mathematics//dynamical systems//finite-time singularity
A finite-time singularity is a solution of a differential equation that becomes infinite at a finite time, and it is the mathematical form behind claims of an intelligence explosion. The simplest case is growth whose rate depends on the current level:
A finite-time singularity is a solution of a differential equation that becomes infinite at a finite time, and it is the mathematical form behind claims of an intelligence explosion. The simplest case is growth whose rate depends on the current level:
dxdt=c xα\frac{dx}{dt} = c\,x^{\alpha}dtdx=cxα
With α=1\alpha = 1α=1 this is ordinary exponential growth. With α<1\alpha < 1α<1 growth is polynomial, slower than any exponential. With α>1\alpha > 1α>1 the solution x(t)=[x01−α−(α−1)c t]−1/(α−1)x(t) = \left[x_0^{1-\alpha} - (\alpha-1)c,t\right]^{-1/(\alpha-1)}x(t)=[x01−α−(α−1)ct]−1/(α−1) reaches infinity at t∗=x01−α/((α−1)c)t^* = x_0^{1-\alpha}/\big((\alpha-1)c\big)t∗=x01−α/((α−1)c): each improvement helps more than the last, and the curve is a hyperbola.
It asks more than superexponential growth, which only needs a rising growth rate and can stay finite at every date; a singularity needs the rate to diverge so fast that infinity arrives at a fixed time.
Nothing physical becomes infinite, so the model means that the curve follows a hyperbola for a while until another effect takes over.
In a process that advances in rounds, a singularity needs the duration of the rounds to shrink fast enough that their sum converges (generation time); if the rounds have a floor, the result is at most exponential.
Whether a curve is singular can depend on the scale chosen to measure it (coordinate singularity).