**From:** Ben Goertzel (*ben@goertzel.org*)

**Date:** Sat Oct 08 2005 - 09:25:17 MDT

**Next message:**rpwl@lightlink.com: "Re: Emergence"**Previous message:**Ben Goertzel: "Novamente/AGISIM presentations online"**Next in thread:**rpwl@lightlink.com: "Re: Emergence"**Maybe reply:**rpwl@lightlink.com: "Re: Emergence"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ] [ attachment ]

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

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

of an envelope what seems to be an elegant definition of "emergence" in

terms of basic computational concepts.

The p.s. to this email presents this definition. It's math-y so

non-math-happy readers should just ignore it!

The main point is to show that the notion of emergence can be formulated in

a precise and elegant way. I have made efforts in this direction in some of

my prior writings but I think this one is a little nicer.

The basic intuitive notion pursued is that emergence occurs when one has a

situation so that:

· f is a pattern in the combination of x and y

· There is no g similar to f so that g is a pattern in x or g is a pattern

in y

I just introduce what seems to be the minimal math formalism (or close to

it) needed to make the above two sentences precise.

The next step would be to try to do some real math modeling and see if the

intuitive examples of emergence observed in complex systems can be easily

cast into this kind of formal perspective

-- Ben G

******

P.S.

OK, here goes...

Suppose we have a set E of entities with an operation + defined on it. (For

instance, E might be the set of subsets of some other set of elementary

entities; or it might be the set of multisets formed from some other set of

elementary entities.)

The + operator may be assumed commutative and associative. Examples of

suitable + operators would be operations of set or multiset union.

Consider also a set F of function mapping E into E. Elements of E may be

mapped into elements of F via considering them as “constant functions.”

Define an operation * on F as function composition. Of course, * is neither

commutative nor associative.

The operation + may be extended from E onto F in an obvious way, so that we

may now think about {+,*} as algebraic operations on F.

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

(f+g)(x) = f(x) + g(x)

Within F, we may assume there is a zero element (the constant function

mapping everything to zero) so that

x+0=0+x=x

0*f = 0

and a 1 element (the identity function) so that

1*f = f*1 = f

Finally, let us assume there is a valuation measure H on E, meaning that for

any x in E, H(x) is defined as some nonnegative real number. Also assume a

two-component valuation measure H(x,y) so that H(x,0) = H(x).

Intuitively, H(x) is to be thought of as the complexity of producing x, and

H(x,y) is to be thought of as the complexity of producing x given y as

input. This interpretation makes it plausible to assume that

H(f*g) <= H(f) + H(g,f)

We may now define the notion of pattern as follows:

f is a pattern in x iff:

f(0) = x and H(f) < H(x)

That is, f is a pattern in x if f produces x and yet f is simpler than x.

As a consequence, it follows that

f*g is a pattern in x if:

(f*g)(0) = x and H(f)+H(g,f) < H(x)

(note that this is if, not iff).

We may define a distance measure between elements of F as follows:

D(f,g) = (H(f,g) + H(g,f))/2

If H(f,g) is interpreted as the complexity of producing f from g as an

input, then this is a metric (a mathematically valid measure of distance)

since

H(f,g) <= H(f,w) + H(w,g)

from which it follows that D also obeys the triangle inequality. We may

produce a similarity measure from this by normalizing D to [0,1] via

D1(f,g) = D(f,g) / [1+D(f,g)]

and then setting

sim(f,g) = 1 – D1(f,g)

Now, what about emergence? Emergence occurs when we have a pattern in a

combination of two things, that does not occur in either of the two things

individually. That is, it occurs when one has a situation so that:

· f is a pattern in x + y

· There is no g similar to f so that g is a pattern in x or g is a pattern

in y

Thus we see that a natural notion of emergence arises from simple notions of

computation.

This account can be made probabilistic if we introduce a “reference

dynamical system” and define H(f,g) as the probability of f being produced

within time T by the dynamical system beginning from initial condition g.

For instance, if the reference dynamical system for computing H(f,g)

consists of a system that creates random computer programs taking g as

input, choosing each program with probability proportional to 2-(program

length) , then this probabilistic-dynamic perspective recovers an

approximation of what one gets from defining H(f,g) as relative Kolmogorov

complexity.

-- Ben

**Next message:**rpwl@lightlink.com: "Re: Emergence"**Previous message:**Ben Goertzel: "Novamente/AGISIM presentations online"**Next in thread:**rpwl@lightlink.com: "Re: Emergence"**Maybe reply:**rpwl@lightlink.com: "Re: Emergence"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ] [ attachment ]

*
This archive was generated by hypermail 2.1.5
: Wed Jul 17 2013 - 04:00:52 MDT
*