Simple Equations for Vinge's Technological Singularity
Hans Moravec, February 1999
```Making maximal simplifying assumptions, and shifting and scaling all
quantities to avoid constants:

Let W be "world knowledge", and assume that each additive increment in
W results in a multiplicative improvement in miniaturization, and thus
in computer memory and speed V.  So:

V = exp(W)
------------------

In the old days, assume an essentially constant number of humans
worked unassisted at a steady pace to increase W at a steady rate:

dW/dt = 1

So          W = t    and      V = exp(t)

which is a regular Moore's law.

-------------------

Now, suppose instead W is created soley by computers, and increases at
a rate proportional to computer speed.  Then:

dW/dt = V    giving     dw/exp(W) = dt

This solves to     W = log(-1/t)   and    V = -1/t

W and V rise very slowly when t<<0, might be mistaken for exponential
around t = -1, and have a glorious singularity at t = 0.

-------------------

Most realistically, assume humans keep working at a steady pace, but
are gradually overtaken by contributions from growing computer power:

dW/dt = 1+V    giving     dw/(1+exp(W)) = dt

Which solves to   W = log(1/(exp(-t)-1))  and   V = 1/(exp(-t)-1)

Unsurprisingly, with this equation, W increases linearly when t<<0,
curves up like an exponential around t = -1, rises to a singularity at
t=0.

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The assumption that   V = exp(W)   is surely too optimistic.

I was thinking in terms of independent innovations.  For instance,
one might be an algorithmic discovery (like log N sorting) that lets
you get the same result with half the computation.  Another might be
a computer organization (like RISC) that lets you get twice the
computation with the same number of gates.  Another might be a circuit
advance (like CMOS) that lets you get twice the gates in a given
space. Others might be independent speed-increasing advances, like
size-reducing copper interconnects and capacitance-reducing
silicon-on-insulator channels.  Each of those increments of knowledge
more or less multiplies the effect of all of the others, and
computation would grow exponentially in their number.

But, of course, a lot of new knowledge steps on the toes of other
knowledge, by making it obsolete, or diluting its effect, so the
simple independent model doesn't work in general. Also, simply
searching through an increasing amount of knowledge may take
increasing amounts of computation.  I played with the V=exp(W)
assumption to weaken it, and observed that the singularity remains
if you assume processing increases more slowly, for instance
V = exp(sqrt(W)) or exp(W^1/4).  Only when V = exp(log(W))
(ie. V = W) does the progress curve subside to an exponential.

Actually, the singularity appears somewhere in the
I-would-have-expected tame region between and

V = W and V = W^2        (!)

--------------------

Suppose computing power per computer simply grows linearly with
total world knowledge, but that the number of computers also
grows the same way, so that the total amount of computational
power in the world grows as the square of knowledge:

V = W*W

also     dW/dt = V+1      as before

This solves to   W = tan(t)   and   V = tan(t)^2,

which has lots of singularities (I like the one at t = pi/2).

--------------------

Unfortunately the transitional territory between the merely
exponential V=W and the singularity-causing V=W^2 is analytically hard
to deal with.  I assume just before a singularity appears, you get
non-computably rapid growth!

```