Minimisers and Control Systems

[gabriel to powers, but not in reply to new matl.]

Perhpas the following is an example that makes my point that inside
minimisers of functions where control theory can be used to hunt
for minima, is the following.

Suppose we have a box F that perceives x and generates a behaviour f,
both of which have representations by real scalars. F can be said to
compute f(x), and by computing f(x) and f(x+delta) can provide the
means to estimate df/dx.

Suppose we have another box M, that can stimulate F and observe its
behaviour. M can clearly, given x, measure f(x) and f(x+delta), and
estimate df/dx. Now add an ECS to M having reference 0, perception
of df/dx and x from M, and ouput x' such that either df/dx for x=x'
or f(x') < f(x). This ECS will need a layered protocol with M
if it is to find an x' that does as very good job - e.g. by
having the ECS estimate the second derivative of f(x) at x,
and then fit a quadratic function q(x) to f in the rgion of x,
and output the x' that minimises the value q(x'). Also to minimise
f() to within some acceptable delta will require more than one step.

Well, stick insects have to walk up twigs to find prey, so more than
one step is no problem. The function f() had better be continuous
and have well defined derivatives most places. This is the necessary
condition for successful use of a control system, as distinct from
some other means to drive perceptions towards desires. Not necessary
for minimisers in general though.

This seems very clumsy and expensive for x scalar if you just
want to make a composite system of M F and ECS to generate a
behaviour B that drives a perception P closer to a desired value

But if x is vector of a few hundred components, it's actually a
very good way to drive P(x) closer to D(x) - x is a vector in
the external world that can be changed by the composite M,F,ECS

These are the kind of complex problems where I am interested to apply
the central idea of BCP.

Now, if you look at the ways Governments or Corporations make bad
decisions, it's because the decision makers do not have Ashby's
"Necessary Variety" in their degrees of freedom, even though
there are likely to be unrecognised degrees of freedom that MIGHT
exert great leverage. And this is why being able to make new
percepts in Kanerva space is important to me. It's necessary
for strong inductive reasoning.

And many decision makers have very few important values - power, short
term profit, keeping up appearances, and acceptance by others of their
ilk. The rest of us suffer the consequences.

It is incidentally being able to recognise these degrees of freedom
that makes great fortunes in business, and wins wars if these are
regrettably unavoidable. My combat pistol instructor used to
emphasise that it was preferable to avoid need to use lethal force, but
if it DID happen to be unavoidable there was no medal for coming in

It was this failure by the "Great Society" that set me off on a
path looking for a "better way".