[Martin Taylor 960229 15:00]
Bill Powers (960228.0530 MST)
In order for a cross-connection to create hysteresis and bistable or
multistable states, both sides of the cross-connection must be affected.
When you try to make one side analog and the other bistable, you
"violate the symmetry" that is needed to make a flip-flop. I don't see
how you plan to make the analog side work normally while its perceptual
functions are cross-connected in a flip-flop arrangement with the
category side of the parallel hierarchies.
They aren't. It's hard to draw an ASCII diagram that makes sense, because
even a nicely drawn graphic is fairly complex. Maybe a textual description
will work better, so that you can draw the sketch yourself.
Imagine two ordinary analogue ECUs, with perceptual functions whose outputs
are X and Y. (From here, for a while, we will be dealing only with perceptual
functions, ignoring references and outputs until they are explicitly
mentioned). These analogue ECUs are anywhere in the analogue control
hierarchy. Nothing said from here on affects either the input to or the
output from these perceptual functions.
Now imagine two more perceptual functions, each of which may have several
inputs. One of them has X as an input. Call its output X'. The other has Y
as an input. Call its output Y'. They are cross-connected: The one that
has X as an input also has k*Y' as an input, and the one that has Y as
an input also has j*X' as an input. If k<0, j<0, k*j> 1, the other inputs
don't matter. If these are linear perceptual functions and X and Y are
both greater than zero, either the output of X' goes to infinity while
the output Y' goes to minus infinity, or the reverse.
If, however, the perceptual functions saturate at 0 and max, the infinities
will be clamped, and sufficient opposed values at the other inputs can switch
the states. If X' = max, then the input to the other one will be j*max + Y.
The input to the first will simply be X. If Y > -j*max, then the output
Y' > 0, regardless of X'. If at the same time X is small, -k*Y' may exceed it,
bringing X' down to zero. The states will have switched. I'm assuming here
no bias, thereby permitting the quasi-stable condition X'=Y'=0 in the
absence of other input. In a true flip-flop there is a bias, tending to
make X' or Y' greater than zero in the absence of other input. That
doesn't seem realistic for perceptual systems, which should not "see"
their category in the absence of relevant input.
The X', Y' outputs are connected entirely independently of the X, Y, outputs,
which are unaffected by the flip-flop behaviour of X', Y'.
Now let's consider the case k>0, j>0. I call this "association." When
0, then X'>0, jX'>0 --> Y'>0, which tends to increase X'. In a linear
system, this is a guaranteed runaway, but in a non-linear (logarithmic
or saturating) system it need not be. What it means is if sensory data
lead to positive outputs from either X or Y analogue perceptual functions,
then _both_ X' and Y' signals go positive. I see this as an associative
relationship.
The case with k>0, j<0 provides a negative feedback loop around the two
functions. An increase in X' tends to lead to a decrease in X', and the
same for Y'. The values of X' and Y' at which the pair is stable will
depend on their nonlinearities, I think (I haven't simulated this system,
but it ought to be stable).
These are the cases for two signals. I conceive of there being very many
with interconnects of both positive and negative weight, which have been
learned in a Hebbian manner (or by random reorganization). Call their
outputs Xi, i=1, ....,n.
If the interconnection weights from Xi to Xj are kij, then there will
be some pairs for which the global interconnect weights conform to Kij<0
(Kij = Sum-over-m(kim*kmj)), and some pairs for which Kij>0. If Kij and Kji
are both less than zero, the pair form a flip-flop relationship. If they
are both greater than zero, the pair have an associative relationship.
If one is greater than and one less than zero, the relationship is ambiguous.
About half of the pairwise loops will be of this stable character, and they
will tend to mitigate the strong effects that might otherwise lead to
"hard" flip flops and "unbreakable" associations--or so I should think,
not having simulated the situation.
Since the Kii weights (the reflexive values of an input on itself) will
depend on the actual signal values of lots of different associative and
flip-flop interconnects, most of the individual kij weights are likely to
be quite small, whether they are positive or negative. But there will be
modular groups of things that tend to go together (associative clusters,
and particularly word perceptions associated with perceptions of the word
referent), and other groups of things that exclude one another ("moving"
and "stopped" for example, or "red" "green" and "blue"). A perceived object
may be moving slowly but it will be perceived either as categorically
stopped or as categorically moving. Both the analogue continuously
perceived slow movement and the categorically exclusive perception are
available together.
How
would you contrast your arrangement with mine, in which there is a
single category-perceiving level receiving analog signals from lower-
level systems and emitting signals indicating that specific categories
are present?
Only in that there is no "level" that does it. The "category surface"
probably behaves the same as your "category level" but it is more easily
described as being "beside" than as being "above" the analogue levels.
Under the existing
model it is possible to have categories of intensities, sensations,
configurations .... relationships.
So far, no difference. The X, Y perceptual signals could come from any
level of the hierarchy, in both systems. I conceive of the related categorical
X' Y' signals as participating in a parallel hierarchy, in which categorical
configurations, for example, are perceived as constructions of categorical
sensations. The categorical configuration perceptual input functions could
get their sensory inputs from the analogue configuration perceptions or
from the categorical sensation perceptions, or both.
Now we start talking about the rest of the control systems. In my system
(and, I think, in yours), X' and Y' are the perceptual functions of
perfectly normal control systems with perfectly normal reference inputs.
If the reference input called for X' to be max/2, the control output would
try to bring it to that value, but if X' were in a flip-flop connection
success would be unlikely. More probably, the result would be an oscillation.
But from here I think our proposals differ:
But control involves sending output
signals to the relationship level, and from that level on downward the
control is analog. How does your proposed arrangement work in relation
to continuous control of analog variables?
In my proposal, the output of the X', Y' control units can go to any level,
not just the relationship level, but the outputs would preferentially go where
the inputs came from. An output of a "category Red" perceiving ECU might
go to the reference input of a "perceive redness" analogue ECU, where it
would combine with other outputs from the analogue hierarchy to generate
the reference level. If the category reference were (Boolean: Red = TRUE),
then the output would influence the analogue redness reference level toward
high perceived "redness." If the category reference were (Red = FALSE),
the output would influence the analogue redness reference toward low
perceived "redness."
The same sort of thing applies to, say, a relationship analogue and category
pair of variables. If the category reference is (Boolean: "to the left" = TRUE),
the output would influence an analogue relationship reference variable
toward "high leftness perception".
How does your proposed arrangement work in relation
to continuous control of analog variables?
I know the above is very sketchy. Is it helpful?
Martin