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 from where
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