[Hans Blom, 951004]
(Martin Taylor 951003 14:00) replying to
Erling Jorgensen (950930.2240CDT)
Getting back to the main thrust of your posting, on a single
vs. multiple apices at the very top of the hierarchy...
Now, if there is such a hierarchy (meaning "if HPCT is a
correct approximation to reality"), then for there to be one
apex, there must be a single number that describes EVERYTHING
that really matters to the individual.
A single number, or a single direction -> "Better."
No, not at all. The behaviour of the PRESENT hierarchy is
affected by the PRESENT values of the top-level reference
signals (and all perceptual- sensory inputs). If there is a
single apex, there is only one top-level reference signal, with
one numerical value AT PRESENT. The action of the entire
hierarchy is devoted to bringing the corresponding perceptual
signal to THAT single value. That single value describes
EVERYTHING that really matters to the individual.
Strange enough, maybe, modern control theory strongly supports both
the "single number" and the "better" notions. Let me explain.
Modern control theory demonstrates that (almost; but the exceptions
don't matter here) all control problems can be reframed as optimiza-
tion problems, in which the supreme goal is to minimize (or maximize)
ONE outcome. Whereas mathematics says that we cannot compare apples
with oranges, modern control theory says that we do exactly that all
the time, that there is in fact no alternative. We literally compare
apples with oranges when the fruit bowl is passed around after dinner
and you're supposed to pick only one fruit. In a wider sense, we
always have to choose the better alternative from disparate items.
This implies that everything has to be reduced to one common coinage
(nerve impulses?), and that sub-goals must have "weights" denoting
their relative importance.
The one supreme goal is often expressed as something like
i = N
J = minimum (SUM w1*(x1[i] - r1[i]^2) + w2*(x2[i]-r2[i]^2) + ...)
u[1]..u[N] i = 1
where some (control) mechanism, that can manipulate the "distance"
between x[i] and r[i], is implied. This mechanism must generate
actions u[1] through u[N] which minimize the sum of errors over the
time period from i through N (where N may go to infinity). The
"better" shows in the above equation in that a MINIMUM is sought --
where the seeking (i.e. controlling) continues all the time (or up
till some maximum time). The above expression particularly applies to
multi-input multi-output systems, and the constants c1 and c2 denote
the relative importance of the goals.
Note that the above formula in itself implies a hierarchy, with ONE
top (J) and lower levels SUM (x[i]-r[i])^2.
Does this contradict your statement above? No. It is perfectly
permissible to say that the behaviour of the PRESENT hierarchy (SUM
...) is affected by the PRESENT values of the top-level reference
signals r[i] (and all perceptual-sensory inputs x[i]). But then there
is an even higher level (J), the level which determines the values of
the top-level reference signals. And at THAT level, "better" (finding
an extremum) is the operative word.
Greetings,
Hans