Re: Formalization of the Artificial Life Systems

[Kerry Hanson]

At 07:08 AM 5/21/97 -0600, you wrote:
>
>Hey guys,
>
>Correct me if I'm misguided (which I usually am [grin]); but,
>does it strike anyone that using something as limited as 
>linear algebra might be a tunnel-vision approach to modelling
>these systems?  
>
>It seems to me that there are some very powerful results in 
>algebra (as a whole theory) that are not restricted to linear
>algebra that would fit nicely into ABMs.  Examples, are the
>concepts of:
>
>  posets -- sets ordered by a reflexive, antisymmetric, and 
>            transitive, binary relation
>  groups -- sets under a closed, associative, identity holding,
>            and inverse holding, binary operation
>  quotient groups -- set of equivalence classes of an algebraic
>                     object under an equivalence relation for that
>                     object
>
>  etc...
>
>Granted, the types of results we achieve from linear algebra could
>be *very* useful wherever they can be applied; but, the whole of 
>linear algebra is based on the ability to establish orthogonal
>sets and I'm not sure how such a criterion can be established in 
>ABMs.  If we view orthogonality as a very specific kind of quotient
>group, then we can think about some ABMs, possibly, possessing 
>such a property.... but, I worry that starting at that point might
>make us too focussed.
>
>Ok....Fire away! [grin]
>glen
>
Actually, I agree, if the application needs the higher level of abstraction.
One of the applications we are pursuing is making use of a model we
developed using pattern theory.  For those who aren't familiar with it,
(from Mumford),

"Pattern theory aims at constructing knowledge representations of complex
systems using an algebraic foundation that describes typical pattern
structures in terms of their compositions (configuratin spaces) and
invariances (similarity groups).  To represent the variability of the
patterns probability measures are defined on the resulting image algebras."

It turns out to be a nice framework to implement a complex adaptive system.
The image algebras in question are actually partition quotient groups
defined on equivalence relation identification rules.  For those who know
about pattern theory SFI just released a paper, "Complexity of
Two-Dimensional Patterns", by Lindgren, Morre and Nordahl.  For those who
want to know more, you might try the book, "Elements of Pattern Theory", by
Ulf Grenander, Johns Hopkins University Press, 1996.  

Using this framework has given us both a structure for the problem and more
degrees of freedom in the design of the simulation.

Regards,
Kerry



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