digression: singularities
exponential growth
$$
\dot x = \gamma\, x, \quad\qquad
x(t) = x_0\,\mathrm{e}^{\gamma t}
$$
- $\gamma$<0$\,$: radioactive decay
super-exponential growth
$$
\dot x = \gamma\, x^2, \quad\qquad
\frac{\dot x}{x^2} = \gamma\, \quad\qquad
\frac{-1}{x} = \frac{-1}{x_0} + \gamma\,t\, \quad\qquad
x(t) = \frac{x_0}{1-x_0\gamma t}
$$
- divergence for $t\to 1/(x_0\gamma)$
→ singularities are possible
- observation: AI grows is 'only exponential'
:: no/weak nonlinear scale effects
AI doubing times
- 6-9 months
1.8 years Moore's law (chips)
:: acceleration matters