From: Charles D Hixson (charleshixsn@earthlink.net)
Date: Mon Apr 24 2006 - 09:52:29 MDT
On Sunday 23 April 2006 10:25, Philip Goetz wrote:
> My intuition, based on experience with how much computational power it
> takes to solve a problem of a particular size, and on Rescher's law of
> logarithmic returns, is that exponentially-increasing computational
> power is required to provide linear increase in "smartness", or some
> measure of the problems we can handle. For instance, finding primes
> of 2N bits takes much more than twice the computational power of
> finding primes of N bits.
>
> I also expect that the computational complexity of cognition is an
> exponential function of the size of working memory, so that if we
> currently have a working memory that can store 5 chunks, the amount of
> computation available in the universe limits us to some double-digit
> number of chunks.
Quite plausibly. The question would be, however, "In what chunk complexity do
problems exist?" (tying it back into the thread topic). If problems of
complexity 5 are the highest that exist, then a limit of you only being able
to simultaneously process 5 chunks would not limit the problems whose
solution you could understand (though it *might* slow down the
comprehension). If, however, there were a problem that required
simultaneously dealing with 6 or 7 chunks, then you couldn't understand it no
matter how long you tried.
Now the proof of the four color theorum isn't, itself, proof of such
complexity. It could be that it's the sheer length of the proof that defeats
us. Do note, however, that symbolic logic is so structured that one only
considers two chunks at any one time. I.e., formal logic methods are either
of the form x = f(y) or of the form x = f(y, z). The required number of
parameters is the inherent chunk complexity. If it has been shown that
every function can be reduced to a form that is not more complex than those,
then chunk complexity isn't a limitation. I believe that it's been shown
that boolean functions can always be so decomposed...though I can't remember
*why* I believe this, but not all functions are boolean. (OTOH, computer
programming is a good indication that boolean functions can, at least, be a
good model for all other functions.)
I think that I've just argued that one can always exchange chunk complexity
for serial complexity. And serial complexity can be handled by external
libraries. Given that my (vague) arguments are correct, then the thread
proposal is correct (though "... Just a Bit Slow" seems like an
understatement.
The key question is CAN x = f (y, z, w) always be rewritten as x = g (y, h(z,
w) )? Not can it be done in a reasonably efficient way, but can it always be
done at all. And then can this be generalized to arbitrarily higher
dimensionality. If those are both correct, then the hypothesis is correct,
and my initial reaction was incorrect.
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