From: Eliezer S. Yudkowsky (sentience@pobox.com)
Date: Wed Jan 30 2002 - 22:14:12 MST
Phil Osborn wrote:
>
> There had to be something more. Calculus gave me an insight. Suppose we looked at the convergence in terms of derivatives. The first derivative of convergence - the velocity - might be constantly or mostly positive over time, but if it averaged to a finite limit, then we would hit that boredom phase.
>
> One candy bar is great. The second is good. The fifteenth or one-thousanth incremental candy bar is barely noticed.
>
> But what if ALL the derivatives are positive? I.e., not just velocity, but acceleration, acceleration of acceleration1, ... to acceleration of acceleration(n), where n is infinite. The degree of positiveness doesn't matter. An infinitesimal will do.
>
> The question then becomes, if we assume that the above is fundamentally correct, under what circumstances would we be able to make all the derivatives positive? If this is impossible, then why try living forever?
An exponential function, e^x, trivially satisfies the above requirement.
I don't see why e^x is inherently more fun than x^2, since I would deny
the idea that we become "bored" with derivatives, derivatives of
derivatives, and so on. Singularity Fun Theory says that we become bored
with repetition. There is no inherent reason why e^x would be seen as
less predictable than x^2; they are both functions of roughly the same
complexity.
In Singularity Fun Theory, it's the *content* of the space, not the size
of the space, that determines the amount of "fun" - in other words, as
long as you haven't exhausted the Fun Space for that size limit, you can
go on having fun within limit N. Eventually you must exhaust a Fun Space
of any finite size, and the more intelligent you are, the faster a given
volume of Fun Space will be exhausted (because of your ability to
generalize better). Singularity Fun Theory says, don't worry, the volume
of Fun Space you can understand also gets bigger with intelligence.
Anyway my point is that if you generalize *on the level of the pleasure
counter*, then you'll get bored with a pleasure function of e^x just as
fast as x^2; both are functions that a cognitive system would learn to
predict. Singularity Fun Theory's answer is to exempt this level from
antisphexishness routines, along with the ultimate lowest level of the
system where it's all ones and zeroes, for the same reason in both cases;
there doesn't seem to be even the theoretical possibility of avoiding
"repetition".
If you did want to avoid repetition on the level of the pleasure counter
itself, the best approximation would be to use the busy beaver function,
not an exponential function. The busy beaver function for N being the
largest number of tape marks produceable by a halting Turing machine with
N or fewer states, and hence, a number that "forces" an ascent to the next
level of complexity in order to understand it. And that is a level of
ascending resource requirements that even ontotechnology would be hard-put
to supply.
Busy beaver for small values:
BB(1) = 1
BB(2) = 4
BB(3) = 6
BB(4) = 13
BB(5) >= 4098
BB(6) >= 1.29*10^865
-- -- -- -- --
Eliezer S. Yudkowsky http://singinst.org/
Research Fellow, Singularity Institute for Artificial Intelligence
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