Flip-flops

[Martin Taylor 970630 16:05]

Bill Powers (970629.1649 MDT)

I doubt whether there would be any hysteresis observed if you presented the
sample sounds one day apart.

So do I. And the GFF would not predict it either, unless there were
something to keep alive the earlier category perception.

How do you know that the perceptual lags are on the order of
milliseconds? The person might issue a judgement while the >>perceptual

signal is still changing.

So you _did_ mean what I proposed in jest in my first paragraph above.

Obviously I can't _know_ it, even from intra-personal observation. >But it

seems _highly_ implausible.

Only because it contradicts what you believe to be the case -- and NEED to
believe to rule out the dynamic-lag hypothesis. The correct answer to my
question is, "I don't know -- I guess those experiments will have to be
done again if we are to rule out the dynamic-lag hypothesis."

Any hysteresis at all, in any kind of situation, electronic, biological,
or other, requires some kind of dynamic lag or memory. I thought you were
suggesting a lag on the order of the lag time of control of the relevant
perception. That's what I was (and am) sceptical about.

I don't know how you see me as _ruling out_ the dynamic-lag hypothesis. In
an experiment that involves continuous tracking, it's quite plausible.
In an experiment in which subjects render judgments that specify the
categories of individually presented patterns, it seems less plausible.

I merely pointed out its implausibility on the grounds that a perception
that one has described has at least _been_ perceived, and if the memory
of it changes later, that may affect the perception of a following sensory
input (probably will), but it's not a dynamic lag of the kind you
proposed unless the two sensory inputs somehow get mixed up.

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

HPCT does not rule out level-hopping perceptual signals; only level-hopping
reference signals.

I know that, and it's always seemed to me to be a smudge on a very elegant
picture that you can have these long level-skips one way when, for good
reason, they are clearly prohibited in the other direction. Sure it may
be the way things are, but, like Einstein, I have a predeliction for
elegance. The GFF proposal eliminates this inelegance at the cost of
eliminating a different constraint--that perceptual signals cannot return
to be inputs to other perceptual functions at the same level. Beauty is
in the eye of the beholder, of course, and maybe you see level-skipping
perceptual signals coupled with special-purpose category perceptual
functions as more elegant than having the category perceptions fall out
from permitting same-level cross-connections.

It's clear, despite what you say, that you have forgotten what the
"Grand Flip-Flop" proposal was (and is). You say things like:

Yes, this is all true, but if the situation you describe holds true, the
positive feedback is large enough to exclude any middle values: the output
is either maximum or minimum. You would not get graded outputs if it takes
a large input to produce a switch.

and

Yes. You're assuming that you already have category signals at the inputs
to your flip-flops. That is, one unit senses the category "fruit" and
another the category "vegetable." THEN a flip-flip cross-connection sees to
it that only one of the signals can be output at a time.

and

However, this
leaves out the process by which perceiving an apple, orange, banana, plum,
or pear can give rise to the perceived category "fruit", and other
perceptions give rise to "vegetable." This question is quite aside from the
question of mutual exclusivity -- that is, whether seeing a plum tomato
leads you to perceive only "fruit", only "vegetable," or both.

and

You
haven't said how the category signal itself is generated from multiple
inputs.

and

All you've considered is the _external_ connections among input
functions, not what the input functions themselves do, internally, that
creates categories. That's what I meant by saying ...

It seems to me that
you're ignoring the main design problem and focusing on an >>ancillary

detail. It's like worrying about the design of the >>hubcaps before you
know how to build a wheel.

None of these comments is appropriate. So I guess I'd better recap the
design, which I had not intended to do.

···

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

We start with any of the ordinary perceptual signals at any perceptual
level. These are analogue signals, and it doesn't matter what they
represent. They are the outputs of the perceptual input functions of
some elementary control units, but for the purposes of this discussion
we are going to ignore the rest of the control units, and consider
only the inputs and outputs to the perceptual functions.

To make things simple, we will make an unnecessary assumption, which is
that at least some of the perceptual outputs go two ways: (1) up to the
next analogue perceptual level, and (2) to a set of perceptual functions
that will turn out to be the category perceivers. We will call these
"category perceptual input functions" CIFs. (Each CIF is, of course,
notionally the input function of an elementary control unit).

(The reason this is an unnecessary assumption is that the analogue
perceptual functions can perform the category process themselves, provided
that there are more of them than are necessary to form on orthogonal
basis for the perceptual space--i.e., they are redundant. But it's
much easier to describe if we split off a subset that performs the
category process.)

Each CIF takes input from one or more of the analogue functions, combines
them, and provides an output perception (as before, we will ignore the
rest of the control unit for which the CIF provides the perceptual input).
How it combines them is unimportant (see your comment above about my
not having specified what the category input functions do). What is
important about the operation of the CIF is that there is the possibility
that some of the inputs may be inhibitory, and that the output is not
linearly dependent on the input. It has a low threshold at or near zero,
and saturates smoothly as the excitatory input increases (a logarithmic
function that cuts off at zero, or an arc-tan transformation, or a logistic,
might do).

A simple version of a CIF might be one that produces a weighted sum of the
inputs and then outputs the logarithm of that sum, or zero if the sum is
negative. A more complex one might do the non-linear transform at the
inputs, or allow some inputs to multiply together rather than to add...
it doesn't matter, so long as there are both excitatory and inhibitory
possibilities for the input weights.

As you see, the CIFs are quite ordinary analogue perceptual functions, and
there's nothing special about the chosen function. You could even omit
the smooth saturation, but then you would get a category system that
provided only "yes-no" answers. Any of the functions in the different
analogue levels of the hierarchy would be appropriate, provided they
eventually saturate, smoothly or otherwise (if they didn't, you'd get
a very physical explosion:-).

Now we come to the cross-connections, and why they _produce_ the category
perceptions.

The output of each CIF is connected to several of the others (conceptually,
to all, but most of the weights will be near zero, so we ignore them).
Some of these connections have postive weights, some have negative weights.
Some weights are large, some small. For this discussion we will ignore the
learning procedure that affects the weights (it's an aspect of reorganization
that we have discussed, but the feedback of the cross-linkiing generates
some special effects). Let's just say that if, in the absence of the
cross-linking, two of the CIFs would often have high outputs at the same
time, their mutual weights are more likely to to be positive than negative,
and if one (A) is usually very low when another (B) has high output, then
the weight from B to A is likely to go negative.

If you trace all the loops from a specific CIF back to itself through one
other CIF, there are four possibilities for the pattern of weights. The
weight _to_ the other node may be positive or negative, and the weight
_from_ the other node may be positive or negative. Two positives or
two negatives complete a positive feedback loop, whereas one positive
and one negative complete a negative feedback loop.

Do this with _all_ the feedback loops from a CIF back to itself, and you
get a feedback loop with considerable dynamic interest. If the weights are
high enough, the whole system of CIFs is likely to go into oscillation or
even into chaotic operation. In fact, in our simulations, we could get
multiple different behaviours of these kinds out of a single small network
with a fixed set of weights, just by putting a momentary pulse in at one
node at critical moments. But that kind of operation isn't very useful
for category judgments:-) So we have to assume that effective reorganization
has ensured that the weights are not high enough to make the CIF system
take off on its own. It must respond to its sensory inputs (the perceptual
signals from the analogue hierarchy).

Ignoring the time dimension for the moment, the input to CIF i consists
of the sensory inputs from the analogue hierarchy plus the combined effects
of the outputs of CIFs 1...n, which are, of course, affected by the output
of CIF i. The overall gain of a loop through CIF j is given by the product
of the weights ij and ji and the slope of the outputs Oi and Oj as a function
of their inputs (as the output saturates, the effect of changes in the
input gets less, and so does the loop gain).

The overall gain of the feedback loop through the whole system of CIFs,
back to CIF i, is given by the sum of all these weight and saturation
products. This self-loop gain of CIF i can be greater than unity.
If it is, the arrival of sensory input that would, in the absence of
cross-correlation, bring CIF i up to level xi, now induces a positive
feedback loop that runs away, increasing the output of CIF i. But it
doesn't run away without limit, because of the saturation of the
output of CIF i, which at some point brings the loop gain back down to
unity, no matter what happens elswhere in the loop. The input "worth" xi
in the absence of cross-connection now produces an output of yi > xi.

The value of yi depends slightly on the value of xi, but is more dependent
on the value of the overall loop gain through the other CIFs. If some of
those that have inhibitory connections with CIF i now get sensory input
that would make their values go high, the effective loop gain from CIF i
back to itself is reduced--it takes more input to cause the runaway,
because the inhibiting one is now operating on a part of its curve where
variation of its input matters (previously is was, we assume, at zero,
unaffected by small changes in the inhibition due to CIF i).

On the other hand, if CIF k with which CIF i has mutually positive weights
gets input that would bring its output above zero, the loop gain from
CIF i back to itself will be increased, enhancing the likelihood that
category i will be perceived, or increasing the magnitude with which
the input to CIF i is seen as being category i. And the existence of
the perception of category i enhances the likelihood that the input
at the _associated_ CIF k will be seen as its category, too. In the
extreme case, perceiving category i may be enough to cause perception
of category k, as might be the case if i were a rose and k were the
verbal label "rose" (not the sound or the letters, but the label).

You can see that it is possible for CIF i to sustain high output even when
an inhibiting CIF also has high output (perception of two normally mutually
exclusive categories together), and that the perception of "category i"
is graded in magnitude, being enhanced by the presence of appropriate
context, that there is hysteresis across two mutually exclusive categories
as the sensory input moves from favouring one to favouring the other and
back again. If you recognize the time for the effects to propagate around
the system, you can see why it takes longer to perceive poor representations
as belonging to the category than it takes to perceive good prototypes,
and why category labels are more easily seen in an appropriate context
(an everyday finding in reading research).

Notice the difference between this and your conjecture:

It also implies something else I haven't mentioned so far, which is >that

it takes more time for the category perception to develop than >it does for
the sensory data to arrive, since the category >perception develops in part
from the recursive connections from >other perceptual functions--IF the
sensory data are not clear, >prototypical instances of the category in
question.

Well, that makes a testable proposition. If there are category perceptions
at every level, and if the sensory data are clear and prototypical, then we
should be able to find category perceptions at, say, the configuration
level that are formed faster than relationship perceptions, which by HPCT
are at a higher level. So you should be able to name "square" and "circle"
faster than you could identify whether the square or circle was "above" or
"below" the center of the screen.

My actual prediction is that one would perceive "square" or "circle" quicker
if the drawing were geometricaly precise than if it looked like a squashed
circle or a rounded square. And did you notice that the experiment you
propose is an experiment in the speed of _association_, not of perception?

The prediction of my version of HPCT would be that the fastest speeds of
either judgement would be exactly the same, since the category perceptions
all occur at the same level, just above the level where relationships are
perceived.

I dispute that prediction, since the signals from low levels get to the
category level by bypassing the intermediate levels. As far as that is
concerned, the GFF and the "standard" design make the same prediction.

I maintain that you are stuck at your own category level, which is why you
see categories at EVERY (lower) level. When you look at the world through
category-perceivers, everything is a category.

I'm no different from you. You've acknowledged that you perceive a category
"red", a category "square", and a category "below." You see categories
at every level, too. And as you say, when you are _concerned_ with
the perception of categories, it is category perceivers that you treat.

Anyway, I hope that this all helps provide you some memory stimulus, to
remember why your comments quoted at the start of this were inappropriate.
There's too much of the rest of your message to comment on directly, but
I hope you will see that much of it also missed the point. I hope I have
satisfied the following, at least:

Define "the main design problem". If you mean the hierarchy of >perceptual

control, that's the framework on which the wheels are >fitted.

That's not what I mean. The main design problem is how you can take a set
of input signals each standing for a different lower-level perception, and
derive a single signal that is present to the degree that any one of those
lower-level perceptual signals is a member of the category.

And I hope I have explained what association means, within the GFF at least.

Martin

[From Bill Powers (970701.0515 MDT)]

Martin Taylor 970630 16:05 --

I think your proposal concerning "CIFs" at all levels of the hierarchy
would be more palatable if you would show how a category control system
would work in some specific control task.

I confess that it is hard for me to follow your verbal description of what
goes on among all these mutually-connected CIFs at the same level. I have
never studied perceptrons. But I get the idea that if the analog inputs
satisfy the conditions necessary to produce a category perception, there
will be a perceptual signal generated by a CIF at the same level where
perceptual signals are generated in the usual analog way. A category
perceptual signal is presumably compared with a reference-category signal
from the next level up to generate an error signal, and the error signal
goes through an output function that distributes its effects to the
reference inputs of lower-level control systems. Does this mean that the
lower-level reference signals are the sum of a set of
analog-error-generated outputs from higher analog control systems, and
category-error-generated outputs from a set of category control systems?

Consider a simple compensatory tracking situation. We have a model that
seems to reproduce tracking behavior, in which a perception of the position
of the cursor relative to the target is maintained in a specific reference
state: cursor an inch to the left, on the target, an inch to the right, and
so on. The control equations are fairly straightforward and they seem to
fit what we observe reasonably well.

Now, according to my understanding of your proposal, there will also be
category signals being generated at this same level. We might guess that
the categories are "cursor on target", "left of target" or "right of
target," or any others you might prefer, like "approaching target" or
"departing from target." Whatever they are, as the tracking process
continues we will see changing categories; the situation will cross the
boundaries between categories and the category-perception signals will
change. As the boundaries are crossed, we will see category error signals
being generated. These error signals will generate outputs that go to lower
levels of control, just as the analog error signals do.

Perhaps you will propose that the category error signals go to lower
category control systems. But at some point, the two hierarchies must
merge, even if only at the level of muscle tensions; the muscles are the
only means of producing effects on the external world and closing the
feedback loops. The actual outputs we see will be the sum of category-error
effects and analog-error effects.

So my question is, how will the outputs differ if we consider the
analog-only tracking model, as opposed to the analog-plus-category tracking
model?

Best,

Bill P.

[Martin Taylor 970701 12:40] Canada Day

Bill Powers (970701.0515 MDT)

Martin Taylor 970630 16:05 --

I think your proposal concerning "CIFs" at all levels of the hierarchy
would be more palatable if you would show how a category control system
would work in some specific control task.

I confess that it is hard for me to follow your verbal description of what
goes on among all these mutually-connected CIFs at the same level.

You seem to have made a good start at understanding, judging from what
follows!

I have
never studied perceptrons. But I get the idea that if the analog inputs
satisfy the conditions necessary to produce a category perception, there
will be a perceptual signal generated by a CIF at the same level where
perceptual signals are generated in the usual analog way. A category
perceptual signal is presumably compared with a reference-category signal
from the next level up to generate an error signal, and the error signal
goes through an output function that distributes its effects to the
reference inputs of lower-level control systems.

Yes, except that the reference signal for a category control unit need
not come from another (higher) category control unit. It could come from
a (higher) analogue level. It might be that an analogue system requires
a particular category of context, for example. When a traffic light is
category "red", the perception of non-movement (analogue) provides no
error, but in the same situation with the light being category "green"
the same analogue perception provides substantial error. The reference
perception might be construed as "moving when the light is 'green'".

Does this mean that the
lower-level reference signals are the sum of a set of
analog-error-generated outputs from higher analog control systems, and
category-error-generated outputs from a set of category control systems?

No, not necessarily. It might be so in the majority of cases, but I
envisage the two systems as being largely intertwined, such that
category perceptual signals are at the same level, hierarchically
speaking, as the analogue signals they categorize. It may be that this
is an unstable structure, and that what you suggest is what happens
after reorganization, but that isn't my conception.

I tend to _draw_ a category structure separate from and beside the
analogue hierarchy, but outputs from the category control units go
into the analogue hierarchy as contributions to reference signals,
and vice-versa. The drawing convention is a convenience that may not
represent the way things are--even if the Grand Flip Flop is itself
anything to do with the way things are.

Consider a simple compensatory tracking situation. We have a model that
seems to reproduce tracking behavior, in which a perception of the position
of the cursor relative to the target is maintained in a specific reference
state: cursor an inch to the left, on the target, an inch to the right, and
so on. The control equations are fairly straightforward and they seem to
fit what we observe reasonably well.

Now, according to my understanding of your proposal, there will also be
category signals being generated at this same level. We might guess that
the categories are "cursor on target", "left of target" or "right of
target," or any others you might prefer, like "approaching target" or
"departing from target." Whatever they are, as the tracking process
continues we will see changing categories; the situation will cross the
boundaries between categories and the category-perception signals will
change. As the boundaries are crossed, we will see category error signals
being generated. These error signals will generate outputs that go to lower
levels of control, just as the analog error signals do.

Yes, that may well happen.

Perhaps you will propose that the category error signals go to lower
category control systems.

No, as you see, I don't propose that. Let's propose a slight variant
on the normal tracking study, which shouldn't be too hard to program.
Rick might make it into a java Demo(?). Instead of showing the analogue
deviation of the cursor from the target, show only the category as
you defined first above: show three regions below the target, a small
"on-target" box, and two boxes for "left of" and "right of" target.
Let the box that would contain the analogue cursor be filled, and the
others be empty. Such a demo would illustrate "bang-bang" or category
control of an analogue variable. And it might be interesting to see
how much inferior that kind of control is to full analogue control. It's
almost bound to be a bit worse, but would it be much worse?

So my question is, how will the outputs differ if we consider the
analog-only tracking model, as opposed to the analog-plus-category tracking
model?

I'm not sure how we would conceive analogue-plus-category, if they were
of the same variable, since the analogue error presumably conveys more
information than the category error. Category control is useful when one
wants a category perception, not caring (much) about what analogue
values gave rise to that perception. For example, usually when one is
reading or writing, what matters is the words, or rather their meanings;
what does not matter is the type-face. But if one is laying out an
attractive book page, what counts is the _analogue_ perceptions relating
to the typesetting. To the reader, the typesetting is useful in speeding
(or making difficult) the category perceptions, but is not itself a
perception of interest (usually).

What migesting is to study analogue control performed to satisfy a
category reference at different levels. I'm not clear how we might go
about such a study, but it should be possible.

Martin

[From Bill Powers (970701.1436 MDT)]

Martin Taylor 970701 12:40] Canada Day --

Happy Canada Day!

Yes, except that the reference signal for a category control unit
need not come from another (higher) category control unit. It could
come from a (higher) analogue level. It might be that an analogue
system requires a particular category of context, for example. When
a traffic light is category "red", the perception of non-movement
(analogue) provides no error, but in the same situation with the
light being category "green" the same analogue perception provides
substantial error. The reference perception might be construed as
"moving when the light is 'green'".

You are collapsing into a single level functions that I visualize happening
at different levels in HPCT. In my model, it takes a higher level system to
determine that the color of a light, combined with a state of motion, is an
error, so that "red and moving" or "green and stopped" yield the
appropriate error, and the appropriate change in reference signal for the
motion control system.

With all the connections you are allowing, and all the rather indefinite
ways of combining signals, I don't see how you can predict what a system of
the kind you propose would actually do. There are so many latent loops that
figuring out even the sense of feedback would be difficult. You're starting
with things you hope the system would do, and proposing connections that
seem as if they would accomplish what you want. The real requirement of
modeling is to be able to work the other way: start with the connections
you propose, and deduce what a system designed that way would actually do
-- which may be nothing like what you want it to do. It is possible that
the arrangement you propose would do exactly what you imagine, but it is
also possible -- far more possible -- that it would behave nonsensically or
blow itself up.

Does this mean that the
lower-level reference signals are the sum of a set of
analog-error-generated outputs from higher analog control systems,
and category-error-generated outputs from a set of category control
systems?

No, not necessarily. It might be so in the majority of cases, but I
envisage the two systems as being largely intertwined, such that
category perceptual signals are at the same level, hierarchically
speaking, as the analogue signals they categorize. It may be that
this is an unstable structure, and that what you suggest is what
happens after reorganization, but that isn't my conception.

I'm beginning to think that you don't really have a model here -- only a
doodle. A word like "intertwined" doesn't suggest a functional system to me.

I tend to _draw_ a category structure separate from and beside the
analogue hierarchy, but outputs from the category control units go
into the analogue hierarchy as contributions to reference signals,
and vice-versa. The drawing convention is a convenience that may not
represent the way things are--even if the Grand Flip Flop is itself
anything to do with the way things are.

I think that if you tried to account for the same phenomena you're
imagining, but by using the HPCT structure as it stands, you could come up
with at least as plausible a system. What you're proposing really has no
structure; you're drawing lines over lines in your diagram until the whole
thing becomes a meaningless scribble. I can't take this seriously.

Best,

Bill P.

[Martin Taylor 970701 18:00]

Bill Powers (970701.1436 MDT)

You are collapsing into a single level functions that I visualize happening
at different levels in HPCT. In my model, it takes a higher level system to
determine that the color of a light, combined with a state of motion, is an
error, so that "red and moving" or "green and stopped" yield the
appropriate error, and the appropriate change in reference signal for the
motion control system.

Your description is exactly what I tried to describe. "The colour of a
light" is a category, not a set of three spectral intensity values, and
a "state of motion" is an analogue signal. You describe what I did, in
perhaps better-chosen words.

With all the connections you are allowing, and all the rather indefinite
ways of combining signals,

There's no indefinite way of combining signals. It's the same old way,
using the same old rules.

You're starting
with things you hope the system would do, and proposing connections that
seem as if they would accomplish what you want.

The _only_ change in the connection rules is to allow perceptual signals
at a level sometimes to connect to the inputs of perceptual functions at
the _same_ level as well as to the inputs of perceptual functions at the
next level. _ALL_ other connections are standard HPCT.

I envisage the two systems as being largely intertwined, such that
category perceptual signals are at the same level, hierarchically
speaking, as the analogue signals they categorize. It may be that
this is an unstable structure, and that what you suggest is what
happens after reorganization, but that isn't my conception.

I'm beginning to think that you don't really have a model here -- only a
doodle. A word like "intertwined" doesn't suggest a functional system to me.

Bad choice of words. I think of all the control units at a level as
being "intertwined" in exactly the same way. I don't make a distinction
between those that participate in cross-connections of their perceptual
functions and those that don't.

I think that if you tried to account for the same phenomena you're
imagining, but by using the HPCT structure as it stands, you could come up
with at least as plausible a system. What you're proposing really has no
structure; you're drawing lines over lines in your diagram until the whole
thing becomes a meaningless scribble. I can't take this seriously.

I _thought_ you had understood the connections I proposed. I was obviously
wrong.

Would it help if I said that the rules for reorganization are exactly
the same, but incorporate the possibility of same-level perceptual
interconnection, and that this leaves some of the control units as
what we (external analysts) see as category perceptions and some as
straight analogue perceptions?

Martin

[From Bill Powers (970701.2108 MDT)]

Martin Taylor 970701 18:00 --

You are collapsing into a single level functions that I visualize
happening at different levels in HPCT. In my model, it takes a
higher level system to determine that the color of a light,
combined with a state of motion, is an error, so that "red and
moving" or "green and stopped" yield the appropriate error, and
the appropriate change in reference signal for the
motion control system.

Your description is exactly what I tried to describe. "The colour of
a light" is a category, not a set of three spectral intensity
values, and a "state of motion" is an analogue signal. You describe
what I did, in perhaps better-chosen words.

I should have used smaller steps. The categories "red," "green," "start",
and "stop" are treated at the logic level by a control program: "If red and
moving then stop else if green and stopped then start." The program-level
outputs "start" and "stop" are category reference levels, telling the
category level control system to produce inputs to itself that it can
perceive as belonging to the categories of either "start" or "stop",
depending on the error at the program level. If the reference level is
"start" and the perception is "not start", the category error is then
transformed into reference signals for lower-level analog control systems
that adjust relationships and events to create analog signals seen as
members of the appropriate category at the category level, accelerating the
car so that "start" is perceived at the higher level, matching the "start"
reference category being received from the logic level.

Actually the logic level has to treat all cases, so that a "start"
reference category is accompanied by a "not stop" reference signal sent to
the "stop" category control system, and vice versa. Otherwise it would be
possible to hit the brake and the accelerator at the same time. If the
logic is complete this will not happen.

In my model, programs are always handled at the program level, categories
at the category level, and so on. There is only one level at which
categories are perceived and controlled. I hadn't intended to descibe the
situation in a way you could see as "exactly what I tried to describe," but
I didn't explain at sufficient length.

With all the connections you are allowing, and all the rather
indefinite ways of combining signals,

There's no indefinite way of combining signals. It's the same old
way, using the same old rules.

Does that mean you add the output of the category control system to the
output of an analog control system to give the net reference signal for a
lower level analog (or category) control system? Since categories represent
class membership while analog signals represent continuously-variable
quantities, I have a problem in seeing how this mixture of apples and
"orange soda" can produce any meaningful result. A small category error
would add something like a step function to the value of the lower
reference signal (or subtract it); wouldn't it just change the magnitude of
the controlled perceptual signal at the lower level? And wouldn't this
cause an error in the higher analog system?

You're starting with things you hope the system would do, and
proposing connections that seem as if they would accomplish what
you want.

The _only_ change in the connection rules is to allow perceptual
signals at a level sometimes to connect to the inputs of perceptual
functions at the _same_ level as well as to the inputs of
perceptual functions at the next level. _ALL_ other connections are
standard HPCT.

Yes, but these connections do not yield standard control behavior. If you
add these lateral connections between the input functions of control
systems at the same level, the control action of each system will greatly
disturb the control actions of the other system. In fact, it will be
impossible for both control systems to bring their perceptual signals to
values matching arbitrary reference values. One system or the other, or
both, will experience a large error, and will be generating a large output
resulting in extremes of action by the lower systems. The perceptual signal
in each system will represent the sum of effects from perceptions in lower
systems and a perception from another system at the same level, so the
relationship between levels of perception will become meaningless. What
does a perceptual signal mean at the configuration level if it is a
function of a set of intensity signals PLUS another configuration signal?
What does chair plus brown equal?

Once you set foot on this road, you have abandoned the basic logic of HPCT,
which involves a change of TYPE of perception with every change of level.
Perceptual signals of different types can never mix; the result would be
like adding size to color, or "in" to "salty," or feet to square feet. A
category is a different TYPE of perception from all those at lower levels
in HPCT. EVERY level involves a new type of perception.

You're striking off in a completely different direction, and I don't think
you have reasoned out what the consequences of your proposals really would be.

I _thought_ you had understood the connections I proposed. I was
obviously wrong.

The problem is that I don't think YOU understand them. You know what the
connections are, but not what the consequences of making them would be. You
are imagining that there would be certain consequences, but what guides
your imagination is not deduction from assuming those connections; you
simply assume that the consequences you imagine would in fact follow from
those connections. You seem to be ignoring the fact that these
cross-connected input functions are parts of control loops.

Same-level cross connections of the kind you propose, even if possible in
principle, would be ruled out by reorganization, because they would make
normal control action impossible. Try it: model two analog control systems,
and then add cross connections directly between their input functions. The
result will be _some_ kind of behavior, but not anything like what you
describe. It certainly could not be described as "control."

Best,

Bill P.

[Martin Taylor 970702 15:30]

Bill Powers (970701.2108 MDT)

Your description is exactly what I tried to describe. "The colour of >a

light" is a category, not a set of three spectral intensity >values, and a
"state of motion" is an analogue signal. You describe >what I did, in
perhaps better-chosen words.

I should have used smaller steps. The categories "red," "green," "start",
and "stop" are treated at the logic level by a control program:

Sorry, I was conceiving the relevant variable inputs to a perceptual
input as being category ["red or green"] and magnitudes [forward velocity
and acceleration].
The reference value is the value of a function which the outside analyst
might describe as ["red and zero and zero" or "green and accelerating at a
desired rate to a desired maximum speed"]. That _is_ different from
your function that is restricted to categories for the speed, and seems
to use a category for speed if the light category is "red" and a category
for acceleration if the light category is "green."

..., the category error is then
transformed into reference signals for lower-level analog control systems
that adjust relationships and events to create analog signals seen as
members of the appropriate category at the category level,

That's the way I see it happening, too.

Actually the logic level has to treat all cases, so that a "start"
reference category is accompanied by a "not stop" reference signal sent to
the "stop" category control system, and vice versa. Otherwise it would be
possible to hit the brake and the accelerator at the same time. If the
logic is complete this will not happen.

I agree that mutual exclusion at the logic level probably is important.
At the category level it happens only because that often is the way the
world has been observed to work.

···

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

Does that mean you add the output of the category control system to the
output of an analog control system to give the net reference signal for a
lower level analog (or category) control system?

Ah, you bring up a VERY important issue that has never been effectively
handled in my brief acquaintance with PCT. Particularly at the category
level and above, but also at the lower levels, we have to conceive of
a Reference Input Function that combines the contributions of th higher
level outputs. Whenever you have dealt with this explicitly, you have
always taken this Reference Input Function to be a simple summation,
and this can often work. That's the same kind of thing I do when I
say that all the Perceptual Input Functions together act like a
multi-layer perceptron--each PIF is seem as a sum-and-squash node.
But is it realistic? The MLP view works equally well on the output
side as it does on the input side, and it may well be that sum-and-
squash reference inputs are appropriate (if there's no squash, each
level of references is _only_ equivalent to a rotation of the reference
basis space, which plays havoc with the idea that HPCT is an improvement
on single-level PCT).

I think a thread on reference combination might be a valuable contribution
to the theory of HPCT.

Since categories represent
class membership while analog signals represent continuously-variable
quantities, I have a problem in seeing how this mixture of apples and
"orange soda" can produce any meaningful result. A small category error
would add something like a step function to the value of the lower
reference signal (or subtract it); wouldn't it just change the magnitude of
the controlled perceptual signal at the lower level? And wouldn't this
cause an error in the higher analog system?

If the category were simply summed with the analogue input, I agree
with this with the exception that the last query must be answered "maybe."
But suppose the category reference acted as a switch in the reference
input function, or something like that?

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

The _only_ change in the connection rules is to allow perceptual >>signals

at a level sometimes to connect to the inputs of perceptual >>functions at
the _same_ level as well as to the inputs of >>perceptual functions at the
next level. _ALL_ other connections are >>standard HPCT.

Yes, but these connections do not yield standard control behavior. If you
add these lateral connections between the input functions of control
systems at the same level, the control action of each system will greatly
disturb the control actions of the other system.

Quite true. It's very like the way the output of one control system can
disturb the input of another. But suppose the environmental feedback
function is such that the control of categories is _useful_, and that
it is sometimes good that one does not perceive a light as being both
somewhat red and somewhat yellow. Suppose that it is good that if the
light is seen as "red" it is not seen as "yellow" except under exceptional
circumstances. Then the fact that perceiving "red" has the same effect
on the "yellow" perception as would an actual reddening of the sensory
input might greatly ease control.

The presumption of _all_ category perceptual control (not just of my
particular proposal) is that there is a reference signal for seeing a
particular category, which means bringing the relevant set of analogue
inputs into the range appropriate for that category. Usually, these
category references have some exclusionary properties consequent on
the operations of the logic level (which itself, I believe, relates
to how the world normally behaves as it is perceived). So, it usually
helps category control if, when "red" is the reference, "yellow" is
unlikely to have much magnitude when the sensory data are appropriate
to "red."

If the foregoing is true, then on average the disturbance that undoubtedly
occurs in the way you describe will ordinarily be in a direction to
reduce perceptual error, quite unlike the disturbances that come from
random variation in an unknown environment.

Of course, the reverse _can_ happen; and we know that it is very hard
to train oneself to see both directions of a reversing (Necker) cube
at the same time, though it can be done. The disturbance you mention
makes it hard to see a light as "yellow" and as "red" at the same time
when those are the references you want. But it makes it easy to see "red"
and at the same time identify the label "red" for the category you see.
There are tradeoffs, as there usually are.

In fact, it will be
impossible for both control systems to bring their perceptual signals to
values matching arbitrary reference values.

Quite so. The point is that it will not normally be the case that both
reference signals are independent for categories that do not normally
occur together. One is usually high when the other is low.

One system or the other, or
both, will experience a large error, and will be generating a large output
resulting in extremes of action by the lower systems.

Yes. And have you ever really _tried_ to perceive an ambiguous pattern as
one and another thing at the same time (and I don't mean the kind of
non-conflicting categories you mentioned before, such as "mine" and
"shirt" or whatever it was you used as an example). It's a big effort,
which seems to suggest that what you say is actually what occurs.

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

What does chair plus brown equal?

Something that is not satisfied by a red chair or a brown table. Do you
insist that "plus" means "arithmetic sum"?

Once you set foot on this road, you have abandoned the basic logic of HPCT,
which involves a change of TYPE of perception with every change of level.

The TYPE of "brown-category" is the same as the TYPE of "brown-magnitude"
in all respects. It's the same level, so far as the GFF proposal is concerned.
That's how you get the perception of "yellowish-brown"--as a "brown-category"
and a magnitude of the non-category brown and the non-category yellow.

Remember that when I unwillingly accepted the need to restate the design,
I pointed out that the separation into category-perceiving units and
"standard" units was an arbitrary convention that made it easier to describe.
Putting the category perceptions off to the side is a nice convenience,
but it obscures the fact that the category perceptions are of the same
type as the magnitudes of which they are categories.

Now, in respect of "chair plus brown", the same kind of question can
be raised in respect of the "standard" HPCT structure. A category "chair"
derives from some kind of configuration analogue perception, I suppose,
and a category "brown" derives from contextually related spectral
intensities. These are now both at the "category level" which makes it
OK to create a perception based on both inputs. That perception has a
magnitude that has to be brought to some reference value. What is the
meaning of a reference "chair plus brown"? Outputs to reference inputs
can't skip levels the way the incomping perceptual signals can, can they?
So what sensible intermediate perceptions are controlled to get "chair
plus brown"--perceptions that are NOT involved in the "brown chair"
perception itself?

I'd say that there is complexity hidden there that is not hidden (and
not complex) in the GFF structure.

What I was trying to avoid by segregating the category and analogue
perceptual systems in my description of the GFF was the possibility
that we might get into muddles like this one.

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

You're striking off in a completely different direction,

Not as I understand it at all. All I'm doing is proposing a mechanism
for a level that has never had a mechanism (other than simple
thresholding, which doesn't seem to conform to the facts of category
perception). It's a mechanism that falls out of a simple relaxation of
an arbitrary rule, a rule unsupported by observation or experiment.
It seems to have some behaviours that nicely conform to observations.
Other than this specification of mechanism, I don't see _any_ change
of direction.

and I don't think
you have reasoned out what the consequences of your proposals really would be.

Not in full detail, obviously not. Even after 40 years, you haven't
discovered all the consequences of the "standard" HPCT structure, so
I don't think it unreasonable that I should not have reasoned out all
the consequences of allowing perceptual outputs at a level to reconnect
back to inputs at the same level.

I _thought_ you had understood the connections I proposed. I was
obviously wrong.

The problem is that I don't think YOU understand them. You know what the
connections are, but not what the consequences of making them would be. You
are imagining that there would be certain consequences, but what guides
your imagination is not deduction from assuming those connections;

That's unfair. The imagining may not be complete, but it is buttressed
my a small amount of simulation (less than I would like, but some), and
experience with running experiments using "multiflops" built with the
very first DEC logic modules. And some of it is "deduction" if that's
what you call the kind of analysis you do on steady-state control systems.
The part that's harder to imagine is the short-term dynamics. That needs
simulation.

Same-level cross connections of the kind you propose, even if possible in
principle, would be ruled out by reorganization, because they would make
normal control action impossible.

Actually, I think they are almost inevitable if we have both Hebbian and
anti-Hebbian reorganization along with e-coli type reorganization, and
if the environmental feedback function is such that control of categories
is valuable for intrinsic variable control. (For anti-Hebbian learning
in the neural system, see H. Markram, et al., Science 275, 213 (1997).)

Try it: model two analog control systems,
and then add cross connections directly between their input functions. The
result will be _some_ kind of behavior, but not anything like what you
describe. It certainly could not be described as "control."

Well, you try it, too. But make sure that the reference for one of the
analogue control systems is "high" when the reference for the other is
"low".

It depends a great deal on the mutual gain. As I pointed out in my earlier
message, if the loop gains get too high in a multi-node system, you can get
clamping, oscillation, or chaotic behaviour. If there are only two units,
the entire behaviour can be described in terms of a cusp catastrophe, in
which the control parameter is gain.

------------------
It would be nice if you would contemplate simple ideas simply, rather than
trying to turn them into great complexities which you can then assert to
be unrealistic, impossible, unthought-out, ... so that you can avoid taking
them seriously. This _is_ a simple idea, that has perhaps a bit more
thought behind it than you assert. It derives from contemplating the
effect of reducing slightly the rigidity of the rules of the hierarchy
(actually simplifying the hierarchy), but making _no_ other changes to
standard HPCT.

You are the one who asserted that you made (not found) the rule of no
same-level interconnects because you couldn't see what they would do.
It's not a rule with any great backing, is it? Why not spend a few
moments contemplating what relaxing the rule might do, how a GFF might
come about naturally, and whether it doesn't actually simplify rather
than make more complex the structure of HPCT?

Martin

[Martin Taylor 970702 21:50]

Bill Powers (970702.1747 MDT)]

Martin Taylor 970702 15:30--

Particularly at the category
level and above, but also at the lower levels, we have to conceive
of a Reference Input Function that combines the contributions of the
higher level outputs. Whenever you have dealt with this explicitly,
you have always taken this Reference Input Function to be a simple
summation, and this can often work. That's the same kind of thing I
do when I say that all the Perceptual Input Functions together act
like a multi-layer perceptron--each PIF is seem as a sum-and-squash
node.

Much of what you say, I think stems from your concept of the levels in HPCT
as a big perceptron.

You go exactly the opposite direction to the question I am trying to get you
to address. I acknowledge that there are probably a variety of different
kinds of perceptual input functions for the different levels. But, like
you, I prefer to keep the simplest assumptions until they are shown to
be inadequate.

Now, I was asking you to address the question of how the _outflow_ part
of the control systems might _also_ have different kinds of combining
functions. I used the example of the perceptron here to emphasize how
there is no value in having levels at all if the perceptual inputs are
simple summators. That argument (which I believe you understand and
have long understood) seems also to apply to the combination of
output signals into reference levels.

You faulted the GFF proposal on the grounds that it makes no sense
simply to sum a category reference with an analogue magnitude reference.
I agree, and asked for some discussion on the different ways
outputs may combine to form references for the lower levels.
Simple summation seems unworkable in the general case, just as it
does on the perceptual side.

////

But the problem with the perceptron approach as an
explanation of _everything_ is that there are kinds of perceptions that
"sum-and-squash" doesn't explain. A perceptron seems to be good for one
kind of thing: categorizing.

I'm not at all clear why it is that every few months you return to this
statement. Perceptrons just aren't very good at categorizing, even though
people do use them rather effectively for this purpose. They are much
better at finding quantitative relationships among things, such as the
event that consists of changes in spectral quality that represents a
consonant-vowel transition. For such things, of course the perceptron
input has to include time-derivative as well as state information.
Perceptions are good at determining the degree to which input data
are like arbitrary configurations in space-time. It's the users of
perceptrons who use the outputs to do categorization.

So it's no wonder that you want to see
categories at every level.

Are you now disavowing your earlier claim that you can see a category "red"
a category "above" and a category "chair"? Oh, yes, one can see categories
at every level (except possibly intensity and transition and perhaps
even there).

The reason I harp on category perception in these posts is because these
posts are about category perception. In posts not about category
perception I talk about other things.

However, a perceptron wouldn't be much good at
perceiving, for example, an event, which is simply a space-time package of
lower-level signals.

Oh, sorry for having used this as an example of what perceptrons do well.

Nor would it be much good at doing logic, or grasping
principles.

You are probably right. But why do you spend so much of this post on
a discussion of what perceptrons won't do? I thought from the Subject
line that this would be a message relevant to the discussion of whether
the GFF has any merit in respect of category perception, and if not,
why not. The performance of perceptrons is simply a red herring, making
it unnecessary to address the topic of category perception. I used the
example of the perceptron to make the point that a simple summation of
output signals at the lower reference input is probably untenable. And
I used that only because you had asserted that a simple sum of a category
reference and an analogue reference wouldn't work very well. I said I
agreed, didn't I? And produced the perceptron example to show why.

////

One of the concepts behind HPCT is that each level is characterized by a
kind of input transformation typical of that level, across all modalities.

Fine. As I pointed out several times, the GFF uses whatever kind of
input transformation is typical of that level, across all modalities.

Going from intensities to sensations seems to require nothing more than
simple weighted summations. But going from sensations to configurations
involves extracting invariances with respect to orientation, size, color,
and so forth -- while preserving _continuous_ relationships. The kinds of
computations required at this level must include things approximating
trigonometric functions, and other processes nobody has even guessed at
yet.

Fine. It's not an issue, whether what you say is correct or not. I'm
not disputing it, just saying it's irrelevant and not in question here.

I don't think there is
ANY principle of computation that is common to all the levels. I think that
each new level introduces principles of computation that don't exist at any
lower level.

So, let's proceed as if that were true, and get on with the discussion
on that basis.

I really don't want to get into discussions of detailed models and what is
wrong with them. If you can pull your various ideas together into a
convincing, internally consistent, and workable model, by all means do so.
But I think you have a way to go.

I find it very hard to know wherein I have failed in this, at least insofar
as the model is described to the degree of detail that the standard HPCT
model is described, and has been as little simulated in its "other processes
nobody has even guessed at yet." In our few little simulations the
perceptual side of the model does perform as described. I grant we have
not used it inside a control system, but to do so would require the
construction of an environment rather more complex than your "monster."

(The environment is usually the hardest thing to construct in such
tests, and I admire your efforts on your monster. But it would be
inadequate to test the GFF properly. If there is a real issue with
the GFF, I think the issue is in how it scales, particularly in
reorganization. In that respect, if you have read "At Home
in the Universe", which Mike Acree recommended today, you'll see that
the issue of how many of the perceptual functions within a level are
cross-linked could be a significant issue--but I seem to remember you
did read it, didn't you, after I recommended it a few months ago?)

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

I find it hard to avoid the impression that you really don't want to consider
the possibility that a simplification of "standard" HPCT might perform as
well as the original. Each successive posting avoids entirely the point
of the one to which it purports to respond, even when my postings treat
your comments point by point. Such glancing discussions advance PCT
very little.

I propose your credo--keep it as simple as it can be, until evidence
shows something new is needed. And keeping it simple is what the GFF does.
In the process it proposes an "other process nobody has [previously]
guessed at" for one of the perceptual levels.

Martin

[From Bill Powers (970703.0731 MDT)]

Martin Taylor 970702 21:50] --

You go exactly the opposite direction to the question I am trying to
get you to address.

You're raising a large number of points; I just got hung up on one of them.

The question of how outputs combine to generate reference signals is, I
have thought, somewhat simpler than the question of how perceptions are
formed. Remember that typically one system of a given level adjusts
reference signals for many systems of the next lower level. Different
systems of a given level, however, will in general send outputs to somewhat
different sets of lower-level systems, with different weights (positive or
negative is the main choice). Rick's spreadsheet demo is a good example; it
contains 6 systems at each of 3 levels, and each higher system contributes
to the reference setting of ALL 6 systems of the next lower level, with
weights of only -1 or 1. In his demo there were the same number of outputs
as lower-level systems, which is the worst case; more generally there would
be more lower systems than higher, so the overlap would not be complete.
Note that the highest system in Rick's demo controlled logical functions;
the outputs were either on or off. Nevertheless, these signals were added
together algebraically at the reference inputs of the next lower (analog)
level, so an analog (multiple-valued) reference signal was generated for
each analog system. Nevertheless, the logical functions were controlled.

In Rick's demo as in some I have played with, the only choice of output
weightings was 1 or -1. The reason was that the main problem is assuring
that for a given system, all feedback loops traced from outputs, through
their effects on the lower-order world, and back to that system's
perceptual signal, must result in negative feedback (or most of them
should). This assures negative loop gain in the overall loop. Exact
orthogonality of output effects of different control systems of the same
higher level is not critical on the output side. A considerable departure
from a "90 degree angle" can occur before interactions become prohibitively
strong.

For the analyst, the main question is the relation between the error signal
in a control system and its net effect on the perceptual signal in the same
system. A superposition principle applies; if many higher control systems
share an environment of many lower systems, the feedback effects from error
to perception can remain negative for each of the higher systems, even
though no one control system at a lower level is uniquely influenced by ANY
single higher-level system. We tend to convey the wrong picture when we
draw diagrams with just one higher system and one lower system; in reality,
the "one" lower system is a largish set of lower systems, which are also
part of the loops of other higher systems. The net reference signal
reaching a lower system doesn't reflect the errors in any one higher system.

The main unanswered question on the output side is the nature of the output
functions. Particularly troublesome is how the error signals resulting from
a perceptual signal representing a dynamic perception like a rhythmic
motion (rate of repetition, for example) is converted into the necessary
dynamic changes in lower-level reference signals. Rick and I have come back
to that problem at intervals. Clearly, the output function must be some
sort of pattern generator; a variable-frequency, variable amplitude
oscillator, for example, with its amplitude and frequency varied by error
signals. This implies an output function common to several control systems,
with, for example one control system varying the frequency and another the
amplitude of oscillation.

I grasp these problems only dimly, which is one reason I concentrate mostly
on lower-level control processes, or on strictly behavioral analyses.
Identifying control processes from the external observer's point of view is
one thing; figuring out how the system must be designed in order to do the
things we observe is very much another.

You faulted the GFF proposal on the grounds that it makes no sense
simply to sum a category reference with an analogue magnitude reference.
I agree, and asked for some discussion on the different ways
outputs may combine to form references for the lower levels.
Simple summation seems unworkable in the general case, just as it
does on the perceptual side.

So far, I haven't been able to model higher-level control processes well
enough to know whether simple summation is always enough. In the few cases
tried it does seem sufficient, but that doesn't answer the general
question. I have chosen simply to leave the question open until such time
as simple summation proves inadequate.

-----------------------------
Perceptrons just aren't very good at categorizing, even though
people do use them rather effectively for this purpose. They are
much better at finding quantitative relationships among things, such
as the event that consists of changes in spectral quality that
represents a consonant-vowel transition.

This is not what I think of as "finding a quantitative relationship." The
output of the process is a signal indicating whether or not a specific
event has occurred, a discrete indication and not a continuous
representation. A perception that represents a quantitative relationship
should vary continuously as that relationship changes, indicating its
current state.

Your bringing up this example greatly increases my discomfort with the
so-called "event" level of perception. The fact that an event is either
occurring or not occurring seems to give it the flavor of a discrete
perception, not an analog one. This makes it into an oddity; a discrete
perception located low in the analog part of the hierarchy, below
continuously-variable relationships. Several years ago, at Gary Cziko's
instigation, I split the old "sequence" level, which was then the 5th level
(as in B:CP) into two levels: the supposedly analog event level and a pure
sequence level, in which only ordering matters, which was moved above the
category level. Now I'm wondering whether it wasn't a mistake to leave that
"event" level behind; perhaps the 5th level as then defined should simply
have been dropped out. Maybe something in the back of my mind just didn't
want to have to do any more re-numbering of levels. On such trivia are
great theories formed, and dissolved.
------------------------

So it's no wonder that you want to see
categories at every level.

Are you now disavowing your earlier claim that you can see a
category "red" a category "above" and a category "chair"? Oh, yes,
one can see categories at every level (except possibly intensity and
transition and perhaps even there).

No, I'm not disavowing it, but your putting it this way suggests a way out
of our impasse. I have always assumed (by necessity) that the input
function of a given level can receive perceptual signals from ANY lower
level: there can be relationships, for example between intensity signals.
This is necessary because we can perceive that one light is "brighter than"
another light, which is a relationship between two independent intensities.
We can perceive transitions of sensations: increasing warmth. The only way
to preserve the idea that perceptual signals can reach only the next higher
level would be to assume some sort of "unity transform" at each intervening
level, which simply relays the signal. That seems unnecessary, especially
since we know that even first-order perceptual signals, in some cases at
least, send copies directly, without synapsing, into the cerebral cortex.

The main reason for prohibiting order-skipping in the downward direction is
that if a higher system sends a reference signal to a system several levels
below, directly, the result will be a change in the perception of that
system (through control action), and that will disturb any control systems
at an intervening level that also send reference signals to the lower
system and receive copies of its perceptual signal. So those systems that
were skipped over will simply react to the disturbance by adjusting their
contributions to the lower-level system's reference signal, and nullify the
effects of the signal from the higher system. Such order-skipping outputs
could, anatomically, exist, but since they would accomplish nothing they
would probably not survive reorganization, or evolution.

On the _input_ side, there is nothing to prevent your proposal from working
-- a category-perceiving function might exist at any level, generating a
category signal out of the analog signals at that level. In general this
category input function would have to receive analog inputs from many
systems at the same level, so there couldn't be a category input function
associated with _each_ system at that level; its inputs would be copies of
_many_ perceptual signals at that level. A category of one signal doesn't
make much sense.

The main problem with your proposal comes up with the _output_ side of a
category control system. If there is a category error, the error signal
must alter the reference signals of lower systems. But that amounts to
order-skipping in the downward direction, with the results I have deduced
above. Assuming that there are levels of perception and control between
categories and, say, configurations, the systems higher than the
configuration level, which also receive copies of the perceptual signals
from the lower levels, would be disturbed by the action of the lower-level
category control system, and would act to nullify its effects.

There is another problem with your proposal to allow cross-connections
between the input functions of analog control systems at lower levels. I
tried to point it out yesterday. Your answer was to say that the reference
signals must not be set in a way inappropriate to the effects of the
cross-connection. But the setting of the reference signals is not the
responsibility of the system that receives them; it is the result of
control actions by many higher-level systems which by definition know
nothing of how the perceptual signals they receive are generated, or indeed
about any individual signal received from below.

At best, your flip-flop arrangement can exist only as a _separate_
treatment of the signals in lower-level systems; it must work with _copies_
of these signals, so as not to disrupt the normal control actions that are
going on. In other words, for a simple example, this separate system would
receive copies of the perceptual signals in two lower-level analog control
systems, and using a couple of neurons hooked up as a flip-flop, create the
category or association signals you descibe. But this would be done without
altering the operation of the two analog systems, which would control as
they normally do.

If you could accept this picture, our models would no longer differ
(structurally) at all. The copies of the analog signals from the two (or
multiple) analog control systems at any level would be carried upward, as
far as necessary, to the level in the cerebral cortex where ALL category
perceptions are generated, the flip-flops (if any) being physically
distinct from the input functions of the lower-level systems. At that place
in the cerebral cortex, the category signals would be compared with
reference signals, and the error signals would be transformed into multiple
reference signal settings of the _next lower order of control_, as is
required by the prohibition of level-skipping in the downward direction.
----------------------------------

I thought from the Subject
line that this would be a message relevant to the discussion of
whether the GFF has any merit in respect of category perception, and
if not, why not. The performance of perceptrons is simply a red >herring ...

Your GFF, if conceived of as a mode of operation of a single category level
working with _copies_ of lower-level signals, involves two distinct
operations. First, there must be some sort of "OR"-like function, such that
signals from separate and largely arbitrary sources can lead to one
particular category signal. A wheel, OR a fender, OR a windshield, OR a
gasoline pump, can give rise to the category that is also indicated by the
symbol "car." The second consideration brings in the questions of
association and mutual exclusivity. These would be implemented by
cross-connections between the separate category input functions, so that
presence of one category signal would suppress or enhance the outputs of
other input functions at the same level. Not all categories are mutually
exclusive with all other categories; nor do all categories evoke
associations with all others. The cross-connections would nicely explain
some phenomena, such as hysteresis and mutual exclusivity, but where those
phenomena don't occur the cross-connections would be assumed absent.

So your GFF, with suitable recognition of the way different input signals
are combined to create even single categories, might make a reasonable
model for the category level of perception. I don't know that; some
modeling is called for. It is quite possible that a perceptron model would
also be appropriate at this level; there is, after all, internal structure
in the category level, with classes and subclasses and so on. It is also
possible that what we now think of as a single level is really several
levels collapsed together in our conceptions.

Best,

Bill P.

[Martin Taylor 970704 0:35]

Bill Powers (970703.0731 MDT)

Thanks for providing answers that need chewing on. I shall do so before
I respond fully. But there's one point, still, on perceptrons.

Martin Taylor 970702 21:50] --

-----------------------------
Perceptrons just aren't very good at categorizing, even though
people do use them rather effectively for this purpose. They are
much better at finding quantitative relationships among things, such
as the event that consists of changes in spectral quality that
represents a consonant-vowel transition.

This is not what I think of as "finding a quantitative relationship." The
output of the process is a signal indicating whether or not a specific
event has occurred, a discrete indication and not a continuous
representation.

No, no, no. That's what I was trying to get across. The output of a
perceptron is _always_ a vector of real numbers. The perceptron _does
not_ categorize. The discrete output is created by thresholding the
continuous output of the perceptron, some other device (or person) making
a decision that if one vector element is greater than X, and the others
are less than Y, then a particular input pattern has been seen. But
the perceptron outputs can vary as continuously as their number
representation allows them (i.e. as continuously as any analogue
simulated by floating-point numbers).

A perception that represents a quantitative relationship
should vary continuously as that relationship changes, indicating its
current state.

Yes.

Your bringing up this example greatly increases my discomfort with the
so-called "event" level of perception. The fact that an event is either
occurring or not occurring seems to give it the flavor of a discrete
perception, not an analog one.

Why is an event "occurring or not occurring" any more than a light is
"bright" or "dim"? The event is a value of a perceptual signal that
is the output of a function tuned to perceiving that (class of) event.
That vale goes up and down over time. An external analyst (or category
perceptual function) can take this continually varying analogue value
and turn it into "occurring" (or not). But that dichotomy is not a
property of the event percetion mechanism--at least not as I (or a
perceptron) would see it.

The rest of your message is more germane to the discussion of category
perception, and deserves a response generated in a more wakeful state.
Thanks again for it.

Martin

[Martin Taylor 970704 10:00]

Bill Powers (970703.0731 MDT)

Happy Independence Day to our US readership.

Martin Taylor 970702 21:50] --

You go exactly the opposite direction to the question I am trying to get
you to address.
You're raising a large number of points; I just got hung up on one of them.

In the message to which I am now responding, you answered a lot of my
points and raised others. In today's message I want to address just two.
I'll try to respond to the others later.

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

In the reverse order to that in your message: (1)

At best, your flip-flop arrangement can exist only as a _separate_
treatment of the signals in lower-level systems; it must work with _copies_
of these signals, so as not to disrupt the normal control actions that are
going on. In other words, for a simple example, this separate system would
receive copies of the perceptual signals in two lower-level analog control
systems, and using a couple of neurons hooked up as a flip-flop, create the
category or association signals you descibe. But this would be done without
altering the operation of the two analog systems, which would control as
they normally do.

I _almost_ agree with this, since it is the configuration we settled on
the last time the GFF was discussed. You made the same point then, and
I accepted it as a correct statement. I still do.

The reason I say "almost" is that there is no _necessary_ link between a
specific category perception and a corresponding analogue perception.
It is simplest to assume that there is a one-to-one mapping, and
that the input to any category input function is a single copy
of some analogue perception at that same level. However, I think it more
probable that the input functions of category-control units at one level
will take their inputs from the same set of next-lower-level perceptions
as do the inputs of the analogue-control units. In other words, the
categories at a level may span the same basis space as the analogue
perceptions at that level, without necessarily mapping one-to-one between
category and analogue perception.

Why a one-to-one mapping might come about naturally is that it is not
improbable that those perceptions that prove useful (for intrinsic
variable control) as analogues may also prove to be useful as categories.
If that happened, then the category-connected units would tend to have
the same input functions (same inputs, same weights and functional
connections) as corresponding analogue functions, which would act
exactly as if the category units took copies of the analogue
perceptions as their own inputs.

Which provides a lead-in to the next comment.

There is another problem with your proposal to allow cross-connections
between the input functions of analog control systems at lower levels. I
tried to point it out yesterday. Your answer was to say that the reference
signals must not be set in a way inappropriate to the effects of the
cross-connection. But the setting of the reference signals is not the
responsibility of the system that receives them; it is the result of
control actions by many higher-level systems which by definition know
nothing of how the perceptual signals they receive are generated, or indeed
about any individual signal received from below.

Yes, I agree with all that, except the idea that this is "a problem."

To see why it may not be a problem, we have to enquire why category
perception exists--no matter what the mechanism or structural organization.

Assumption 1: All perceptual types that exist in most individuals of a
species are sustained by reorganization that occurs as these individuals
exercise control within their particular environment.

Assumption 2: Only those perceptual types for which control enhances
the stability of intrinisic variables will be sustained by reorganization.
By "intrinsic variables" here, I do not mean any function of perceptual
control error. I mean those variables whose effective control enhances
the likelihood that the genetic structure will propagate to the next
generation. Crudely but inaccurately, this can be stated as: an intrinsic
variable is one for which control failure is likely to lead to the
death of the organism.

Now we ask why there should be category control _at all_. Under these
two assumptions, it must mean that to control categories is at least
sometimes better for the control of intrinisic variables than to control
the analogue variables that correspond to the categories--that it is
sometimes better to identify "friend" or "predator" and to control for
the perception of the one and the non-perception of the other than it
is to control the exact magnitudes of the perception of "friendness"
and "danger-to-me-ness".

If it is sometimes better to control for category perception than to
control for the corresponding analogue perception, something about the
environment must be such that performing category control has the side-
effect of stabilizing the intrinsic variables better than does analogue
control of the perceptions. (All "control" of intrinsic variables
comes about because of the side-effects of controlling perceptual
variables _in the environment of the moment_.) Perception of the
category "predator" is likely to result in the imminent disruption
of the values of intrinsic variables, when the perceiver gets eaten.
It doesn't matter what kind of predator it is, or whether the tiger
has a bushy tail. The environmentally important reference is that
the sensory array should not include "something-that-might-eat-me".

If the control of categories matters in this way, it is because the
side-effects of one course of action may well differ discontinuously
from the side-effects of another course or courses. To continue to
browse when a predator is perceived may be incompatible with running
away to reduce the predator perception to a safe low value. One runs
or one browses, but there is no continuum of half-running, half-browsing.
"Fight or flight" is another often-cited discontinuous action choice,
which depends on a category perception.

Choice, here, does not mean explicit, logical, choice. It means that
there are reference values for the perceptions "predator" (low) and
"friend" (high in social animals), or for "defeatable if opponent"
(high) and "too-strong if opponent" (low). (Notice that this last pair
is a _logic-level_ pair of perceptions).

The conclusion I draw is that category perceptions exist because they
work in the real world. They work because the world is _really_
structured in such a way that dichotomous courses of actions sometimes
lead to qualitatively different results, and that those results have
corresponding reference values which often are high or low in
mutual exclusion. That is why I suggested to you that you test for
category perception control using only reference values that had
a kind of Xor relationship. It is because I believe category
perceptions can exist stably only in an environment in which
discretely different results come from perceptions of one category
or the other in the presence of references which seldom can usefully
be high together.

What this implies is that the mutual exclusion that seems to exist
in the perceptions of categories within a "class" (e.g. "red" vs "green",
"mine" vs "yours", "chair vs table", "democracy vs dictatorship") exist
for two reasons that sustain each other in a positive feedback loop
(in the reorganization process). The world is such that there exist
situations in which it is useful to set a reference value for perceiving
one member of the class and not the others; and the actions that bring
about the desired reference (categorical) perceptions are dichotomous
and cannot be performed together.

There's no need for any one level to "know" the meanings of the signals
they receive from lower or higher levels. But the world is such that
reorganization stabilizes those perceptual types that work. If reference
signals "demanded" the simultaneous perception of "red" and "green"
much of the time, it is unlikely that those categories would be
perceived as categories, at least not as belonging within a mutually
exclusionary class--provided that the real-world environment collaborated
by providing sensory inputs that could be construed as both at the same
time.

So we come to the conclusion you earlier stated: mutual exclusion
is the province of the logic level, and not of the category level as
such. But this conclusion can be taken one step further, suggesting
that the mutual exclusion that _works_ at the logic level works because
of the way the real-world environment is structured, and that permits
reorganization to build category perceiving systems that themselves
exhibit tendencies to mutual exclusion.

Hence, returning to your comment, I do not see a problem in the claim
that category perceptions will usually occur with reference values
that are as mutually exclusive as are the category perceptions themselves.

Other points in your message later.

Martin

[From Bill Powers (970704.1140 MDT)]

MArs Pathfinder is on the surface of Mars, apparently upright on its base
petal (the best case), and in process of retracting its airbags.

Martin Taylor 970704 10:00] --

It is simplest to assume that there is a one-to-one mapping, and
that the input to any category input function is a single copy
of some analogue perception at that same level. However, I think it
more probable that the input functions of category-control units at
one level will take their inputs from the same set of next-lower-
level perceptions as do the inputs of the analogue-control units. In
other words, the categories at a level may span the same basis space
as the analogue perceptions at that level, without necessarily
mapping one-to-one between category and analogue perception.

I don't understand your concept of what a category perception is. In mine,
a category indicates that _some member of a set_ is present. I suppose that
the least set is a single member, but in general there would be many
members, ANY ONE OF WHICH would indicate presence of the same category. Any
dog, dachshund or Great Dane, is "a dog."

Why a one-to-one mapping might come about naturally is that it is
not improbable that those perceptions that prove useful (for >intrinsic
variable control) as analogues may also prove to be useful >as categories.
If that happened, then the category-connected units >would tend to have the
same input functions (same inputs, same >weights and functional
connections) as corresponding analogue >functions, which would act exactly
as if the category units took >copies of the analogue perceptions as their
own inputs.

This seems unlikely _IN THE EXTREME_. One of the guiding principles in my
conception of the hierarchy has been complete avoidance of duplication of
functions, most particularly when two physically separate systems must
coordinate with each other. It is very unlikely that two input functions
could have the same inputs, same weights, and same functional connections
as any other input function. And in a modular hierarchical system,
duplication of function is all but unnecessary.

I don't see why you would want to duplicate the input functions of, say, a
configuration recognizer just to create a class of configuration-category
signals, when you have available copies of the _outputs_ of the input
functions, the perceptual signals indicating the configurations that are to
be categorized together. If you have separate signals indicating a square,
a rectangle, a parallelopiped, and other four-sided configurations, all
these signals can be input to a configuration-perceiver that produces an
output that might be labelled "quadrilateral."

And I don't see how you could ever create a configuration-level categorizer
whose inputs were sensation-signals, which would generate a
configuration-category: the result, by my way of understanding categories,
would be a _sensation_ category: "blue", for example. Blue is the name of
an array of sensation signals, each representing a different weighted sum
of color intensities, but called by the same name (or at least represented
by the same category signal). Any signal from that array would give the
same color-category signal that any other signal from that array would
give. That is the essence of the categorization process as I see it.

Which provides a lead-in to the next comment.

To see why it may not be a problem, we have to enquire why category
perception exists--no matter what the mechanism or structural organization.

Assumption 1: All perceptual types that exist in most individuals of
a species are sustained by reorganization that occurs as these
individuals exercise control within their particular environment.

Assumption 2: Only those perceptual types for which control enhances
the stability of intrinisic variables will be sustained by reorganization.

You’re making a third assumption that is not mentioned:

Assumption 3: the basic neural computing capabilities and connectivities
needed to construct a perception of the category type are present. This
considerably modifies you next statements:

If it is sometimes better to control for category perception than to
control for the corresponding analogue perception, something about
the environment must be such that performing category control has
the side-effect of stabilizing the intrinsic variables better than
does analogue control of the perceptions.

So your assumption is that if it would be better for an organism to control
for category perception, it will develop the ability to do so. This is a
non-sequitur; it would perhaps be better for bacteria to have systems
concepts, but if they don’t have the evolved structure in which system
concepts can develop, they’re not going to learn them no matter how long
they reorganize. There is no physical possibility of developing that kind
of control.

All "control" of intrinsic variables
comes about because of the side-effects of controlling perceptual
variables _in the environment of the moment_.) Perception of the
category "predator" is likely to result in the imminent disruption
of the values of intrinsic variables, when the perceiver gets eaten.
It doesn't matter what kind of predator it is, or whether the tiger
has a bushy tail. The environmentally important reference is that
the sensory array should not include "something-that-might-eat-me".

That is anthropomorphic reasoning, which you can carry out because you have
a category level. But an organism can avoid things that might eat it
without perceiving that they might eat it, and find food without
categorizing it as "food." You can categorize behaviors of another organism
that it cannot itself categorize. You can even believe that the other
organism is categorizing, because you can place some of the things it
controls into a recognizeable category, like "fleeing predators." But this
does not mean that the other organism perceives "fleeing" or "predators" as
categories; it may simply avoid the perception of particular shapes, one at
a time, without perceiving them as members of a single category as you do.

The implication is that some organisms can and do function without a
category level, just as they can function without perceiving or controlling
principles or system concepts. And this argues that category perception IS
a level, a coherent level of perceptual computation that has a physically
distinct existence from the lower levels, and is specialized to deal in
terms of categories.

I think your basic mistake is in assuming that simply because an organism
might be better off with any given level, it will therefore reorganize
until it possess that level of perception and control. We have to consider
evolution and phylogenetic differences; not all organisms have all levels
of control, nor are all levels necessary for survival. While many kinds of
organisms can reorganize, not all kinds begin with the right
pre-organization to allow developing all levels of control.

Best,

Bill P.

[Martin Taylor 970705 16:10]

Bill Powers (970704.1140 MDT)

I don't understand your concept of what a category perception is.

Perhaps that's the reason why you have so much trouble understanding
that I conceive of no such thing as a "category type perceptual input
function", and why you say things like:

You're making a third assumption that is not mentioned:

Assumption 3: the basic neural computing capabilities and connectivities
needed to construct a perception of the category type are present.

You could not make such an assertion if you took seriously what I have
been trying to present.

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

In mine [concept of what a category perception is],
a category indicates that _some member of a set_ is present. I suppose that
the least set is a single member, but in general there would be many
members, ANY ONE OF WHICH would indicate presence of the same category.
Any dog, dachshund or Great Dane, is "a dog."

"Great Dane," then, is not a "category perception" for you, I suppose.

For me, what you describe as a "category perception" is a logical
function on what I think of as basic category perceptions. For me,
a basic category perception is derived from _analogue_ inputs, at least
in part. The next level is the logic level, at which "Or" operations
such as you describe are performed. The output of a logic-level operation
is easily described as a category in everyday parlance (it's always
category "1" or category "0" or the equivalent), but it doesn't
help clarity of discourse to label the outputs of two different levels
with the same word.

As soon as you say "member of a set" you are describing functions on
categories, not the contruction of the primary categories themselves.
For me, a category is based on the values of analogue variables, with
perhaps the values of other category members thrown in (remember that
this latter is inherent in the cross-connection structure of the GFF,
so it isn't an ad-hoc imposition to account for data).

Thinking of categories as defined by set membership lands you in the
same deep water that the AI-based thinkers have trouble swimming in.
Is a penguin a bird, if birds fly?

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

Why a one-to-one mapping might come about naturally is that it is not
improbable that those perceptions that prove useful (for intrinsic
variable control) as analogues may also prove to be useful
as categories. If that happened, then the category-connected units
would tend to have the same input functions (same inputs, same
weights and functional connections) as corresponding analogue
functions, which would act exactly as if the category units took
copies of the analogue perceptions as their own inputs.

This seems unlikely _IN THE EXTREME_. One of the guiding principles in my
conception of the hierarchy has been complete avoidance of duplication of
functions, most particularly when two physically separate systems must
coordinate with each other. It is very unlikely that two input functions
could have the same inputs, same weights, and same functional connections
as any other input function.

Right, which is why I assumed it was not a necessary feature of the
category process. The passage you quote was describing a possible exception
to the assertion that the categories would ordinarily not be in a
one-to-one relationship with corresponding analogue variables. It
presupposed a possibility that (to quote): "it is not improbable that
those perceptions that prove useful (for intrinsic variable control)
as analogues may also prove to be useful as categories. If that
happened..."

If you are correct that "that happening" is unlikely _IN THE EXTREME_,
then it is equally unlikely that the categories deveoped by the GFF
would have a one-to-one relationship with analogue variables at the
same level. That's hardly a comment on the category perception
process, is it? -- unless you demand that each category has a
corresponding analogue signal.

And in a modular hierarchical system,
duplication of function is all but unnecessary.

If you ignore the possibility of damage to the structure, and assume that
all components perform preperly at all times. But that's a red herring,
worth a discussion only in a different context. In practice, duplication
of function exists almost everywhere: one muscle fibre pulls almost always
at the same time as its neighbour, and not when a fibre on the opposite
side of a joint is pulling; hundreds of auditory fibres fire in near
synchrony for a particular auditory signal; and it would be very
surprising if we were to find no such duplication in the higher levels
of the brain that are harder to study. But this isn't the thread in
which to pursue that issue.

I don't see why you would want to duplicate the input functions of, say, a
configuration recognizer just to create a class of configuration-category
signals, when you have available copies of the _outputs_ of the input
functions, the perceptual signals indicating the configurations that are to
be categorized together.

_I_ don't "want to duplicate the input functions of" anything. _I_ don't
want to "create a class of configuration-category signals". But, if
the perceptual input functions of the configuration level are not all
orthogonal, they are redundant. If there are cross-connections at random
from perceptual outputs to perceptual inputs at this same level, what
falls out is a system of configuration-category signals, some of which
may have close correspondences among the configuration-analogue signals,
and some of which don't, and few of which interfere with the analogue
perceptions (none, after effective reorganization).

It's a consequence of relaxing the strict rule about perceptual signals
going upward, not a means to force a pre-desired result, which is what
your language seems to assert. We look at this simpification of the
structure, observe that it has a lot of properties that correspond to
experience, and suggest that it might be worth pursuing a bit further
to see whether it remains plausible.

To use _copies_ of the perceptual signals is to impose a separate
system especially in order to create category perceptions. I prefer
not to make presuppositions that separate systems and special
mechanisms are required to do what comes naturally without them.
They work, but they are extra baggage. They are useful for didactic
purposes, because one can draw GFF units that have one analogue input
from below to go along with the cross-connections, which makes
the drawing easier to follow. But copies are nice, not required.

If you have separate signals indicating a square,
a rectangle, a parallelopiped, and other four-sided configurations, all
these signals can be input to a configuration-perceiver that produces an
output that might be labelled "quadrilateral."

Yep. A logic-level "or", generating what I would call a derived
category. Or else there could be an input function that takes the
analogue input from the perceptions below the configuration level
to generate a perception of "four-sidedness." This would not, however,
be a category perception unless there were a mechanism that enhanced
or inhibited the "four-sidedness" perception in different contexts.
It would otherwise be a perception with continuous variation in its
possible magitude.

And I don't see how you could ever create a configuration-level categorizer
whose inputs were sensation-signals, which would generate a
configuration-category: the result, by my way of understanding categories,
would be a _sensation_ category: "blue", for example.

Why would "blue" be a configuration-level category? Why would you want
it to be?

Blue is the name of
an array of sensation signals, each representing a different weighted sum
of color intensities, but called by the same name (or at least represented
by the same category signal). Any signal from that array would give the
same color-category signal that any other signal from that array would
give. That is the essence of the categorization process as I see it.

Yes. What's wrong with that? You seem to be using this as an example
to counter something I've said, but I can't determine what that might be.

I'm glad you softened "called by the same name" into "represented by the
same category signal." "Called by the same name" is a consequence of
association of a label category to the sensation-category signal, and
that's a different (though related) process.

Actually, it's a good example of what I have been talking about, because
the sensory array parameters that lead to "blue" category in one context
may lead to "green" category in another, because of cross-connections
we know to exist.

Which provides a lead-in to the next comment.

Which I felt, when I wrote it, was in the mood you call "truth-saying",
taking things which are incontrovertibly so, and making simple, clear
deductions from them. Obviously you did not agree.

To see why it may not be a problem, we have to enquire why category
perception exists--no matter what the mechanism or structural organization.

Assumption 1: All perceptual types that exist in most individuals of a
species are sustained by reorganization that occurs as these
individuals exercise control within their particular environment.

Assumption 2: Only those perceptual types for which control enhances
the stability of intrinisic variables will be sustained by reorganization.

You're making a third assumption that is not mentioned:

Assumption 3: the basic neural computing capabilities and connectivities
needed to construct a perception of the category type are present. This
considerably modifies you next statements:

Assumption 3 is equivalent to an assumption that the outputs of
neurons carrying perceptual signals can be used as inputs to other
neurons that produce perceptual signals. It is an assumption without
which HPCT falls utterly, so I hardly felt it worth mentioning. There
are lots of such "other" assumptions I didn't mention. In fact, I'm
not sure that we need the assumption on which Assumption 3 is based,
that the computing is done neurally. Assumptions 1 and 2 are enough
if we also assume what is normally assumed to allow HPCT to be plausible.

Assumptions 1 and 2 are statements about reorganization that are
ordinarily unstated, but which were explicitly needed for this
particular discussion.

If it is sometimes better to control for category perception than to
control for the corresponding analogue perception, something about the
environment must be such that performing category control has the
side-effect of stabilizing the intrinsic variables better than does
analogue control of the perceptions.

So your assumption is that if it would be better for an organism to control
for category perception, it will develop the ability to do so.

How do you read this into what I wrote? Reorganization doesn't often
invent new kinds of perception does it? Let me rephrase what I wrote:

"In an environment in which category control would have the side-effect
of stabilizing the intrinsic variables better than does analogue control,
it is sometimes better to control for category perception."

Is this not tautological? How can you say:

This is a non-sequitur; it would perhaps be better for bacteria to have systems
concepts, but if they don't have the evolved structure in which system
concepts can develop, they're not going to learn them no matter how long
they reorganize. There is no physical possibility of developing that kind
of control.

No. and they won't. And systems without sufficient cross-connections
within a level of the hierarchy won't develop category perception at
that level either, at least not using the GFF mechanism. What's that
got to do with the _obvious_ truth of what I said?

All "control" of intrinsic variables
comes about because of the side-effects of controlling perceptual
variables _in the environment of the moment_.) Perception of the
category "predator" is likely to result in the imminent disruption
of the values of intrinsic variables, when the perceiver gets eaten.
It doesn't matter what kind of predator it is, or whether the tiger
has a bushy tail. The environmentally important reference is that
the sensory array should not include "something-that-might-eat-me".

That is anthropomorphic reasoning, which you can carry out because you have
a category level. But an organism can avoid things that might eat it
without perceiving that they might eat it, and find food without
categorizing it as "food."

Indeed they can. No problems here.

You can categorize behaviors of another organism
that it cannot itself categorize.

Of course, and that's often a major problem of ethologists and
anthropologists, let alone psychologists. But I didn't do it in this
argument, not at all.

You can even believe that the other
organism is categorizing, because you can place some of the things it
controls into a recognizeable category, like "fleeing predators." But this
does not mean that the other organism perceives "fleeing" or "predators" as
categories; it may simply avoid the perception of particular shapes, one at
a time, without perceiving them as members of a single category as you do.

Right. All I needed for the argument is a pair of action behaviours that
serve to control the values of some perception to two different reference
levels. If there is no intermediate action such that the two actions
can be continuously transformed one into the other, and they can't be
performed together, that's enough. One action serves to increase or
decrease the value of the perception, the other either has the
opposite effect or has no effect. I used "browsing" and "running away"
which are category labels an observer might put on them, not categories
the antelope necessarily uses.

The implication is that some organisms can and do function without a
category level, just as they can function without perceiving or controlling
principles or system concepts. And this argues that category perception IS
a level, a coherent level of perceptual computation that has a physically
distinct existence from the lower levels, and is specialized to deal in
terms of categories.

You go a stage too far here. You are doing what they used to call re-ifying,
taking the word "level" to assert what it is supposed to describe. I'd
restate what you say as "The implication is that some organisms can and
do function without perceiving categories. This argues that category
perception is a coherent kind of perceptual computation that can be
distinguished from analogue computation, and is specialized to deal
in terms of categories." This restatement does not exclude the possibility
that category perception is a level in the sense you intend, but it
does not require it either.

However, the argument is really the reverse, that category perception
will not be stable unless there are dichotomous actions that have
stabilizing effects on intrinsic variables while they influence a
controlled perceptual variable. The argument is _not_ that category
perception inevitably occurs if there is the possibility of such
dichotomous actions that influence the perception.

I think your basic mistake is in assuming that simply because an organism
might be better off with any given level, it will therefore reorganize
until it possess that level of perception and control.

I still cannot imagine where you get the idea I either make that assumption
or believe that reorganization has that kind of magical property. Nothing
I wrote was written under such a misapprehension, and I am rather
surprised that after all these years (including technical discussions
on reorganization) that you would still believe that I am capable of
such silliness.

Now, how about re-reading the passage I thought was "truth-saying"?

I realize you are juggling a lot of balls, and can't spend time making
sense of long messages; but at the same time, you might consider the
possibility that some messages might make better sense if you avoid
starting with the assumption that the writer is a buffoon.

Martin

[From Bill Powers (970705.1722 MDT)]

Martin Taylor 970705 16:10

I don't understand your concept of what a category perception is.

Perhaps that's the reason why you have so much trouble understanding
that I conceive of no such thing as a "category type perceptual
input function", and why you say things like:

You're making a third assumption that is not mentioned:

Assumption 3: the basic neural computing capabilities and
connectivities needed to construct a perception of the category
type are present.

You could not make such an assertion if you took seriously what I
have been trying to present.
----------------------------

That was a pretty uninformative reply to my statement that I don't
understand your concept of category perception. I still don't understand
it. What IS a category perception as you conceive it?

I suppose that
the least set is a single member, but in general there would be
many members, ANY ONE OF WHICH would indicate presence of the same
category. Any dog, dachshund or Great Dane, is "a dog."

"Great Dane," then, is not a "category perception" for you, I suppose.

In quotes, as the name of a category of perceptions, yes. Not in quotes,
indicating an analog configuration perception, no. A set of configuration
signals from an analog level can all become inputs to a given
category-perceiver, the output of which is the same signal no matter which
of the inputs is present; in my example we call that category signal by the
name "dog." It turns out that subsets of this same set of configuration
signals also enter other category-perceivers, whose outputs are called such
things as "Great Dane", "dachshund," and so forth. The referent of a
category signal is not another category, but any direct analog experience
that is among the inputs to the category perceiver.

That is my conception of category perception. What is yours?

For me, what you describe as a "category perception" is a logical
function on what I think of as basic category perceptions.

Then you misunderstand my proposal; I propose that the category perception
is a quasi-logical function of basic _analog_ signals. The category level
is the interface between the analog world and the (partially) discrete
world of higher levels.

For me,
a basic category perception is derived from _analogue_ inputs, at
least in part.

That is exactly what I am proposing.

As soon as you say "member of a set" you are describing functions on
categories, not the contruction of the primary categories themselves.

That is why I say that the category level uses a quasi-logical input
function; the input is presented not with 1s and 0s, but with analog
signals, from which it generates category signals. Categories are functions
of analog signals.

For me, a category is based on the values of analogue variables,
with perhaps the values of other category members thrown in
(remember that this latter is inherent in the cross-connection
structure of the GFF, so it isn't an ad-hoc imposition to account
for data).

You have still not addressed my objection that cross-connections would
disrupt control. You merely say that they wouldn't.

Thinking of categories as defined by set membership lands you in the
same deep water that the AI-based thinkers have trouble swimming in.
Is a penguin a bird, if birds fly?

That's a logical problem, not a category problem. Anyway, I don't think of
categories as defined by set membership. I think of them as signals that
indicate the presence of any of the inputs that, by virtue of being
perceived as members of the same category, produce the same perceptual
result. The reference to set membership was not intended to be technical.

----------------
If you are correct that "that happening" is unlikely _IN THE
EXTREME_, then it is equally unlikely that the categories deveoped by the GFF
would have a one-to-one relationship with analogue variables at the
same level. That's hardly a comment on the category perception
process, is it? -- unless you demand that each category has a
corresponding analogue signal.

No, I demand that each category have a corresponding _set_ of analog
signals, any one or more of which leads to perception of the same category.
There is no requirement that all the analog signals come from the same
lower level: neither do categories correspond to any particular analog
perception or perceptual level.

And in a modular hierarchical system, duplication of function is
all but unnecessary.

In practice, duplication
of function exists almost everywhere: one muscle fibre pulls almost
always at the same time as its neighbour, and not when a fibre on
the opposite side of a joint is pulling;

That is not determined by the muscle fibers, is it? One muscle fiber exerts
a certain torque at a joint; there is no need for a completely different
set of muscle fibers to exert the same torque at the same joint. If you
have a perceptual function producing a signal representing the angle of a
joint, you don't need to construct a new perceptual function receiving the
same intensity information if another system needs to perceive the angle at
that joint. The perceptual signal that already exists is enough.
Furthermore, a "duplicate" perceptual function would probably not come near
reproducing the first one, so the different systems using the different
"joint angle perceptions" would not be compatible with each other.

hundreds of auditory fibres fire in near
synchrony for a particular auditory signal; and it would be very
surprising if we were to find no such duplication in the higher
levels of the brain that are harder to study. But this isn't the
thread in which to pursue that issue.

You're confusing redundancy (where a large number of parallel channels
carry similar information) with duplication of function (in which
computations done in one part of the brain are duplicated in different
neurons in a different part of the brain).

I don't see why you would want to duplicate the input functions of,
say, a configuration recognizer just to create a class of
configuration-category signals, when you have available copies of
the _outputs_ of the input functions, the perceptual signals
indicating the configurations that are to be categorized together.

If there are cross-connections at random
from perceptual outputs to perceptual inputs at this same level,
what falls out is a system of configuration-category signals, some
of which may have close correspondences among the configuration-
analogue signals, and some of which don't, and few of which
interfere with the analogue perceptions (none, after effective
reorganization).

I'm sorry, Martin, but this is as far as I go with you in this fantasy. You
tell me that all these things happen, but what you mean by "happen" is a
long way from what I mean. What you mean is that you can imagine such
things happening without creating any obvious conflicts with your general
mental picture. What I mean is that you can observe them happening in a
working model. I find myself getting sucked deeper and deeper into your way
of playing this game, and I don't want to play it at all. I will be
satisfied if you can tell me what you think a category perception is. How
it's implemented is, for me, a totally open question (regardless of your GFF).

Best,

Bill P.

[Martin Taylor 970706 10:00]

Bill Powers (970703.0731 MDT)

Continuing the discussion of various points from your long, careful
message of three days ago.

I had raised the issue of how output signals of control units
at level N are coordinated to become reference signals for
level N-1. I had argued that if the coordination was done by
simple summation, there would be no point in having multiple
levels of control, because each output stage would represent
a simple rotation in environmental variable space of the actions
of any other output stage, meaning that all the output effects
could, in principle, be accomplished in one stage with different
weights. I used this to argue that simple summation would not
be the mechanism for combining output signals into next-lower
reference signals, at least not in most cases.

The question of how outputs combine to generate reference
signals is, I have thought, somewhat simpler than the question
of how perceptions are formed. Remember that typically one
system of a given level adjusts reference signals for many
systems of the next lower level. Different systems of a
given level, however, will in general send outputs to somewhat
different sets of lower-level systems, with different weights
(positive or negative is the main choice).

Yes, we take that as common ground. However "somewhat simpler"
is a relative phrase.

What else is common ground? That if an elementary control unit
(ECU) at level N has non-zero error, it will alter its output
in such as way as to reduce the error. This output alteration
affects the reference values of all lower-level systems at level
N-1 to which it is connected. They therefore attempt to alter
their perceptual values, which produces disturbances to all
the other ECUs at level N. Provided there are at least as many
units at level N-1 as there are at level N, these disturbances
can be countered, but it takes time for the whole system to
settle down again.

I think all the last paragraph is also common ground, is it not?

Rick's spreadsheet demo is a good example; it
contains 6 systems at each of 3 levels, and each higher system
contributes to the reference setting of ALL 6 systems of the next
lower level, with weights of only -1 or 1. In his demo there
were the same number of outputs as lower-level systems, which
is the worst case; more generally there would be more lower
systems than higher, so the overlap would not be complete.

Apparently it is.

Note that the highest system in Rick's demo controlled logical
functions; the outputs were either on or off. Nevertheless, these
signals were added together algebraically at the reference
inputs of the next lower (analog) level, so an analog
(multiple-valued) reference signal was generated for
each analog system. Nevertheless, the logical functions were
controlled.

I think this comment addresses the opposite issue. My original
point was that in a linear system, what is easily done in a
multi-level hierarchy could also be done by a single-level
set of control units, not that hierarchies don't work with
summation reference inputs. Also, I did not argue that a single
level system would be easier to build or would work better.

I was thinking from an evolutionary standpoint: if a single
level system works, why would a multi-level system having the
same functionality ever evolve?

Exact orthogonality of output effects of different control systems of the
same higher level is not critical on the output side. A considerable
departure from a "90 degree angle" can occur before interactions
become prohibitively strong.

Yes. It's a dynamical issue. But it's not the issue I was querying.
Actually, its a dynamical issue relevant to your problems with the
"monster," even though the monster is a one-level system.

Although it wastes net bandwidth to do so, since I agree with
it completely, the following is worth repeating.

For the analyst, the main question is the relation between the error
signal in a control system and its net effect on the perceptual
signal in the same system. A superposition principle applies; if
many higher control systems share an environment of many lower
systems, the feedback effects from error to perception can remain
negative for each of the higher systems, even though no one control
system at a lower level is uniquely influenced by ANY single
higher-level system. We tend to convey the wrong picture when we
draw diagrams with just one higher system and one lower system; in
reality, the "one" lower system is a largish set of lower systems,
which are also part of the loops of other higher systems. The net
reference signal reaching a lower system doesn't reflect the
errors in any one higher system.

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

The main unanswered question on the output side is the nature of the
output functions.

And therein may lie a resolution of the dilemma originally posed.
If the output function at level N is non-linear, then the the lower
level outputs no longer represent simple rotations of the upper
level outputs in the space of environmental variables. (Integrators,
amplifiers, and differentiators are all linear operators, and
I guess I had implictly assumed that the output functions would
be constructed from them, as has been the case in all the
simulations I remember seeing.)

Particularly troublesome is how the error signals
resulting from a perceptual signal representing a dynamic perception
like a rhythmic motion (rate of repetition, for example) is
converted into the necessary dynamic changes in lower-level
reference signals. Rick and I have come back to that problem at
intervals. Clearly, the output function must be some
sort of pattern generator; a variable-frequency, variable amplitude
oscillator, for example, with its amplitude and frequency varied by
error signals.

These involve inherently non-linear operators (a linear oscillator
runs away to infinite amplitude). And such output stages would
indeed make a multi-level hierarchy act differently from one in
which the references summed and the output stages were linear.

Nevertheless, though it no longer seems a necessity, I wonder
whether reference inputs should not be regarded as perceptual
inputs are, as combining functions with properties that differ
at different levels--for example,whether at one level the reference
value might be the maximum of the outputs from levels above
(i.e. selecting the course of action for which the higher-level
error*gain was greatest), and at another level the Euclidean
sum (sqrt(sum of squares)), which might be relevant to what
Garner used to call "separable" or "non-separable (integral?)"
perceptual dimensions.
-----------------

I think this message completes my response to yours of July 3.
Thank you for it. It generated considerable thought.

Martin

[Martin Taylor 970706 10:50]

Bill Powers (970705.1722 MDT)

I will be
satisfied if you can tell me what you think a category perception is.

So far as I can understand it from your new description of what
you think a category perception is, we think the same.

How
it's implemented is, for me, a totally open question (regardless of your GFF).

So it is for me, too. I've said in almost every message in this
thread that I don't claim the GFF _is_ the mechanism.

To restate for the umpteenth time: If cross-connections from the
perceptual outputs at one level sometimes go to perceptual inputs
at the same level, the GFF is what results.

That much is a straight statement of fact. The rest is analysis,
since there's been hardly any simulation (though there has been
some on the perceptual side, not in a control environment). If
your analysis differs from mine in what the GFF would actually
do, there are two ways to address the issue: (1) clarify the
assumptions and analytic procedures so that each step is well
understood, and (2) simulate.

Method (2) is clearly preferable, but the problem with setting up
a suitable environment is much harder than it is for your "monster"
problem. Hence we resort to sub-part simulation to augment analysis.
I can do no better.

There are issues relating to cross-connection density and gain
that have been simulated at least in part, but not in a control
environment, for reasons that your "monster" experience must help
you understand.

···

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

I suppose that
the least set is a single member, but in general there would be >>many

members, ANY ONE OF WHICH would indicate presence of the same >>category.
Any dog, dachshund or Great Dane, is "a dog."

"Great Dane," then, is not a "category perception" for you, I suppose.

In quotes, as the name of a category of perceptions, yes. Not in quotes,
indicating an analog configuration perception, no. A set of configuration
signals from an analog level can all become inputs to a given
category-perceiver, the output of which is the same signal no matter which
of the inputs is present; in my example we call that category signal by the
name "dog." It turns out that subsets of this same set of configuration
signals also enter other category-perceivers, whose outputs are called such
things as "Great Dane", "dachshund," and so forth. The referent of a
category signal is not another category, but any direct analog experience
that is among the inputs to the category perceiver.

That is my conception of category perception. What is yours?

The same, if these inputs are analogue.

But the problem I see in the back of your mind is one that
occurs at _every analogue_ level of perception. This is how
the perceptual output can be the same for a wide variety of
analogue input signals. This happens whenever there is more than
one input to the perceptual input function.

It's true for the perception of a particular shade of pink that
an infinite variety of spectra will generate the _same_ perceptual
values. It's true for every level that a particular perceptual
value can be generated by a wide variety of inputs. Where I
think you may be getting hung up is in how it is possible that
a particular perceptual output value can be produced by inputs
from a discontinuous region--input(x1,...xn) produces perceptual
value Pk, and so does input (y1,...yn), but no input of the type
(a1*x1+b1*y1, ....,an*xn+bn*yn) produces output Pk. It's not
a big deal, and to me the perceptual output from this kind of patchy
distribution of inputs is not adequate to define a category.

Even the much-maligned multi-layer perceptron (MLP) can produce
the same output for inputs in any member of a set of arbitrarily
placed patches of the possible input values. That's exactly the
situation of the "Analogue-daschund OR Analogue-Great Dane" =
"Analogue Dog" problem you pose. It's a characteristic problem
soluble by an MLP. A theorem by Richard Lippman shows it can
be achieved in (I think) four layers. (That's not to say we have
MLPs that can look at arbitrary photographs of different dogs and
identify them well. That's a different problem, as is the problem
of how to train the MLP to achieve its potential).

The catgory, for me, doesn't become a category until the perceptual
output has a gap of inaccessible values between a range of values
representing "is a member of the category" and "is not a member
of the category." Otherwise, what we have is just an analogue
perception of the ordinary kind (which is what the MLP provides
at its output).
------------------
Next point.

For me, a category is based on the values of analogue variables, >with

perhaps the values of other category members thrown in >(remember that this
latter is inherent in the cross-connection >structure of the GFF, so it
isn't an ad-hoc imposition to account >for data).

You have still not addressed my objection that cross-connections would
disrupt control. You merely say that they wouldn't.

I thought I had addressed it. Maybe you could restate wherein I
failed to do so? You made one such comment, which I answered.
You have not criticised the answer. Are there other points I have
not considered?

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

I will be
satisfied if you can tell me what you think a category perception is. How
it's implemented is, for me, a totally open question (regardless of your GFF).

Have I satisfied this request?

Martin

Will you geniuses please tell me what GFF means?
Ellery (Now a Ph.D.)

To Martin Taylor:
Many thanks for a most helpful response.
Ellery

[From Bill Powers (970707.0200 MDT)]

Martin Taylor 970706 10:50--

But the problem I see in the back of your mind is one that
occurs at _every analogue_ level of perception. This is how
the perceptual output can be the same for a wide variety of
analogue input signals. This happens whenever there is more than
one input to the perceptual input function.

It's true for the perception of a particular shade of pink that
an infinite variety of spectra will generate the _same_ perceptual
values. It's true for every level that a particular perceptual
value can be generated by a wide variety of inputs. Where I
think you may be getting hung up is in how it is possible that
a particular perceptual output value can be produced by inputs
from a discontinuous region--input(x1,...xn) produces perceptual
value Pk, and so does input (y1,...yn), but no input of the type
(a1*x1+b1*y1, ....,an*xn+bn*yn) produces output Pk. It's not
a big deal, and to me the perceptual output from this kind of patchy
distribution of inputs is not adequate to define a category.

When you say "a wide variety of inputs" you are speaking categorically. At
the category level we do not distinguish differences that make a difference
at the analog levels.

At the analog levels, the perceptual signal is the same only if the inputs
vary (if they vary) in such a way as to keep the value of the perceptual
function the same. The perception y = x1+x2 remains the same only as long
as x1 and x2 vary in a particular linear relationship in the x1-x2 plane. y
= x1^2 + x2^2 remains the same only as long as x1 and x2 lie on a circle.
There are far more ways in which the inputs can vary so as NOT to maintain
a constant perceptual signal in an analog system.

The inputs to a category function need only exceed a threshold to produce a
category signal, and it doesn't matter which inputs do this. The inputs do
not have to bear any particular functional relationship to each other as
they do for an analog function. So while it is true that the inputs of both
category and analog perceptual functions can vary in a "wide variety of
ways" while the perceptual signal remains constant, the conditions under
which category and analog perceptual signals remain the same are very
different.

Even the much-maligned multi-layer perceptron (MLP) can produce
the same output for inputs in any member of a set of arbitrarily
placed patches of the possible input values. That's exactly the
situation of the "Analogue-daschund OR Analogue-Great Dane" =
"Analogue Dog" problem you pose.

This could mean that the MLP is a good candidate for a model of the
category perception level. However, you have mis-stated the problem I
posed: I was speaking of "Analogue-daschund OR Analogue-Great Dane" =
"Categorical Dog".

It's a characteristic problem
soluble by an MLP. A theorem by Richard Lippman shows it can
be achieved in (I think) four layers. (That's not to say we have
MLPs that can look at arbitrary photographs of different dogs and
identify them well. That's a different problem, as is the problem
of how to train the MLP to achieve its potential).

The category level does not create distinctions; it erases them. To model
the situation I describe, the aim would have to be to produce an MLP that
will give the same signal for any input configuration that a human being
would call "a dog," but not for any input configuration we would all "a
horse" or "a person."

The category, for me, doesn't become a category until the perceptual
output has a gap of inaccessible values between a range of values
representing "is a member of the category" and "is not a member
of the category." Otherwise, what we have is just an analogue
perception of the ordinary kind (which is what the MLP provides
at its output).

In my conception of a category, there is no category-level signal saying
"not this category." Absence of a signal simply fails to give an impression
of any category; it does not imply the presence of some other category. The
mutually-exclusive relation between "A" and "not-A" is a property of the
logic level, not the category level.

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

You have still not addressed my objection that cross-connections
would disrupt control. You merely say that they wouldn't.

I thought I had addressed it. Maybe you could restate wherein I
failed to do so? You made one such comment, which I answered.
You have not criticised the answer. Are there other points I have
not considered?

You said that in two control systems with cross-connected input functions,
the reference signals would have to be properly adjusted to take the
flip-flop effect into account, so the reference signal for the system in
which the input is flipped LOW would not demand that it be HIGH.
Technically, that does constitute an answer to my objection. However, it
does not resolve my objection, so it is a non-responsive answer. Your
general answer to these objections seems to be "Reorganization will take
care of all that, so don't worry about it." This will answer any objection
to any proposal, but it is not the kind of answer I find very useful.

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

I will be
satisfied if you can tell me what you think a category perception is.
How it's implemented is, for me, a totally open question
(regardless of your GFF).

Have I satisfied this request?

No. I still don't know what aspect of human experience you are modeling
when you speak of categories.

Best,

Bill P.

[From Bill Powers (970707.0300 MDT)]

Martin Taylor 970706 10:00--

I had raised the issue of how output signals of control units
at level N are coordinated to become reference signals for
level N-1. I had argued that if the coordination was done by
simple summation, there would be no point in having multiple
levels of control, because each output stage would represent
a simple rotation in environmental variable space of the actions
of any other output stage, meaning that all the output effects
could, in principle, be accomplished in one stage with different
weights. I used this to argue that simple summation would not
be the mechanism for combining output signals into next-lower
reference signals, at least not in most cases.

You are ignoring the fact that simple summation is not sufficient to
describe the computations on perceptual input signals, either. The problem
is in your assuming one type of computation that is simply repeated at
different levels. I know of no evidence to support this idea, either
behaviorally or anatomically. As it happens, simple summation of outputs,
with only the sign being variable (inhibition versus excitation) is enough
to provide negative loop gain in all arrangements I have investigated.
Since we're dealing with scalar signals throughout, a change in an output
signal can have only a negative, positive, or zero effect on a perceptual
signal, even if the input function is non-linear. Even a two-valued input
function will work with this arrangement; if there is positive feedback
over some portion of the input range, the control system will simply skip
to the next negative feedback region (see experimental example in the
"Spadework" paper).

The question of how outputs combine to generate reference
signals is, I have thought, somewhat simpler than the question
of how perceptions are formed. Remember that typically one
system of a given level adjusts reference signals for many
systems of the next lower level. Different systems of a
given level, however, will in general send outputs to somewhat
different sets of lower-level systems, with different weights
(positive or negative is the main choice).

Yes, we take that as common ground. However "somewhat simpler"
is a relative phrase.

What else is common ground? That if an elementary control unit
(ECU) at level N has non-zero error, it will alter its output
in such as way as to reduce the error. This output alteration
affects the reference values of all lower-level systems at level
N-1 to which it is connected. They therefore attempt to alter
their perceptual values, which produces disturbances to all
the other ECUs at level N. Provided there are at least as many
units at level N-1 as there are at level N, these disturbances
can be countered, but it takes time for the whole system to
settle down again.

You assume that every output at level N affects all systems at level N-1.
This is not true in the human system. This is only the worst case for
specific subsections of the hierarchy, which we have invesgitated in models
mainly to see if that case could be handled, as it can.

I think all the last paragraph is also common ground, is it not?

No.

Rick's spreadsheet demo is a good example; it
contains 6 systems at each of 3 levels, and each higher system

contributes to the reference setting of ALL 6 systems of the next

lower level, with weights of only -1 or 1. In his demo there
were the same number of outputs as lower-level systems, which
is the worst case; more generally there would be more lower
systems than higher, so the overlap would not be complete.

Apparently it is.

The overlap is complete in Rick's model because he deliberately made it so,
not because it is always so, or even often, in the real system.

Note that the highest system in Rick's demo controlled logical
functions; the outputs were either on or off. Nevertheless, these
signals were added together algebraically at the reference
inputs of the next lower (analog) level, so an analog
(multiple-valued) reference signal was generated for
each analog system. Nevertheless, the logical functions were
controlled.

I think this comment addresses the opposite issue. My original
point was that in a linear system, what is easily done in a
multi-level hierarchy could also be done by a single-level
set of control units, not that hierarchies don't work with
summation reference inputs.

But that point is true only if each new level is simply a further summation
of already-existing signals. You require that all perceptual functions be
linear, apparently (except when you talk about "squashed" input functions).
But we know this is not the case; some perceptual functions, like the one
that computes the invariants of a configuration with respect to rotation,
require computations that are something like sines and cosines, which
aren't even single-valued, much less linear.

I was thinking from an evolutionary standpoint: if a single
level system works, why would a multi-level system having the
same functionality ever evolve?

Perhaps it wouldn't. On the other hand, you're assuming it is possible that
a single-level system could have the same functionality as a multiple-level
system. This might be true in a simple homogeneous brain in which any part
can handle the computing processes taking place in any other part, and
where all the computing functions are basically the same. But this is not
true of the real brain. The physical levels in the brain have different
capabilities; cut the brain off at the thalamus, and the person will no
longer be able to do algebra. The physical levels in the brain are
specialized to do specific kinds of computations, different from those done
in other layers of the brain.

Exact
orthogonality of output effects of different control systems of the
same higher level is not critical on the output side. A considerable
departure from a "90 degree angle" can occur before interactions
become prohibitively strong.

Yes. It's a dynamical issue. But it's not the issue I was querying.
Actually, its a dynamical issue relevant to your problems with the
"monster," even though the monster is a one-level system.

No, this is not a dynamical issue. It's an issue of the solutions to a
simultaneous set of algebraic equations, which yield the output values
required to maintain each perception at an arbitrary reference level.

Although it wastes net bandwidth to do so, since I agree with
it completely, the following is worth repeating.

For the analyst, the main question is the relation between the error
signal in a control system and its net effect on the perceptual
signal in the same system. A superposition principle applies; if
many higher control systems share an environment of many lower
systems, the feedback effects from error to perception can remain
negative for each of the higher systems, even though no one control
system at a lower level is uniquely influenced by ANY single
higher-level system. We tend to convey the wrong picture when we
draw diagrams with just one higher system and one lower system; in
reality, the "one" lower system is a largish set of lower systems,
which are also part of the loops of other higher systems. The net
reference signal reaching a lower system doesn't reflect the
errors in any one higher system.

Martin, when you "agree completely" with something I write, I can be pretty
sure that you're putting your own twist on it and not actually agreeing
with my point at all. If the above was supposed to illustrate your point
about "dynamical issues", that is not the point I was making, which was
about static relationships.

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

The main unanswered question on the output side is the nature of
the output functions.

And therein may lie a resolution of the dilemma originally posed.
If the output function at level N is non-linear, then the the lower
level outputs no longer represent simple rotations of the upper
level outputs in the space of environmental variables. (Integrators,
amplifiers, and differentiators are all linear operators, and
I guess I had implictly assumed that the output functions would
be constructed from them, as has been the case in all the
simulations I remember seeing.)

Fine. Now extend the same assumption to input functions, which are also, in
general, both nonlinear and multiple-valued. I have never assumed linearity
in the real system; I use linear models because they are easiest to
implement, not because I think the real system is linear.

Any way, a simple rotation is not a linear function if the angle of
rotation is the variable (as in turning your hand from prone to supine
while watching it). It's only linear if you assume that the entries in the
rotation matrix are constants. In a rotation matrix like

   Sin(theta) 1
      -1 Cos(theta)

this is certainly not the case.

Particularly troublesome is how the error signals
resulting from a perceptual signal representing a dynamic perception
like a rhythmic motion (rate of repetition, for example) is
converted into the necessary dynamic changes in lower-level
reference signals. Rick and I have come back to that problem at
intervals. Clearly, the output function must be some
sort of pattern generator; a variable-frequency, variable amplitude
oscillator, for example, with its amplitude and frequency varied by
error signals.

These involve inherently non-linear operators (a linear oscillator
runs away to infinite amplitude). And such output stages would
indeed make a multi-level hierarchy act differently from one in
which the references summed and the output stages were linear.

Not at all. The outputs of the oscillators are, as usual, scalar signals.
It would be possible for the x position of an arm to be set by reference
signals proportional to cos(t) while the y position is set by sin(t). The
muscles contributing to x motion need not be aligned with the x axis, and
so forth, so the net reference signal entering the individual control
systems would be the sums of the x and y reference signals given different
projections onto the axes. The result would be a circular motion of the
arm. See my Byte articles for a specific example.

Nevertheless, though it no longer seems a necessity, I wonder
whether reference inputs should not be regarded as perceptual
inputs are, as combining functions with properties that differ
at different levels--for example,whether at one level the reference
value might be the maximum of the outputs from levels above
(i.e. selecting the course of action for which the higher-level
error*gain was greatest), and at another level the Euclidean
sum (sqrt(sum of squares)), which might be relevant to what
Garner used to call "separable" or "non-separable (integral?)"
perceptual dimensions.

You can imagine anything you want, but as far as I can see you're doing it
at random.

Best,

Bill P.