[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?

## ···

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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.

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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.

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Best to all,

Bill P.