**From:** Phil Goetz (*philgoetz@yahoo.com*)

**Date:** Tue Aug 23 2005 - 22:05:26 MDT

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--- "Eliezer S. Yudkowsky" <sentience@pobox.com> wrote:

*> Phil Goetz wrote:
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*> >
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*> > CST might say things such as
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*> >
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*> > - a plot of the number of goals of the system vs. the importance of
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*> > those goals would show a power-law distribution
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*> >
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*> > - there is some critical number of average possible action
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*> transitions
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*> > above which the behavior of the system leads to an expansion rather
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*> > than a contraction in state space
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*> >
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*> > - there is a ratio of exploration of new hypotheses over
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*> exploitation
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*> > of confirmed hypotheses, and there are two values for this ratio
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*> that
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*> > locate phase shifts between "static", "dynamic", and
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*> > "unstable/devolving" modes of operation
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*>
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*> Phil, I think those are the first three interesting (falsifiable)
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*> things I've
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*> ever heard anyone say about CST and intelligence. Did you make them
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*> up on the
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*> spot, or would you seriously advocate/support any of them? Are there
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*> relevant
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*> papers/experiments?
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I just made them up.

- Plotting number of goals per importance level: There are

numerous examples in the CST literature about systems that

have events of different sizes. Classic examples include

earthquakes, sandpile avalanches, percolation lattices,

and cellular automata (e.g., length of time that an initial

configuration in Conway's game of Life takes to converge).

For certain systems - which appear to be the systems with

the most computational power in information-theoretic terms

- the number of events of size s is described by the equation

P(size = s) = k / (s^c).

These systems may have three modes of operation: mode 1

("solid"), in which P(size = s) has something like a Poisson

distribution; mode 2 ("liquid"), in which P(size=s) = k/(s^c),

and mode 3 ("gaseous"), in which all events have infinite size

(never stop, or have no gaps in continuity, like an infinite

percolation lattice that is fully-connected). In many cases,

specific numbers can be found that delineate the transition

between these nodes. For infinite 2-dimensional percolation

lattices where each point has 8 neighbors, for instance,

the first infinite-size connected group occurs when the

lattice density (probability of a site being occupied) is

approximately .59275.

I did some analysis which suggests that there is a single

distribution underlying all three phases, which is dominated

by a power-law term within the "liquid" region.

I have no good reason to think that the importance of goals

would have such a distribution. I would expect that the number

of inferences made to plan for a goal, including dead-end inferences,

could have such a distribution, depending on how many possible

inferences can be made from each new fact. The average number of

possible inferences to make from a just-derived fact plays

the same role as the average number of neighbors that an occupied

point in a percolation lattice, or the probability of turning

a randomly-chosen cell on in the next iteration of a Life game.

- there is some critical number of average possible action

transitions: That wasn't stated well. I was thinking of

behavior networks, like Pattie Maes' Do the Right Thing

network, in which each behavior enables some other behaviors,

and of probabilistic finite-state automata. But the notion

of an organism's state space isn't well-defined enough for

real organisms for the statement to make sense. For simple

simulated organisms, the state space is finite, so again it

doesn't make sense.

A better use of the ideas going into it (stuff from

Stu Kauffman's 1993 book The Origins of Order on networks

constructed from random Boolean transition tables)

might be to say:

Suppose a reactive organism observes v variables

at each timestep, and is trying to learn which n of these

v variables it should pay attention to in order to choose

its next action. Let H be the average information content,

in bits, of a proposed set of n variables (the entropy of

the distribution of possible next actions based on them).

There is some value c such that, for H << c,

the organism always takes (uninteresting) short action

sequences; for H >> c, the set of outcomes to explore

will be too large for learning to take place. The number

of variables n to consider should be chosen so as to set H = c.

One might do this by using PCA on your original v variables,

and pulling off the highest-ranked principal components

as your operational variables until their entropy sums to c.

This brings us back to the utility of signal processing.

And information theory. :)

- ratio of exploration of new hypotheses over exploitation

of confirmed hypotheses: The language comes from Holland's

genetic algorithm theory, which shows that the genetic

algorithm (without mutation) leads to an optimal

balance between exploration and exploitation (provided

the evaluation function provides scores for an organism

with a normal distribution around its average value).

The idea comes from simulations of evolution, or from

any other optimization method, in which, if you keep

mutation (or, say, the temperature in simulated annealing)

too low, you get too-slow convergence on a good solution,

but if you crank it up too high, you get poor solutions.

- Phil Goetz

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