[Martin Taylor 991211 11:37]
[From Rick Marken (991210.0900)]
It is true that a fixed reference will create no problem if the
system with the fixed reference is not used by any higher level
system as the means of controlling its perception. And it is,
indeed, possible to remove the problem by reorganization.
Actually, it is true that a fixed intermediate reference value will create
no problem that perception _is_ used by higher level systems, if the
higher perceptions are based on more than that single intermediate
high level perceptions: a = x + y + z, b = x - y + 3z
mid-level perceptions x, y, z.
Now set reference values for a = 3, b = 3
One way for the high-level perceptions to satisfy their reference values
is for x, y, and z all to come to the value 1.
Now set a reference for the x-control system so that x is forced to be 2.
According to Rick, this means that a and b cannot be controlled so as to
regain their reference values of 3 each. But in fact they can. All that
is necessary is for y and z each to take on the value 0.5. There is no
loss of higher-level control.
Higher-level control is lost only when the fixing of an intermediate
reference level means that there are fewer remaining intermediate
degrees of freedom than are necessary to avoid conflict among the
higher-level control systems. Without analyzing the details of the
published spreadsheet, Rick's claim (that the high-level variables
can't be controlled if an intermediate one is fixed) suggests that
there really are 6 degrees of freedom at each level. If you fix one
reference value in the middle level, this would allow the 6 high-level
systems only 5 remaining degrees of freedom for control. If those
high-level systems had been linear control systems, this would immediately
mean that they were inevitably in a conflict situation.
However, as I show below, the published spreadsheet does not behave
like this, and in fact all six third-level variables do retain control
even when the reference level "F7" is fixed at 50.
Conflict occurs when N analogue control systems have at some point in their
control loops fewer than N degrees of freedom (the famous 2 systems setting
the same perception to 2 different reference values is a special case of
this more general statement). But conflict is NOT assured when N _logical_
(category-perceiving) control systems attempt to control through fewer
than N degrees of freedom at lower levels, because there exist regions
in which intermediate analogue perceptual values are compatible with
different higher logical-valued references.
To see this, let's recast the example above, leaving "z" out of the
higher-level perceptual signals.
(1) Analogue high-level perceptions: a = x + y, b = x - y
Reference values: a = 3, b = 3
Both a and b can be brought to their reference values if x = 3, y = 0.
Now fix the reference value for x to be 2, as before. When you
do this, a and b find themselves in conflict, since a is at its reference
value only if y is 1, and b is at its reference value only if y is -1.
(2) Categorical (logical) high-level perceptions: a = (x+y)>2, b = (x-y)<4
Reference values: true, true.
As before, these can be satisfied by x = 3, y = 0, when x+y=3 and
x-y=3 (and by a wide range of other values for x and y).
Now fix the reference value for x to be 2, as above. This time there is
no conflict between a and b. Both achieve their reference values when
y = 1, and x+y = 3, x-y = 1 (and for a range of y values either side of 1).
Conflict need not happen between higher-level categorical control units
when the reference value for an intermediate-level analogue control unit
is fixed. But it may happen (not, however, with a and b defined as above,
since there is a range of values of y that satisfies both inequalities
no matter what value x is fixed at, and vice-versa).
Rick's spreadsheet provides a more complicated example. Rick said
(Rick Marken 991208.2130)
I understand "keeping a commitment" to mean keeping a perceptual
variable in some agreed to state. A child who is keeping a
commitment to be quiet in class, say, is maintaining a perception
of his own noise level in an agreed to state of "quiet". A child
who keeps such a commitment is doing the equivalent of setting
the reference value for one of the intermediate level perceptual
variables in my spreadsheet hierarchy to a constant.
For example, change the reference value in cell F7 to a constant,
like 50. This control system, at level two of the hierarchy model,
is now _commited_ to keeping the perception it is controlling
at 50; and you will see that the system will be able to do
this successfully; the perceptual variable (in cell F8) is
brought to and maintained at 50, protected from disturbance.
The system keeps its commitment.
But you will also find that some of the level three systems
(the ones in rows 3 through 5) can no longer keep the perceptual
variables they control (the values in row 4) matching the
references for these perceptions (the values in row 5).
I wonder if Rick (or anyone that subsequently argued the point) ever
actually tried this experiment? I did, and I find Rick's statement
to be false. The level 3 systems in Rick's spreadsheet as published
do keep their perceptual variables matching the references for these
I set the reference value in F7 to 50 (wildly different from its
prior value of around -41). I then watched the hierarchy try to come
to terms with this fixed setting. It takes a long time (several thousand
iterations because of the large slowing factors), but eventually
everything stabilizes with all the level 3 control perceptions matching
their reference values.
What the spreadsheet _actually_ demonstrates is that if the higher-level
variables are logical ones, then control can be maintained _despite_
fixing an intermediate level reference value.
Analyzing the nature of the logical variables used by Rick shows why this
is possible. The six level 3 perceptual functions D3-I3 are:
D3. if ((0.2*D8) > E8), reference value = true
E3. if ((0.5*E8) > F8), reference value = true
(in other words, if F7 is set to 50, and F8 therefore is near 50, the
E3 perception must be over 100, and the D3 perception over 500 (or possibly
even more, depending on where E3 actually settles).
F3. if ((0.1*F8) > G8), reference value false (i.e. G8 > 5)
G3. if ((0.3*G8) > H8), reference value true (i.e. H8 < 0.3*G8, which is
always satisfied by H8 < 1.5)
H3. if ((0.4*H8) > I8), reference value false (i.e. I8 > H8)
I3. if ((2*I8) > 0), reference value false (i.e. I8 < 0).
It's clear that all these conditions can be satisfied by some set of
values for D, E, G, H, and I8 no matter what the value of F8. One such
is what the spreadsheet found: 508.95 101.31 (50.00) 24.23 -55.18 -14.89.
If the higher level perceptions are analogue, they will be able
to maintain control if and only if there are enough intermediate
degrees of freedom remaining after some have been removed by the external
forcing of some intermediate reference values.
If the higher-level perceptions are categorical (logical) they will always
be able to maintain control under conditions that would allow analogue
perceptions to be controlled, but additionally they may be able to maintain
control even when there are fewer intermediate degrees of freedom than
controlled logical perceptions. Whether the logical units can in practice
maintain control will depend both on the reference value set by external
intervention and by the relationships among the logical variables
I leave it to the reader to assess the relevance of this to the issue of