[From Bill Powers (930623.0930 MDT)]
Bruce Nevin (930622.1316) --
RE: phonemes, words, etc.
In the PCT model, a perception is a signal; it is a function
either of signals generated directly by sensory endings, or of
signals that are outputs of lower-level perceptual functions.
Some of the signals in lower-level perceptual functions are under
feedback control; some are not.
These functions, at the lower levels, are primarily analogue in
nature. That is, they handle continuous variables, making one
output variable a function of some set of input variables. The
following discussion, however, applies to any kind of function.
A variable y is a function of a set of x's: y = f(x1,x2,...xn). Y
represents the magnitude of the perceptual signal output by a
perceptual function, the x's represent the magnitudes of
perceptual signals collected from lower perceptual functions and
serving as input to, or arguments of, the function.
For any such function, there are many sets of values of the x's
that will produce the same value of y. All sets of x's that
produce the same y are therefore perceived as the same state of
the same variable. If the x's vary in combinations that leave y
unchanged, the changes in the x's are not perceived at the level
of y. Any other combinations of changes in the x's result in a
changed value of y, and the perception y is then affected by the
changes in the x's. The entity perceived as y is then perceived
in a new state. If the function is continuous, the states may be
indicated on the real number scale. If the function is discrete,
the states may be indicated on an integer scale, including the
binary scale.
Two perceptions, y1 and y2, may be derived through two different
functions f1 and f2, from the same set of x's, x1 .. xn. If the
functions are different, then combinations of changes in the x's
that leave y1 constant may alter y2, and vice versa. Other
changes may alter both y1 and y2. Still other patterns of change
in the x's may leave both y1 and y2 the same. This general
relationship between x's and y's holds true for any number of
y's, although the maximum number of y's cannot exceed the number
of x's if independent changes in the y's are to result.
If we could plot the dependence of y on its corresponding set of
input x's in as many dimensions as desired, we could draw
"surfaces of indifference" in this space passing through all the
values of the x's that left the value of y unchanged. Any or all
x's could change without altering y if the values lay in the same
surface of indifference. Changes in y result from the x's
changing in a way that moves the result to a new surface of
indifference: a change normal to the surface of indifference in
hyperspace.
In two dimensions, if the functions are y1 = x1 + x2 and y2 = x1
- x2, we would have _lines_ of indifference. If x1 decreased by
the same amount that x2 increased, the value of y1 would remain
unchanged but y2 would change. If x1 and x2 increased or
decreased by the same amount, y2 would remain constant and y1
would change. Any function of x1 and x2 would create surfaces of
indifference: for y3 = x1^2 + x2^2, the surface of indifference
would be a circle. The perceptual signal y would thus represent
only the radius of the circle, and would be indifferent to
changes in x1 and x2 that represented different points on the
same circle.
If x changed uniformly from 1 to -1 while y changed uniformly
from -1 to 1, we would find that y1 remained constant throughout,
y2 changed uniformly from -2 to 2, and y3 declined from 1 to 0
and then increased to 1 again. Three different aspects of the
behavior of x1 and x2 would be perceived simultaneously, and all
the behaviors through time would be different. Only two of these
aspects could be controlled at the same time; the other would
simply follow as a consequence.
Each x is derived in a similar way from a set w1...wn at a lower
level (if one exists). This is the basic hierarchical relation
between levels of perception in the hierarchical PCT model.
When some physical change occurs that affects sensory receptors,
it has a systematic effect on many perceptual signals. Different
kinds of physical changes will have different patterns of
systematic effects on individual perceptual signals. Some
perceptual signals will be affected more than others; some will
be increased while others will be decreased.
These patterns of change can be resolved into two components for
any one higher-level perceptual function. One component alters
the set of lower-level perceptions parallel to a surface of
indifference. That component has no effect on the higher
perception. The other component is normal to the surface of
indifference; it results in a change in the output perceptual
signal. The magnitude of the output perceptual signal thus
represents a particular aspect of the external event that altered
the lower-level perceptions. What that aspect is depends entirely
on the organization of the perceptual function involved.
In general, any external event that alters a set of perceptual
signals at one level will have differential effects of either
sign on perceptual signals at the next level. Some signals at the
next level will not be changed at all. Some will change slightly.
Others will change greatly. Each perceptual function defines its
own set of surfaces of indifference in the space of the lower-
level perceptions (each lower-level perception representing one
dimension of the space). A path orthogonal to the surface of
indifference created by any function represents the range of the
variable being perceived, and the position along that path
represents the magnitude of that variable. The common external
event is thus first represented as a set of states in the lower-
level collection of x's, and is then rerepresented at the higher
level by a new set of signals indicating many aspects of that
event, each aspect being perceived in some state between zero and
maximum.
In passing upward through the hierarchy of perceptions, this
representation and rerepresentation occurs over and over, new
types of functions being introduced and new aspects of the lower-
level world being created and signified by perceptual signals in
the next higher level. Some functions may compute weighted sums.
Some may compute spatial or time derivatives. Some may compute
logical relationships. In HPCT, an attempt has been made to
define the functions that are typical of particular levels, where
the functions are only suggested by the classes of perceptions
that they relate. The primary task of perceptual research in HPCT
is to discover, through finding what appear to be
hierarchically-related types of perceptions, the kind of
functions that must be performed to create the observed
relationships between adjacent levels. Perceptions can be defined
through control tasks.
The research question is thus not so much whether configurations
are derived from sensations, but what kind of perceptual function
would be required to create that apparent relationship. Of course
it is also important to establish the actual existence of
specific pairs of hierarchically-related perceptions. The two
questions must be answered together.
This is what PCT has to say about perceptions. Applying this
model to the perceptions of phonemes and words is a matter of
proposing appropriate perceptual variables and showing that one
type depends on other types. Indentifying surfaces of
indifference amounts to finding changes in lower perceptions that
make no difference to higher ones, and defining the particular
function of the lower perceptions that is orthogonal to those
surfaces of indifference. Identifying perceptions is a matter of
setting up appropriate control tasks and applying the test for
the controlled variable. Go to it.
···
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Best,
Bill P.