**From:** *rpwl@lightlink.com*

**Date:** Sat Oct 08 2005 - 19:35:30 MDT

**Next message:**Ben Goertzel: "RE: Emergence"**Previous message:**Ben Goertzel: "Emergence"**Maybe in reply to:**Ben Goertzel: "Emergence"**Next in thread:**Ben Goertzel: "RE: Emergence"**Reply:**Ben Goertzel: "RE: Emergence"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ] [ attachment ]

Ben Goertzel wrote:

*> Hi,
*

*>
*

*> I remember there was some discussion on this list a while back regarding
*

*> complex-systems concepts like "emergence" and their utility (or otherwise)
*

*> for understanding AI and the world in general.
*

*>
*

*> Well, I stupidly left the lights on in my car yesterday (while it was parked
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*> for a few hours) and wound up sitting for 90 minutes waiting for AAA to come
*

*> give me a jumpstart. In the meantime I sketched out (literally) on the back
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*> of an envelope what seems to be an elegant definition of "emergence" in
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*> terms of basic computational concepts.
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*>
*

*> The p.s. to this email presents this definition. It's math-y so
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*> non-math-happy readers should just ignore it!
*

*>
*

*> The main point is to show that the notion of emergence can be formulated in
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*> a precise and elegant way. I have made efforts in this direction in some of
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*> my prior writings but I think this one is a little nicer.
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*>
*

*> The basic intuitive notion pursued is that emergence occurs when one has a
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*> situation so that:
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*>
*

*> · f is a pattern in the combination of x and y
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*> · There is no g similar to f so that g is a pattern in x or g is a pattern
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*> in y
*

*>
*

*> I just introduce what seems to be the minimal math formalism (or close to
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*> it) needed to make the above two sentences precise.
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*>
*

*> The next step would be to try to do some real math modeling and see if the
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*> intuitive examples of emergence observed in complex systems can be easily
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*> cast into this kind of formal perspective
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*>
*

*> -- Ben G
*

*>
*

*> ******
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*>
*

*> P.S.
*

*>
*

*> OK, here goes...
*

*>
*

*> Suppose we have a set E of entities with an operation + defined on it. (For
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*> instance, E might be the set of subsets of some other set of elementary
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*> entities; or it might be the set of multisets formed from some other set of
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*> elementary entities.)
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*>
*

*> The + operator may be assumed commutative and associative. Examples of
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*> suitable + operators would be operations of set or multiset union.
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*>
*

*> Consider also a set F of function mapping E into E. Elements of E may be
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*> mapped into elements of F via considering them as “constant functions.”
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*> Define an operation * on F as function composition. Of course, * is neither
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*> commutative nor associative.
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*>
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*> The operation + may be extended from E onto F in an obvious way, so that we
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*> may now think about {+,*} as algebraic operations on F.
*

Wait! You can extend + from E into F only if there is a unique x in E that

is associated with each f in F. To do this you need to map elements of F

into elements of E ... and the mapping has to be injective if you want to

extend + as you suggest.

But in the previous paragraph you talked about mapping in the opposite

direction, from E to F, and that would not, in general, buy you an

injective mapping.

I think you must have meant something a lot more specific (a restricted

notion of what is supposed to be in F). Could you clarify what you mean by

"constant functions" and what F is (surely it cannot be the set of ALL

possible functions from E to E?), and how this allows you to extend + to F?

Many thanks.

Richard Loosemore

*> Distributivity may be assumed with regard to function application, i.e.
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*>
*

*> (f+g)(x) = f(x) + g(x)
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*>
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*> Within F, we may assume there is a zero element (the constant function
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*> mapping everything to zero) so that
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*>
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*> x+0=0+x=x
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*>
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*> 0*f = 0
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*>
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*> and a 1 element (the identity function) so that
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*>
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*> 1*f = f*1 = f
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*>
*

*> Finally, let us assume there is a valuation measure H on E, meaning that for
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*> any x in E, H(x) is defined as some nonnegative real number. Also assume a
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*> two-component valuation measure H(x,y) so that H(x,0) = H(x).
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*>
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*> Intuitively, H(x) is to be thought of as the complexity of producing x, and
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*> H(x,y) is to be thought of as the complexity of producing x given y as
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*> input. This interpretation makes it plausible to assume that
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*>
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*> H(f*g) <= H(f) + H(g,f)
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*>
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*> We may now define the notion of pattern as follows:
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*>
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*> f is a pattern in x iff:
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*> f(0) = x and H(f) < H(x)
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*>
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*> That is, f is a pattern in x if f produces x and yet f is simpler than x.
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*>
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*> As a consequence, it follows that
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*>
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*> f*g is a pattern in x if:
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*> (f*g)(0) = x and H(f)+H(g,f) < H(x)
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*>
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*> (note that this is if, not iff).
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*>
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*> We may define a distance measure between elements of F as follows:
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*>
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*> D(f,g) = (H(f,g) + H(g,f))/2
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*>
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*> If H(f,g) is interpreted as the complexity of producing f from g as an
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*> input, then this is a metric (a mathematically valid measure of distance)
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*> since
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*>
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*> H(f,g) <= H(f,w) + H(w,g)
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*>
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*> from which it follows that D also obeys the triangle inequality. We may
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*> produce a similarity measure from this by normalizing D to [0,1] via
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*>
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*> D1(f,g) = D(f,g) / [1+D(f,g)]
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*>
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*> and then setting
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*>
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*> sim(f,g) = 1 – D1(f,g)
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*>
*

*> Now, what about emergence? Emergence occurs when we have a pattern in a
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*> combination of two things, that does not occur in either of the two things
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*> individually. That is, it occurs when one has a situation so that:
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*>
*

*> · f is a pattern in x + y
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*> · There is no g similar to f so that g is a pattern in x or g is a pattern
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*> in y
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*>
*

*> Thus we see that a natural notion of emergence arises from simple notions of
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*> computation.
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*>
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*> This account can be made probabilistic if we introduce a “reference
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*> dynamical system” and define H(f,g) as the probability of f being produced
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*> within time T by the dynamical system beginning from initial condition g.
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*> For instance, if the reference dynamical system for computing H(f,g)
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*> consists of a system that creates random computer programs taking g as
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*> input, choosing each program with probability proportional to 2-(program
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*> length) , then this probabilistic-dynamic perspective recovers an
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*> approximation of what one gets from defining H(f,g) as relative Kolmogorov
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*> complexity.
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*>
*

*> -- Ben
*

*>
*

*>
*

*>
*

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