Re: Formalization of the Artificial Life Systems

[Vladimir Jojic]

On Wed, 21 May 1997, glen e. p. ropella wrote:

> 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:

What is ABM?

>   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

When I say linear algebra, I mean:
- groups
- vector (and affine spaces)
- linear operators
- linear, quadratic and multilinear forms (I am not sure about the terms)   
etc.

What do You mean, when You say linear algebra? 

> 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.

This is a problem. But, since you can define various operators, you can
define orthogonality any way you like it. Finding operators with different
qualities, to support our models, will be challenge ...
 
I don't expect to get all done in just one pass, so using different
starting points is the part of the `big plan' ... but for start (some time
next week) just a high dimensional (Euclidean, meaning) space, one agent,
then alter the operations, add agents etc. and see what happens ... 

Regards,
Vladimir




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