Hans Moravec, October 2003

These equations supersede and expand on the more confusing February 1999 "Simple Equations for Vinge's Technological Singularity" Express an exponent a bit above 1 as (n+1)/n The equation dx/dt = x^((n+1)/n) with initial condition x(0) = 1 can be solved in Maple dsolve({ diff(x(t),t)=x(t)^((n+1)/n), x(0)=1 } ,x(t)); to produce the solution x(t) = (n/(n-t))^n This has a singularity at t=n. We discussed exponents in the range 1 to 2. 1 resulted in an exponential, I was surprised that merely setting it to 2 produced a singularity. In the solution above, exponent 2 happens when n = 1, and gives a singularity at t=1. The exponent approaches 1 as n is made large. The singularity recedes farther and farther into the future, but never actually goes away. We reach an exponent of 1 (i.e. dx/dt = x) as n approaches infinity. In that limit (n/(n-t))^n does indeed become e^t (the limit is the classic definition of e^t), and the singularity vanishes to infinitely far into the future. But for any exponent even slightly greater than 1 there is a singularity, just far away. ------------------ Then I tried a functions sub-polynomial (i.e. a smidge) faster than linear dx/dt = x*(1+log(x)) -> x(t) = exp(exp(t)-1) vastly fast double exponential, but without a singularity But merely square the log, or the log factor dx/dt = x*(1+log(x)^2) -> x(t) = exp(tan(t)) dx/dt = x*(1+log(x(t)))^2 -> x(t) = exp(t/(1-t)) and the singularity is back Generalizing dx/dt = x*(1+log(x(t)))^((n+1)/n) gives x(t) = exp((n/(n-t))^n-1) and again we have a singularity that recedes ------------------ Tickling the tail of the singularity, I see dsolve({diff(x(t),t)= x(t)*(1+log(x(t)))*(1+log(1+log(x(t)))),x(0)=1},x(t)); gets x(t) = exp(exp(exp(t)-1)-1) raising the log log term to (n+1)/n power brings in a singularity at t=n as before. So instead layer on another log in the next term Let lp(x) = 1+log(x) dsolve({diff(x(t),t)= x(t)*lp(x(t))*lp(lp(x(t)))*lp(lp(lp(x(t)))),x(0)=1},x(t)); -> x(t) = exp(exp(exp(exp(t)-1)-1)-1) Seems to be a pattern here. Makes me wonder what the limit of x*lp(x)*lp^2(x)*lp^3(x)*lp^4(x)... might be a function still barely faster than linear, yet only infinitesimally away from triggering a singularity