Re: Formalization of the Artificial Life Systems

[glen e. p. ropella]

Kerry Hanson writes:
 > 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.  

Just to add to Kerry's citation, the SFI working paper # for that
paper is 97-03-023 and you can get a copy at

  http://www.santafe.edu/sfi/publications/Working-Papers/97-03-023.ps

Vladimir Jojic writes:
 > What is ABM?

Sorry, you might not have been on the list when we were last talking
about ABMs and IBMs.  It is an acronym for Agent Based Model[ling] and
IBM is an acronym for Individual Based Model[ling].

This brings up another worrisome point for me.  You used the blanket
term "artificial life systems" to indicate what we've been calling
ABMs.  Now, I don't really want to start a discussion about the
semantic meaning of these terms... But, I think it's useful to note
that I think of ABMs being a bit orthgonal to "artificial life".  The
fundamental property (in my opinion) of alife systems is this
iteration through time.  I'm not sure it should be restricted to the
agent/behaviour point-of-view.

Of course, that comment has very little bearing on what we're really
discussing, which is the formalization of ABMs and the attempts to
build a set of results for those systems.

 > 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? 

Well, to me, linear algebra is the subsection of algebra that
addresses strictly linear algebraic objects, which means any algebraic
object (a set and a relation) that is orthogonalizable (is that a
word?) or can be divided into an orthogonal set of basis
elements.... basically, linear algebra is the study of vector spaces.
It's unclear to me whether an affine group is a part of linear
algebra, certainly it's "linear" in some sense! [grin] But, still the
requirement of an affine group to consist of 1 to 1 mappings between
the elements of the set seems to disallow some common relationships
between agents.  For instance, if we attempted to model 4 agents in a
space where each agent could communicate with any of the other agents,
the communications *channels* or pathways could be modelled as an
affine group.  But, to abstract to variable or multiple types of
messages that can be sent along those channels, one needs something
significantly more powerful than an affine group.

One thing that seems to me to make Swarm map more nicely onto the
general body of abstract algebraic theory as opposed to just linear
algebra is the simple notion that Swarm is based on message passing.
Message passing can be seen as the atomic operation in any of these
models.  And a message is simply a type of binary relation (which is
intuitionally embedded in the wonderful [receiver message] notation).
I don't think it's *as* useful to think in terms of an objective
operator because of polymorphism.  I.e. It truly is a binary operation
because:

   [x m];  //from inside object y

is not the same as

   [x m];  //from inside object z

One really ends up with the following construct:

   let m be a message relation
   let x, y, & z be members of the set O of objects
   then

   "[x m]; //from object y"  means  (y m x)
 and
   "[x m]; //from object z"  means  (z m x)

In many of our models, (z m x) == (y m x), but, it's not necessarily
the case.  This is more obviously tied to polymorphism in the other
situation where the receiver of the message is swapped, (x m y) ?= (x
m z).

 > 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 ...

Well, orthogonality will be impossible in most models, I think.  I
take orthoganality to mean "a set, U, of elements in a larger set, O,
is orthogonal to a set, V, of elements in O if and only if, with a
pair of elements (ui,vi), changes in the selection of ui do not affect
the selection of vi."  This is an abstraction of the concept of linear
independence (probably a bad abstraction [grin]).  And I think it
applies regardless of the measure on the set O.

The reason I think it will be difficult to establish orthogonality in
most of these ABMs is because it is rarely the case that an "action"
on object, ui, will not affect the future actions on another object,
vi.

Now, there are some instances where it seems clear that this is the
case, at least in time.  And that's the purpose of the concurrent
group type used in the activity library.  But, establishing this kind
of property when all possible global and local states will be even
harder than it is for time.  One of the purposes for Swarm (the 
software package) is to explore systems and experimentally determine
which systems might be orthogonalizable.

 > 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 ... 

And I don't intend to discourage you!.  I think what you're doing is
wonderful.  I wish I had more time to help pursue it!  But, I do want
to caution you against creating a formalism that only applies to a
very small subset of these systems.  Often, as valuable as that type
of approach is, it leaves one dissatisfied.  [grin]

glen
-- 
{glen e. p. ropella <address@hidden> |                                  }
{Hive Drone, SFI Swarm Project         |            Hail Eris!            }
{http://www.trail.com/~gepr/home.html  |               =><=               }


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