[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

perception. Consider:

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

perceptions.

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.

-------------

Summary:

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

controlled.

I leave it to the reader to assess the relevance of this to the issue of

"commitment."

Martin