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Date:         Sun, 23 Aug 1992 09:08:27 -0600
From:         "William T. Powers" <POWERS_W%FLC@VAXF.COLORADO.EDU>
Subject:      Hierarchies
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[from Mary Powers 9208.22]
(penni sibun)

The "hierarchy" words, sentences, paragraphs:

A sentence is a collection of words, a paragraph a collection of
sentences.

A word can be a sentence or a paragraph, a sentence a word or a
paragraph, a paragraph a word or a sentence.

Cell, organ, organism, group:

An organ is a collection of cells, an organism a collection of
organs, a group a collection of organisms. At least in the
opinion of some people.

A cell can be an organism, or vv. However an organ is not viable
on its own, and (most?) PCTers don't buy groups as
superorganisms. But whoever made this list, and the many who have
gone along with it, probably look at it like word-sentence-
paragraph.

The problem in each case is thinking of larger and larger
collections as hierarchies. What Bill has tried to do in HPCT is
develop a hierarchy of levels of perception in which the level
below is necessary for the level above, but the higher level is
of a different type. You must have sensations in order to
perceive a configuration, but a configuration isn't simply a
collection of sensations.

I'm not really equipped to get further into this. However, by a
happy coincidence, Bill was writing a commentary to Martin Taylor
which covers this very subject, and I've asked him to put it on
the net here:
------------------------------------------------------------------
[Slightly edited ...WTP]

You'll remember that some time ago I raised the point that not all
levels of the hierarchy would work with weighted sums of lower-level
signals. The logical levels obviously don't. This has a bearing on
your discussion of conflict, cooperation, and coordination.

The levels as I have defined them have a characteristic that's not
immediately obvious. There is no generating principle which, given the
definition of one level, would predict the definition of the next.
These levels were found empirically, by a slow process of observing
myself perceiving and doing things that weren't in the model yet. Each
level came as a surprise and even a shock as I realized how obvious
was the type of perception that I'd been taking for granted.

I think that my levels are orthogonal to the levels below them. This
means that each new level introduces new degrees of freedom. If a new
level used an input computation of the same type as that of the lower
level, this would not be true. But when a new type of perception is
brought in, the same algebraic considerations cease to hold.

Consider sensations versus configurations. A sensation is a weighted
sum of intensity signals. So the number of possible sensations is the
number of different and independent weighted sums. If there are nine
intensity signals, there can be nine independent sensation signals at
a given time. Let's suppose that these nine sensations are
discriminable tones of a scale. Clearly, only nine notes at a time can
behave as independent tones.

A configuration in the auditory modality is a static pattern of
sensations. For example, a major triad chord would qualify as an
auditory configuration. With nine notes available, we could have three
simultaneous independent triad chords, sometimes overlapping some
notes and sometimes not. But these triads are not weighted sums; they
are weighted products.

If the sensations designate frequencies, a chord is defined by f2 =
K1*f1 and f3 = k2*f2. Actually, it's probably more like f2 = (K1 mod
2)*f1 and so on, because of octaves. So the rules of linear
combination no longer hold true. It's possible for a given set of nine
configuration-perceivers to respond to different subsets of nine
sensation-signals depending on the frequency range in question. When
you introduce higher powers of the variables, the number of degrees of
freedom multiplies (no pun intended).

Some of these degrees of freedom consist of variations at the
sensation level that make no difference at the configuration level,
but nontheless are present in it, or pass through it.

If the next level is transitions, or paths, we now have what seem like
infinite possibilities in the ten configuration signals. We have beat
notes, glissandos of whole chords or single notes, vibrato, tremolo,
accelerando, diminuendo, and other variations that mean nothing at the
configuration level. We now have combinations of the various
derivatives which are not sensed at the configuration level. Each
derivative adds a new degree of freedom.

And if the next level is events, we have phrases that include all the
possible combinations of the variables at the transition,
configuration, sensation, and intensity levels -- which looks like a
very great deal more than nine degrees of freedom altogether. We have
all kinds of frequency-time patterns, and at each level something new
is introduced that is not constrained at the lower levels.

So I think that the degrees-of-freedom argument based on repeated
application of weighted summations to previous linear combinations of
weighted summations does not apply to the levels of perception as I
have defined them. This is because with each new level, as I have
defined levels, there is a change of logical type; no new level of
perception is commensurate with any lower level.

At each new level, a new consideration is introduced: the most obvious
example is the sequence level. What are the possible dimensions of
ordering? I think that they are probably of a different order of
infinity from the dimensions of lower-level perceptions. The number of
possible orderings of three triads is limitless, not even considering
possible orderings of possible time-frequency patterns. This has
become a combinatorial problem, not an algebraic one: there are as
many dimensions are there are elements to be ordered. Any set of
configurations can represent a snapshot of an infinity of different
sequential progressions. And when the logic level is introduced, we
have 2^(2^n) possible Boolean expressions with n elements; elements
connected by "and", "or", and "not" can't be expressed in the same
terms as Ax + By + Cz. And that only takes care of logical rules --
there are many other kinds, such as grammatical rules.

Most hierarchical systems I have seen have depended on some generative
principle that applies to the relationships between all adjacent
levels. If my analysis of perception is reasonably correct, evolution
doesn't work that way. A new level is added only when the type of
computing process typical of the highest existing level, elaborated
horizontally and perhaps vertically too as far as possible, is no
longer sufficient to preserve the organism. The next true level that
is added is of a different type, using a different principle for
computing perceptions.

I would have said all this earlier if I had thought of it.

Best,

Bill P.
Comments: Gated by NETNEWS@AUVM.AMERICAN.EDU