Complexity change in top-down design

[Øyvind Teig <Oyvind.Teig@xxxxxxxxxxxx>]

Dear all
22Jun98

Complexity change in top-down design
====================================


In "Is "the therory of everything" merely the ultimate ensemble
theory" Max Tegmark writes on page 22.:

  We noted that if one keeps adding additional axioms to a
  formal system in attempt to increase its complexity; one
  generically reaches point where the ballon bursts: the
  formal system becomes inconsistent, all WFFs become
  theorems, and the mathematical structure becomes trivial
  and looses all its complexity. (p22)

He continues:

  In Section IV, we replaced this "top-down" approach with a
  "bottom-up" approach, making an overview of our local
  neighbourhood  in "mathematical space". (p22)

In the paper of which I understand one ppm, he describes an
example of top-down approach, and I started thinking about what
we do every day: both top-down and bottom-up approaches are used
for software design. Even if this isn't what he has in mind, it 
is interesting to see if it fits into my world as well. I can
understand that we "attempt to increase complexity" in top-down
(see New Scientist excerpt). But 1) when does our balloon burst,
2) what becomes theorems (WFFs: I'm not sure what it is, not 
the least I cannot see what it is that in our case becomes
theorems) and, 3) which mathematical structure in our case
becomes trivial and loses its complexity? In other words,
please explain, anyone!

"Anything goes" in New Scientist, June 6 excerpt:

  Another difficulty with the ultimate ensemble theory is that
  it appears very wasteful. However, Tegmark has an extraordinary
  argument with which to counter his critics. He says there is
  actually less information in the multiverse than in an individual
  universe. 
    To illustrate his argument, Tegmark gives the example of the
  numbers between 0 and 1. A useful definition of something's
  complexity is the length of a computer program needed to generate
  it. Imagine trying to generate a single number between 0 and 1,
  specified by an infinite number of decimal places. Expressing it
  would take an infinitely long computer program. But to generate all
  numbers between 0 and 1, all you would have to do is start at 0,
  step through 0·1, 0·2 and so on, then 0·01, 0·11, 0·21 and so
  on--an easy program to write. In other words, creating all
  possibilities is much simpler than creating one very specific
  one. 

References
  New Scientist: 
    http://www.newscientist.com/ns/980606/features.html
  Max Tegmark's abstract: 
    http://www.sns.ias.edu/~max/toe.tml

------
Cheers,  

Oyvind Teig, Autronica, Trondheim, Norway
Oyvind.Teig@xxxxxxxxxxxx
Tel.: +47 73 58 12 68
Fax.: +47 73 91 93 20