Re: Bomb in the Hierarchy Simulation (was Back in

control)

[Martin Taylor 2005.02.22.0929]

[From Bill Powers (2005.02.19.1604

MST)]Martin Taylor 2005.01.19.15.59–

They are only numbers, whatever the

external analyst might think they represent. The one level doesn’t

know it controls the amount of sensation. It only controls the value

of a perceptual variable. Likewise for the level that the external

observer thinks is controlling the shape of a configuration. It’s just

another value of a scalar variable. The control system doesn’t

“know” the meaning of what it controls.No, but the computations that take place determine that meaning. Each

level of perception computes invariants from the level below, so if a

particular perception at the level below somehow goes out of control,

there is no reason to suppose that anything at the next level will

also go out of control. Perceptions at a given level are functions of

many at a lower level, not just one.

That last was exactly the point I was making, wasn’t it? That if

one of the inputs ran away, there was a possibility that the

higher-level system might not be able to sustain control by means of

its other inputs?

You say “there is no reason to suppose that anything at the

next level will also go out of control.” Of course there isn’t.

But there’s also no reason to suppose that nothing at the next level

will go out of control.

The Bomb is (necessarily in a functioning hierarchy) something

that happens only rarely, and that rarity is increased the larger the

explosion under consideration. Trivially, assuming equal probabilities

everywhere, if the probability of it propagating through one level is

p, then the probability of it propagating throung n levels is

p^n.

(Of course that probability will be different for different

control units, whether they are at the same level or at different

levels, but the same principle applies; it would be interesting,

though, to determine whether in a heterogeneous hierarchy, the Bombs

follow paths of least resistance in the way landslides and snow

avalanches do).

Furthermore, each level introduces new

information. The shape of a cube is computed fromwhere

sensations occur in the visual map, not on the magnitudes of the

sensations, and the whereness is not indicated in the magnitude of any

sensation signal. Sensations must be controlled to change whereness,

but the magnitude of a given sensation is not the critical variable.

The spatial relationship between two objects is left unchanged if both

objects move in the same way, or if they change brightness or

orientation or color. A runaway magnitude at one level does not

necessarily imply a runaway magnitude at the next level. There can,

perhaps, be special cases where that link between levels might exist,

but as a general rule I don’t think it does.

I illustrated the Bomb algebraically in a linear system (as has

[Erling Jorgensen (2005.02.22 0100 EST)] numerically). That is the

kind of system usually used in simulations to demonstrate the

viability of PCT (e.g. Rick’s spreadsheet). We all know that

mathematically a hierarchy of linear control systems is exactly

equivalent to a one-level control system, so demonstrating the Bomb in

a linear hierarchy really is no demonstration at all.

In general, P(n) = p(P(n-1,1)…P(n-1,k)) where P(n) is the

perceptual signal of some level n control unit, p is an arbitrary

function, and P(n-1,m) is the contribution of the m’th level n-1

perceptual signal to P(n).

The Bomb can explode if for a particular function p, the level n

control unit has a finite probability of being unable to compensate

for a runaway in any of the P(n-1,m). Only if there is NO

combination of input values for which the control unit cannot

compensate will the Bomb explosion definitively stop if an explosion

reaches that control unit.

Well, I still believe that this sort of

verbal argumentation doesn’t get us anywhere. Better to produce a

mathematical demonstration; then the outcome won’t depend on who finds

the cleverest argument.

There are two kinds of simulation that I can imagine being

useful. One demands that a heterogeneous hierarchy (one not involving

linear systems at more than one consecutive level) be natively

reorganized in a sufficiently complex varying environment, in which

feedback strands turn positive from time to time. That would be the

ideal case, but a difficult one to set up and run, and even then,

whatever the result, the verbal discussion would continue. If no Bombs

showed up, I could argue both that it’s just a matter of time, and

that the PIFs had been inappropriately chosen. If the Bomb did show

up, you could argue that a better choice of PIFs would have eliminated

the Bomb.

The second kind of simulation would replace actual control

systems by a network of nodes and links in which the propoagation of a

signal (the Bomb explosion front) out of a node was a probabilistic

function of the inputs. Such a netwrok could be tested analytically or

by Monte Carlo simulations, and parametric variation of the different

node probabilities could show how the Bomb propagation and

level-by-level damping were affected by probability variation. That

experiment would also be subject to verbal argument, both as to its

applicability to the real world, and as to the ways of choosing the

probabilistic functions at issue.

“Cleverest argument” determines only who of an audience

comes to believe something. It doesn’t determine the reality behind

the argument.

Martin