# [sl4] simplistic models of capability growth

From: Johnicholas Hines (johnicholas.hines@gmail.com)
Date: Wed Feb 04 2009 - 18:27:36 MST

I think it is valuable to specify models and make them concrete. Then
we can ask each other "Are you using THIS mental model? How exactly
People might, with justification, criticize me for offering models
which are simplistic. However, I hope this message spurs people to
offer countermodels in a comparably concrete and specific format.

1. Here is one model, which you might call "constant capability
increase per effort".
There is one endogenous variable "capability" (c).
There is one exogenous variable "effort" (e).
There is one parameter, "capability growth per effort" (cpe).
The single equation is that the time derivative of capability is
proportional to effort (D[c]=e * cpe).
The story for this model is something like: "Once upon a time, there
was an engineer, and she worked every day on her system. It gradually
got more and more capable."

2. Here is a second model, which you might call "constant capability
increase per capability".
There is one endogenous variable "capability" (c).
There is one parameter, "capability growth per capability" (cpc).
The single equation is that the time derivative of capability is
proportional to capability (D[c]=c * cpc).
The story for this model is something like: "Once upon a time, a
system tried to improve itself. It became more and more capable, and
was able to improve itself faster and faster."

3. Here is a third model, which you might call "diminishing capability
increase per capability".
There is one endogenous variable "capability" (c).
There is one parameter, "capability growth constant" (cgc).
The single equation is that the time derivative of capability is
inversely proportional to capability (D[c]=cgc / c).
The story for this model is something like: "Once upon a time, a
system tried to improve itself. It became more and more capable, but
each increment took more and more time."

4. You can add the first two models together.
There is one endogenous variable "capability" (c).
There is one exogenous variable "effort" (e).
There are two parameters, "capability growth per effort" (cpe) and
"capability growth per capability" (cpc).
The single equation is that the time derivative of capability is the
sum of two components, one proportional to effort, and one
proportional to capability (D[c]=e * cpe + c * cpc).
The story for this model is something like: "Once upon a time, there
was an engineer, and her system. They worked together on improving the
system's capabilities, and the system gradually got more and more
capable, until her improvements were eclipsed by its
self-improvements."

5. You can add a threshold to get a model with linear engineering,
exponential growth, and a threshold (D[c]=e*cpe + (c>ct?c*cpc:0)).
There is one endogenous variable "capability" (c).
There is one exogenous variable "effort" (e).
There are three parameters, "capability growth per effort" (cpe),
"capability growth per capability" (cpc), and "capability threshold"
(ct).
The single equation is that the time derivative of capability is the
sum of two components, one proportional to effort, and one which is
proportional to capability if capability is above the capability
theshold, and zero otherwise.
(I think this is what most AGI rhetoric uses.)

6. This last one takes a bit of motivation. Suppose the system in
question has two parts, one which is modifiable by the system and one
which is not. The total capability of the system is a combination of
the two parts. But the combination is not a simple addition or
multiplication of figures of merit. Rather, the total capability of
the system is measured in "speed", and each "elementary cycle" needs
to pass through both the modifiable part and the nonmodifiable part.
There is one endogenous variable "modifiable part's speed" (m).
There are two parameters, "modifiable part's improvement per
capability" (mpc), and "nonmodifiable part's speed" (n).
There are two equations:
The total capability of the system, the speed, is the inverse of the
sum of the inverses of the modifiable and nonmodifiable parts' speeds
(c = 1/((1/n)+(1/m))).
The time derivative of the modifiable part's speed is proportional to
the total capability (D[m]=mpc * c).

Note: ALL of these are one-dimensional models. The world of
higher-dimensional models (e.g. separate capabilities for chess and
learning to learn) is entirely open.

Johnicholas

This archive was generated by hypermail 2.1.5 : Wed Jul 17 2013 - 04:01:04 MDT