physical variables. Specifically,

p = (o+d)^3/10000 = q^3/10000

I had to divide by that big number so that the existing control parameters
would still work. A cubic function is pretty non-linear but everything
continued to work like a charm.

Best

Rick

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[Martin Taylor 2005.02.22.20.05]

[From Rick Marken (2005.02.22.1540)]

Martin Taylor (2005.02.22.17.32) --

So, if you can, why not try sgn(f)*log(abs(f) + k)?

Actually, I had already tried this! It doesn't work, though, because some of
the higher level perceptions require that some of the lower level inputs go
negative.

This goes negative exactly when the original function does. Are you sure you tried just what I suggested?

I just tried making the level one perceptions a cubic function of the
physical variables. Specifically,

p = (o+d)^3/10000 = q^3/10000

I had to divide by that big number so that the existing control parameters
would still work. A cubic function is pretty non-linear but everything
continued to work like a charm.

It's got the opposite nonlinearity to a log, though. Not that this would make it stop working. Perhaps you meant "^-3" rather than "^3". Square root and log look much alike over a reasonable range. maybe cube root and log do, too.

Martin

[From Bill Powers (2005.02.23.0802 MST)]

Martin Taylor 2005.02.22.20.05 --

[From Rick Marken (2005.02.22.1540)]

Martin Taylor (2005.02.22.17.32) --

So, if you can, why not try sgn(f)*log(abs(f) + k)?

Actually, I had already tried this! It doesn't work, though, because some of
the higher level perceptions require that some of the lower level inputs go
negative.

This goes negative exactly when the original function does. Are you sure you tried just what I suggested?

Perhaps the meaning of "sgn" wasn't clear. That's the "signum" function, meaning its value is 1 or -1 according to the sign of its argument, f.

If f is the linear form of the controlled variable, then the logarithmic form is p = sgn(f)*log(abs(f) + k), and when f is negative the value of p is negative.

In a program you'd just say

if f > 0 then p = log(f + k) else p = -log(-f + k)

where k = some small positive constant like 0.001.

Probably a more important factor than nonlinearity in this context is the physiological limit on the magnitude of a perception (at most, a few thousand impulses per second). Positive feedback runaway is limited by that. On the output side, runaway is limited by the maximum output (signal or force) that can be produced. Once the maximum is reached no further increases can occur.

Best,

Bill P.

[From Erling Jorgensen (2005.02.23 1225 EST)]

Bill Powers (2005.02.23.0802 MST)

Probably a more important factor than nonlinearity in this context is the
physiological limit on the magnitude of a perception (at most, a few
thousand impulses per second). Positive feedback runaway is limited by
that. On the output side, runaway is limited by the maximum output (signal
or force) that can be produced. Once the maximum is reached no further
increases can occur.

On the output side, I agree that the signal or force would have a physiological

limit. But our customary way of writing the output equation for the control loop

does not usually separate out the environmental feedback function as separate

from the overall output gain. Doesn’t that feedback function out in the

environment provide quite a substantial source for increasing the positive

feedback effect, even when some kind of behavioral maximum has been reached?

(And isn’t this what “terrorists” and others typically exploit? A literal bomb with

its explosive effect does what the person’s own muscles would not be able to

accomplish.) I think we need to be careful about pre-deciding the limits on

a phenomenon like positive feedback.

All the best,

Erling

[From Bill Powers (2005.02.23.1123 MST)]

Erling Jorgensen (2005.02.23 1225 EST)–

On the output side, I agree that
the signal or force would have a physiological limit. But our
customary way of writing the output equation for the control loop does
not usually separate out the environmental feedback function as separate
from the overall output gain. Doesn’t that feedback function out in
the environment provide quite a substantial source for increasing the
positive feedback effect, even when some kind of behavioral maximum has
been reached?

Usually the environmental feedback function involves a loss of energy.
But not always, as in driving a car or when a real bomb is involved, as
you mention. However, loop gain is the product of all gains encountered
in one trip around the loop, so if any gain goes to zero the loop gain is
zero.
The main limits are the limit of a neural signal, the limit of muscle
strength, and the limit of movement (for example, you can extend an arm
or leg only until it is straight). Once any function in the loop
saturates (reaches a maximum of output beyond which it can’t increase)
the positive feedback ends – that is, there can be no further
runaway effect. The output may still be large, and environmental effects
may continue to change, but the control system is no longer increasing
its activity in a positive exponential manner, without limit.

The primary consequence of positive feedback is loss of control, whether
during the runaway or after a limit has been reached. Models should put
limits on the variables inside the control system and wherever else is
appropriate, to show this effect.

Best,

Bill P.

[Martin Taylor 2005.02.23.16.50]

[From Bill Powers (2005.02.23.0802 MST)]

Martin Taylor 2005.02.22.20.05 --

[From Rick Marken (2005.02.22.1540)]

Martin Taylor (2005.02.22.17.32) --

So, if you can, why not try sgn(f)*log(abs(f) + k)?

Actually, I had already tried this! It doesn't work, though, because some of
the higher level perceptions require that some of the lower level inputs go
negative.

This goes negative exactly when the original function does. Are you sure you tried just what I suggested?

Perhaps the meaning of "sgn" wasn't clear. That's the "signum" function, meaning its value is 1 or -1 according to the sign of its argument, f.

If f is the linear form of the controlled variable, then the logarithmic form is p = sgn(f)*log(abs(f) + k), and when f is negative the value of p is negative.

In a program you'd just say

if f > 0 then p = log(f + k) else p = -log(-f + k)

where k = some small positive constant like 0.001.

unless your language has the "sgn" function!

Probably a more important factor than nonlinearity in this context is the physiological limit on the magnitude of a perception (at most, a few thousand impulses per second). Positive feedback runaway is limited by that. On the output side, runaway is limited by the maximum output (signal or force) that can be produced. Once the maximum is reached no further increases can occur.

Quite so.

I should have pointed out to Rick that the logarithmic function itself will have some limiting effect on the runaway, as the greater the value of the perceptual signal, the lower the loop gain. In itself, that may be a factor in limiting the size of avalanches -- at least in a simulated hierarchy. In a real physiological network, one might expect bifurcations to occur occasionally, which could do drastic things to functions of which their variables are arguments. Flip-flops are one example, but they don't exist in a pure hierarchy. Category perceptual functions do, though!

Martin

[From Rick Marken (2005.02.24.0900)]

Martin Taylor 2005.02.22.20.05]

Rick Marken (2005.02.22.1540)

Martin Taylor (2005.02.22.17.32) --

So, if you can, why not try sgn(f)*log(abs(f) + k)?

Actually, I had already tried this! It doesn't work, though, because some of
the higher level perceptions require that some of the lower level inputs go
negative.

This goes negative exactly when the original function does. Are you
sure you tried just what I suggested?

Yes. I think the reason it doesn't work is because there is a little
positive feedback zone between q = -1 and q = 1, where q = abs(f)+k. The
log of q is negative when q<1 (for log base 10) so q is of the opposite sign
that it should be for positive f<1 and negative f<-1. There is also the
problem the the log is not defined when abs(f) = k.

I think it might be possible to do this using a table that maps the input
(f) values from -n to n into logarithmically spaced perceptual values from
-n to n. I'll try to do it that way of I get the chance.

Best

Rick

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[From Bill Powers (2005.02.24.1352 MST)]

Rick Marken (2005.02.24.0900)--

Yes. I think the reason it doesn't work is because there is a little
positive feedback zone between q = -1 and q = 1, where q = abs(f)+k. The
log of q is negative when q<1 (for log base 10) so q is of the opposite sign
that it should be for positive f<1 and negative f<-1. There is also the
problem the the log is not defined when abs(f) = k.

You're right, I hadn't caught that region between 0 and 1. You can handle that by making the total range of the controlled variable large (like 1000 units), and then setting k to be 1.01.

However, if the log form is too unwieldy you can always use something like a square root that goes to zero at zero input, with the same tricks for handling negative inputs.

Suppose one of the levels controls for a path through a series of locations. By selecting x-y-z triads of points Axyz, Bxyz, Cxyz, ... Nxyz as target locations, the higher system moves the system along a route through space. If now the position-seeking system develops positive feedback, will this lead to positive feedback in the route-following system? It will surely cause a problem, but I don't see how you could define the problem as constituting positive feedback at the route-following level. An error in the route does not lead to a change in the next designated goal that takes the route farther from the desired pattern instead of closer to it. All that happens is that when the next point is designated, the lower system avoids it instead of seeking it, which acts like a large disturbance of the higher system. But a large disturbance is not the same thing as positive feedback.

Best,

Bill P.

···

I think it might be possible to do this using a table that maps the input
(f) values from -n to n into logarithmically spaced perceptual values from
-n to n. I'll try to do it that way of I get the chance.

Best

Rick
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[From Rick Marken (2005.02.25.1700)]

Bill Powers (2005.02.24.1352 MST)--

Suppose one of the levels controls for a path through a series of
locations. By selecting x-y-z triads of points Axyz, Bxyz, Cxyz, ... Nxyz
as target locations, the higher system moves the system along a route
through space. If now the position-seeking system develops positive
feedback, will this lead to positive feedback in the route-following
system? It will surely cause a problem, but I don't see how you could
define the problem as constituting positive feedback at the route-following
level. An error in the route does not lead to a change in the next
designated goal that takes the route farther from the desired pattern
instead of closer to it. All that happens is that when the next point is
designated, the lower system avoids it instead of seeking it, which acts
like a large disturbance of the higher system. But a large disturbance is
not the same thing as positive feedback.

That's what I thought. But what happened in my hierarchical model seemed to
be positive feedback in the level 2 system producing what looked like
positive feedback in the level 3 system that controlled a perception that
included the level 2 perception. When I put the level 2 system into a mild
positive feedback loop, the system above it (which was controlling for
p21-p22>n, where p21 is the perception in the positive feedback loop) went
into what appeared to be positive feedback as well; that is, p21-p22 kept
_decreasing_ rather than increasing. Because this could have just been a
result of the disturbance created by p21 I upped the gain of the level 3
system and decreased the gain of the positive feedback loop but these things
didn't help. Still apparent positive feedback. Then I reversed the sign of
the feedback in the level 3 system. And it still looks like positive
feedback at level 3. Since it _looks like_ positive feedback no matter what
(p21-p22 keeps _decreasing_ rather than increasing or staying the same) I
think it might actually be true that positive feedback in the level n+1
system is created by positive feedback in the level n system. But a
mathematical proof would certainly be more convincing.

Best

Rick

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[Martin Taylor 2005.02.25.21.10]

[From Bill Powers (2005.02.24.1352 MST)]

Suppose one of the levels controls for a path through a series of locations. By selecting x-y-z triads of points Axyz, Bxyz, Cxyz, ... Nxyz as target locations, the higher system moves the system along a route through space. If now the position-seeking system develops positive feedback, will this lead to positive feedback in the route-following system?

I have some difficulty imagining what positive feedback in the route following system would look like. Would it be going to other places, going off the road, going backwards, or what? I'm having a problem, in othe words, in translating the analogue concept into a discrete world.

What would an external observer see if the route following control system went into positive feedback?

Martin

[From Bill Powers (2005.02.26.0240 MST)]

Rick Marken (2005.02.25.1700)--

That's what I thought. But what happened in my hierarchical model seemed to
be positive feedback in the level 2 system producing what looked like
positive feedback in the level 3 system that controlled a perception that
included the level 2 perception. When I put the level 2 system into a mild
positive feedback loop, the system above it (which was controlling for
p21-p22>n, where p21 is the perception in the positive feedback loop) went
into what appeared to be positive feedback as well; that is, p21-p22 kept
_decreasing_ rather than increasing.

Was that because of positive feedback or because the disturbance was increasing? I can't really follow this, however, without going through the whole simulation myself, and I just find spreadsheets too confusing with only cell designations to work with. My memory isn't that good. I thought level 3 was logical in nature -- how can anything "increase" or "decrease" in a logical level where variables are either true or false?

Best,

Bill P.

···

  Because this could have just been a
result of the disturbance created by p21 I upped the gain of the level 3
system and decreased the gain of the positive feedback loop but these things
didn't help. Still apparent positive feedback. Then I reversed the sign of
the feedback in the level 3 system. And it still looks like positive
feedback at level 3. Since it _looks like_ positive feedback no matter what
(p21-p22 keeps _decreasing_ rather than increasing or staying the same) I
think it might actually be true that positive feedback in the level n+1
system is created by positive feedback in the level n system. But a
mathematical proof would certainly be more convincing.

Best

Rick
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MindReadings.com
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[From Bill Powers (2005.02.26.0245 MST)]

Martin Taylor 2005.02.25.21.10 --

I have some difficulty imagining what positive feedback in the route following system would look like. Would it be going to other places, going off the road, going backwards, or what? I'm having a problem, in othe words, in translating the analogue concept into a discrete world.

Me, too. That's my basis for wondering whether positive feedback could propagate from a lower level to a higher one that deals in an entirely different kind of variable.

Best,

Bill P.

[Martin Taylor 2005.02.26.11.03]

[From Bill Powers (2005.02.26.0240 MST)] (to Rick marken)

how can anything "increase" or "decrease" in a logical level where variables are either true or false?

[From Bill Powers (2005.02.26.0245 MST)]

Martin Taylor 2005.02.25.21.10 --

>I have some difficulty imagining what positive feedback in the route >following system would look like. Would it be going to other places, going >off the road, going backwards, or what? I'm having a problem, in othe >words, in translating the analogue concept into a discrete world.

Me, too. That's my basis for wondering whether positive feedback could >propagate from a lower level to a higher one that deals in an entirely >different kind of variable.

I have no problem with "an entirely different kind of variable" provided that the different kind of variable has an analogue continuum representation. Where I do have a problem is in the transition to discrete representation, where, for example, something is perceived as a member of class A or of class B. That representation doesn't change with continuous variation in an underlying variable except when the class perception switches from one to the other.

Having said that, I can think of two concepts that might be worth pursuing.

One is the notion of fuzzy class membership. Much of the time we use a category perception such as "the light is red" versus "the light is green", in which case the problem Bill mentioned to Rick exists. But sometimes we say to ourselves "that's a poor sort of red" or "that's a real true red". In more mathematical language, we not only perceieve it as having a class label, but also as having a class membership value. Whether this relates to the issue, I don't know, and can't spare the time to examine. But it might be an avenue of thought.

The other potential avenue of thought is phenomenological. We do see escalating behavioural explosions that seem to be related to categorical misperception: the escalating interpersonal conflict started by a perception that someone did an action from malice, rather than by accident, for example; or the fit of temper based on the "innate perversity of inanimate objects" such as the screwdriver that maliciously persists in slipping out of the screw slot. I'm sure all of you can think of many kinds of such example. These seem to show that the Bomb can propagate within the "logical" (i.e. category-based) levels. That being so, the questions are how that propagation happens, and how it crosses the analogue-to-discrete perceptual boundary.

Wish I had more time for this. But now I have to work.

Martin

[From Rick Marken (2005.02.26.0820)]

Bill Powers (2005.02.26.0240 MST)--

Rick Marken (2005.02.25.1700)--

When I put the level 2 system into a mild
positive feedback loop, the system above it (which was controlling for
p21-p22>n, where p21 is the perception in the positive feedback loop) went
into what appeared to be positive feedback as well; that is, p21-p22 kept
_decreasing_ rather than increasing.

Was that because of positive feedback or because the disturbance was increasing?

I can't be sure. But I originally though it was because the disturbance is increasing. I think that might be it, since changing the polarity of the higher level system doesn't fix things (if it were positive feedback in the higher level system I think changing the polarity of that system should bring back negative feedback, but it doesn't).

I can't really follow this, however, without going through the whole simulation myself, and I just find spreadsheets too confusing with only cell designations to work with. My memory isn't that good. I thought level 3 was logical in nature -- how can anything "increase" or "decrease" in a logical level where variables are either true or false?

Your memory is excellent. Level 3 perceptions were logical. The one in question, p31, (first subscript is the level, second is the system at that level) is :

p31 = If (p21-p22)> 0 then 1 else -1

So the perception is only 1 or -1. With positive feedback in the p21 control loop, p31 remains -1. What continues to decrease, regardless of the sign of the loop gain for the system controlling p31, is p21-p22, the basis for the computation of the logical perception.

But if positive feedback in a higher level system (the one controlling p31 in this case) is measured in terms of the effect of output on the controlled perception (p31) then there is no positive feedback in the higher level system in this case. It's just a system that doesn't work.

Best

Rick

···

---
Richard S. Marken
marken@mindreadings.com
Home 310 474-0313
Cell 310 729-1400

[From Bill Powers (2005.02.26,1201 MST)]
Martin Taylor 2005.02.26.11.03 –
I think you, Rick, and I are all talking about the same thing now. When
discrete control systems are introduced, runaway becomes less likely. As
you say, if there is an underlying continuum, as in the case of better
and worse members of a category, there is still the possibility of
propagation of the positive feedback up the levels. But when the
transition to discrete variables occurs, it’s harder to imagine how that
would happen.
The same thing would happen when we look at different levels of
discreteness. What would positive feedback look like at the level where
we name categories? At the level where we control sequential order? At
the level where we control the values of logical functions, or programs?
I think the context changes with every level.
The question remains, how would reversing the sign of feedback at one
level have the effect of reversing the sign of feedback at a higher
level? I’m sure there are cases where this would happen, but I can’t
think of any just now. If a lower-level system runs away, it would do so
by making the perceptual signal avoid the reference setting, but that
could go in either direction; p < r and p getting smaller, or p > r
and p getting larger. One of those cases would make p change the right
way for the next system up to retain negative feedback.
But as Rick now concludes, all we can really say is that the higher-level
system simply stops working. There is an effect of positive
feedback at lower levels on systems at higher levels, but the nature of
that effect is not necessarily to change the sign of feedback for the
loop passing through the higher level. I know that all you talked about
was a probability that some system would have its feedback sign reversed,
so the probability of a blowup was the product of the probabilities that
it would happen at each consecutive level. I’ll concede that. The
significance of the Bomb effect, however, depends critically on just what
that probability is. If it’s 0.0000000000001 per year, we can ignore
it.

I also point out that there can be positive feedback at level N without
there being positive feedback at any lower level. All it takes is that
the sign of the relationship between output and input reverse at the
higher level. This does not alter the sign of any lower loop. So a Bomb
can exist at a higher level if there are many paths through lower
systems, but some entail positive feedback for the higher loop. A change
of circumstances can set off that bomb without causing any lower system
to run away. Consider Dr. Strangelove. There the Bomb, literally, was the
arms race that led to the Doomsday Machine. Everything was under control
until a maverick launched an attack, which triggered the positive
feedback loop and breached all the peacemaking efforts.

Blooie. But all the lower systems remained negative feedback systems,
faithfully achieving each goal they were given.

Best.

Bill P.