# Killeen; math; symmetry

[From Bill Powers (960116.1300 MST)]

Bruce Abbott (960116.1100 EST) --

Another equation of interest in Killeen's system is the one he uses
to adjust predicted response rates to take "ceiling constraints"
into account -- the fact that the minimum interresponse time,
delta, imposes a maximum of 1/delta responses per second on the
response rate. The equation is:

B' = B/(1 + delta*B),

where B' is the adjusted value of B.

At first I was a bit puzzled by this equation ...

Well, so am I. If there is a "response constraint" such that the maximum
behavior rate is 1/delta, then it is simply a constraint -- you don't
need to propose an asymptotic approach to that limit as B goes to
infinity. You just write

if B > 1/delta then B := 1/delta.

Using the Killen equation, we predict that if B is half the maximum
rate, B' = (0.5/delta)/(1 + delta * (0.5/delta)) = (1/3)(1/delta). So
this equation influences B when it is still far from the limit. Why do
that? Why not just put a limit on B?

In fact, if a response takes a fixed time, then when responses are
spaced a little farther apart than that fixed time, there should be no
difference between actual and observed response rates.

Signals ---> | |
><-------- IRT -------->|
---------------- ----------------- ------------
closure level - \ - / - - - - - \ - /- - -
\/ \/
Movements ---> |<--------- IRT ------->|

In this diagram, each V is a down-up movement taking a certain fixed
length of time. The time between similar parts of each movement is the
inter-response time: the signals initiating the movements are separated
by the actual IRT. If the signals get very close together, you have

Signals ---> | |
--> | | <-- IRT
---------------- -------------
closure level - \- - / - - -
- \/\/
Movements-> ->| |-- IRT

So the IRT may not allow enough movement to release and close the
contact, but it is still the same IRT. I think Killeen is way off base
here.

To my mind, this is a clumsy way to deal with the limitations
imposed by the fact that responses have finite duration; it seems
odd that the basic equation for response rate would not predict the
actual IRT directly and impose any minimum IRT limit by taking
account of the actual dynamics involved in responding.

I completely agree. How often do we see responses occurring at the
maximum physically possible rate? And how reliable are contact closures
as a measure of behavior rate when the time needed to depress and
release is comparable to the time between responses?

Anyhow, in your computer model the IRT and IIT are constants, so what's
the point of all this?

···

-----------------------------------------------------------------------
Rick Marken (960116.0800) --

1. X(n) = g (Xn-k) does not describe a closed loop system and

2. When loop gain in a positive feedback system is >= 1.0 the
system becomes unstable.

Do these statements make it through the first pass?

Yes. I agree with you that an equation which involves TWO or more passes
around the loop (1) is not the right equation for a closed-loop system.
In a digital system, k has to be 1. Also, without any time in the
equation, this is not an equation about behavior. The rate at which X
changes depends only on whether you have a 8086 or a Pentium. In
modeling a real system you have to take into account how fast the
variables can change. When you leave out time you have a successive-
state equation, but it isn't tied to the behavior of physical variables.

Oops. So my simulation is a proportional rather than an integral
control system.

Actually, I'm surprised that your simulation worked at all, unless you
had a slowing factor somewhere else in the loop. It should have run away
if the loop gain was less than -1 or greater than 1.
-----------------------------------------------------------------------
Shannon Williams (960113) --

The power of PCT is in the concept of the control loop. No one
aspect of that control loop should be emphasized more than the
others.

Therefore, if you think that it is important to emphasize that
behaviors exist only to control perception then you must equally
emphasize that perceptions exist only to elicit behaviors. If you
don't, then you emphasize one biased and limited aspect of the
loop, rather than the concept of 'loop'.

This is incorrect. Behavior controls inputs but inputs do not control
behaviors. The situation is not symmetrical, because the control system
has gain: the output has a greater effect on the input than the original
effect of the input on the behaviuor. In general, the environmental part
of the loop involves a loss; some of the output energy is dissipated.
The organism contains power amplification; the output contains more
energy than the input. See my short paper, "The assymetry of control" on
p. 251 of Living Control Systems I.

If you try to reason about a control loop the same way you reason about
digital S-R effects, you will think that a loop with a gain greater than
-1 must run away. A couple of famous cyberneticists, Warren McCullouch
and (on one occasion at least) W. Ross Ashby, thought that was true, and
this was the main reason they missed the boat on control theory. In
fact, control systems exist with gains of more than minus one million,
where the minus sign indicates that the output effect on the input
opposes the initial deviation of the input. These loops are perfectly
stable. When you understand how this stability is produced despite the
large amplification factors, you will have a pretty good understanding
of PCT.

I do agree that saying "behaviors exist to control perceptions" is
somewhat fuzzy. The prime mover is the reference signal, not the
behavior. But the statement does have this meaning: there is no reason
to think that any behavior exists except as part of a negative feedback
control system that maintains its perceptual input near the level
specified by a reference signal (even if the reference signal is zero).
The only exception to this is during reorganization, when behaviors may
be emitted which do not succeed in controlling an input. But those
behaviors soon go away.
----------------------------------------------------------------------
Best to all,

Bill P.