From: Andy Brice (andyb1@btclick.com)
Date: Fri Jun 28 2002 - 16:51:51 MDT
> > I disagree. Both are different representations of the same thing. It's
> like
> > using two different languages to describe a chair. Both are equally
> accurate.
>
> Not true. If one is a subset of the other, the superset is more powerful.
> For example, 2nd order (predicate) logic
> (http://www.earlham.edu/~peters/courses/log/terms3.htm) is a more powerful
> description language than 1st order (propositional) logic
> (http://www.earlham.edu/~peters/courses/log/terms2.htm) because it
> incorporates the
> latter. In the same way, algorithms are a superset of math.
I tend to think of an algorithm as a description of how to evolve a discrete
set of states, e.g. cellular automata, and mathematics as describing
continuous functions, e.g. y=f(x). Of course, not all mathematics is
continuous, so I am over-simplifying.
I believe that it has been shown that cellular automata can perform any
calculation that a Turing (universal) machine can perform. So it should
theoretically be possible (although probably not very convenient) to
calculate functions, e.g. a planetary orbit, with cellular automata. However
the accuracy of a cellular automata will always be restricted by its finite
size.
A continuous function doesn't suffer from this. But I doubt you can describe
a seashell pattern adequately in terms of continuous functions.
In this simplified case it seems that cellular automata is more powerful,
but neither is able to completely represent the other.
Andy Brice
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