Newsgroups: bit.listserv.csg-l
Date:         Fri, 18 Dec 1992 20:27:37 EST
From:         mmt@BEN.DCIEM.DND.CA
Subject: Re: Misstatements; Latin mottos; Knowel,ed

[Martin Taylor 921218 19:45]
(Bill Powers 921218.1500)

Well, I just put a long posting to Tom Bourbon to bed, saying I have to go
home, and as I was putting on my outdoor shoes, along comes this posting
from Bill, demanding a response before the weekend...

(To Bruce Nevin)
>>It seems to me that, as the strength of some category
>>perception grows, it becomes easier (more acceptable?) to fill
>>in missing category-attribute perceptions by imagination.  At
>>some threshold, or seeming threshold, where imagined
>>perceptions are integrated with real-time perceptions, it is as
>>though the exemplar of the category is perceived as fully or
>>truly present, whereas before there were only unsupported signs
>>or symptoms that fostered a belief, readiness, or expectation.
>
>Another phenomenon of categories, well known in experimental
>psychology, is the hysteresis effect. As a figure changes shape,
>say between a rectangle and an ellipse, there is a point where a
>person switches from one label to the other. When the change is
>carried out in the reverse direction, the switch-point is delayed
>so it occurs well into the region where the other label was used
>when the change was going the other way. This is explained by
>calling it "perseveration." The person who exhibits this
>phenomenon has perseverosia. I suppose that curing it would call
>for taking an antiperseverant.

I had almost responded to this posting by Bruce, because it bears on a
pretty fundamental issue that relates category perception to continuous
perception (and to the information-theoretic ideas in my posting to Tom).

Before we knew about PCT, we had been developing concepts in cognition
based on the requirement for quick but stable transitions among ideas
(such as the recognition that a stimulus pattern represents a particular
class of object).  Also we were concerned with how people build on existing
knowledge and how they develop analogies.  All of this was within a
geometric framework.  I'm not sure whether I'll be any better at explaining
this here than I have been at the information theory, but never mind.  I
wrote a 20-page draft of an introduction to the ideas a couple of years ago
and quit because there was too much background to cover if the ideas were
to be given justice.  So...

The fundamental geometric object in our construction is a fold catastrophe.
The reason is that things in the world can change continuously and (despite
Bill's claim) most perceptions are based on very poor information--the SNR
is very low.  So if a structure is near the boundary between two categories,
it might oscillate between them if the boundary were just a threshold.
The way around this is that category boundaries are not thresholds, but
fold catastrophes.  As the sensory data move along from, say, ellipse toward
rectangle, the perception eventually changes to the rectangle branch
of the fold.  As the data move back the other way, the perception tracks
along the rectangle branch beyond the original transition point until
eventually it falls onto the other branch.  Between the two, the perception
is reasonably stable at whatever it had been (I say reasonably, because
there is a second catastrophe function that causes a temporal hysteresis
here as well, but I don't want to go into that).

Here's my best shot at an ASCII view:

      rectangle
________________________________
                               /|
                           ^  / |
                           | /  V       ellipse
                           |/______________________
           <-----roundendness--------->


Now we have to add another dimension.  In fact we do it twice, and once
we call the added dimension "stress" and once "learning."  The picture
looks the same for each.

Consider the "roundedness" axis.  If you don't know about ellipses and
rectangles as categories, you could describe the object as "a kind of
squarish circle" or something like that, with adjectives to describe the
degree of squarishness.  There's no fold catastrophe.  If you begin to
learn, a precise rectangle might be a "rectangle" and a clean ellipse
an "ellipse", but in between, it wouldn't really be anything.  You have
something a bit like:

     rectangle
___________________
                   \
                    \           ellipse
                     \___________________

(I can't draw a shallowly sloping line, which would be the early stage I
described. This is a later stage, when the categories are moderatly developed).

Under no stress to categorize, behaviour with respect to the pattern will
be along the shallow slope, varying continuously as the roundedness varies.
But under strong stress, either task dependent (rectangles give you money,
ellipses don't) or context dependent (like alphabetic symbols rather than
ellipses and rectangles), the behaviour varies on the fold pattern,  The whole
structure is a cusp catastrophe, in which the object is primarily a category
to which adjectives can be applied when it is on the fold, or primarily
an uncategorized entity when it is on the slope.  The catastrophe allows
a different kind of perceptual control--going up a level.

When the category is identified, the other attributes that go along with it
are subordinated to the fold.  What does that mean?  The first thing to
note about a catastrophe is that it requires some kind of POSITIVE feedback
to maintain it.  There must be a memory, of cross-feeds from related
perceptual functions, or something like that.  What this means is that
teh existence of the category in part substitutes for sensory information.
The effect Bruce mentions is a natural consequence, and what Bill mentions
is a simple description of a necessary phenomenon, of having the ability
to form categories.
=======================
Bill to Tom Bourbon (921218.1438 CST) --
>
>>In the past, both of us wondered how, specifically, Shannon's
>>ideas, or any of the major concepts from information theory,
>>would improve any of the quantitative predictions we make with
>>our simple PCT models.
>
>This is the right question about information theory -- not "does
>it apply?" but "what does it add?" The basic problem I see is
>that information theory would apply equally well to an S-R model
>or a plan-then-execute cognitive model -- there's nothing unique
>about control theory as a place to apply it. Information theory
>says nothing about closed loops or their properties OTHER THAN
>what it has to say about information-carrying capacity of the
>various signal paths.

You are right, but that "OTHER THAN" is a pretty big place to hide very
important stuff.  I had not previously realized that you wanted me to use
Shannon to differentiate between S-R and Plan-then-execute. I think I did
incidentally make that discrimination in my posting in response to the
same posting by Tom.  At least I think I showed how applying Shannon
demonstrated that neither S-R nor Plan-then-execute could be viable.  But
we knew that already, so I didn't play it up.

> All working models have signal paths of
>some sort, but only the control-system model puts those paths
>together in a way that results in control.

Yes, you found the importance of that long ago without recourse to the
necessity arguments that come from from information theory, but that
doesn't make the arguments less relevant.  Newcomen and Watt built steam
engines of some power without worrying about the proper thermodynamic cycles.
But we build more efficient ones nowadays.

>And in a control
>model, the signals in the various paths normally carry far less
>information than the theoretical limits allow.

Dubious.  I would like to be able to figure out how to test that assertion.
My intuition/prejudice about evolution is that it seldom wastes resources
very badly, and I would be surprised if your statement is true about the
real live control systems that have evolved (as opposed to control "models").

> But
>information theory doesn't tell you there will BE an error
>signal.

As I pointed out to Tom, it does.

>In all these cases, something has to be added to
>get a workable system. And I don't think that this something
>comes from the abstract principles involved, however convincingly
>one can prove that they apply.

Yes, that's absolutely right.  Natural laws are no use without boundary
conditions to describe particular situations.  But if you understand the
abstract principles, you can make better [bridges/kettles/radios/control
systems].

Martin
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