Newsgroups: comp.ai.genetic
From: stevem@comtch.iea.com (Steve McGrew)
Subject: Re: Fitness Landscapes
Organization: New Light Industries, Ltd.
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In article <32752831.41C67EA6@santafe.edu>, Erik Van Nimwegen 
<erik@santafe.edu> wrote:
>Steve McGrew <stevem@comtch.iea.com> wrote:
>> Something that almost everybody seems to overlook is the fact that
>> the landscape is itself determined by the representation, the mutation 
>> operators and the recombination operator, as much as by the fitness 
>> function. 
>
>Yes, I agree but I think that even if one takes account of the genetic
>operators in defining a distance metric on a landscape the picture
>of a population evolving on that landscape can be misleading. [snip]

>But if you're interested in the flow of a fluid (HUGE number of
>marbles that collide all the time) on that landscape the information
>contained in the shape of that landscape is not very informative at all.
>Just think of fluid dynamics phenomena such a turbulence. They are not
>determined by the shape of the landscape at all but can be very
>important for the dynamics of the flow. 

        A sentence got garbled there, I suspect.  But  it looks like you said 
that turbulence in the flow of an evolving population is not determined by the 
shape of the landscape.  That's going to take some serious thinking!  The 
fluid flow analogy has certainly occurred to me, but not turbulence.  My 
knee-jerk reaction is that what goes on in a GA is much more complicated than 
what goes on in a fluid -- even a turbulent fluid.  If you back up and unfocus 
your eyes, maybe a large enough population looks a little like a heavy, 
diffuse gas flowing generally downhill over a landscape-- but unless the gas 
is already concentrated into a blob that is all flowing towards the same 
general point,  it is going to behave in very un-gaslike, un-fluidlike ways. 
        With just the standard bit-flip mutations, a "particle" in the gas 
doesn't move continuously; it can pop up anywhere. Of course you can get 
around this particular difficulty mathematically by giving your space as many 
dimensions as there are bits in the representation, but it's very cumbersome 
and stretches intuition close to the breaking point.
        With recombination, it's worse because there are very long-range 
interactions between pairs of  "particles" in the gas, such that the identity 
of a particle gets lost: it is only identified by its position, but its 
position is not traceable to a single particle in the previous generation.  
And, the correlations between particles in generation N and generation (N+1) 
are dependent on the relationship between a particle's position and its 
fitness.
        So I'm having a hard time grasping the idea of turbulence in an 
evolving population.  To use the concept of turbulence, you must need to 
define some kind of "momentum density" and "population density" in the 
solution space, as well as several "forces": selection pressure that 
annihilates low-fitness particles and preserves high-fitness particles; forces 
that create new particles at random when needed; reproductive fitness forces 
that bias probability of interaction between particles; forces that define the 
nature of the interaction (recombination) between particles, and so on.  All 
this will have to take place in a representation space, in which each location 
has a fitness associated with it.   
        The first difficulty with the model is the concept of "density".  I'm 
not saying it can't be done, but how can you define density in a 
representation space that is multiply connected to a huge degree (say, 
2^100,000), and contains only 20 to 100 particles?
        "Momentum density" is equally hard for me.  In a space where a 
particle can disappear from one location and appear instantly at a distant 
location at random,  or even be annihilated along with all traces of its 
former existence, what can be the meaning of particle "motion" or even of 
"flow"?
        In Fogel's Evolutionary Algorithms, I can accept the fluid flow 
analogy because the representation space is "almost" simply connected and you 
can always trace the geneology of a population member back through a chain of 
single individuals.  There, the interaction between individuals is effected 
only by a creation/annihilation operator that gives high-fitness individuals 
opportunities to produce mutant offspring while giving low-fitness individuals 
a high probability of producing no offspring.  In an EA there is no 
recombination operator and the distance between a parent and its offspring 
(and thus between any two points in the space) is a useful concept.
        In a  GA that has a recombination operator, though, the situation 
seems immensely more complicated.  Guess I'll have to get your paper; maybe 
the answers are there.  Thanks for the response & the lead to your paper.
Steve

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