Suppose we conceive intelligence as "the ability to achieve complex goals in complex environments."
With finite computational resources there are always going to be some complex goals that one can achieve better than others....
Hutter and Schmidhuber's mathematical approach to general intelligence
basically verifies this idea, in a more formal & theoretical way...
There are many ways to do this. But, for many purposes, any definition of intelligence that has the general form "Intelligence is the maximization of a certain quantity, by a system interacting with a dynamic environment" can be handled in roughly the same way. It doesn't always matter exactly what the quantity being maximized is (whether it's "complexity of goals achieved" , for instance, or something else). My own definition of intelligence as "the ability to achieve complex goals in complex environments" -- which I've also formalized mathematically -- fits in here.
Let's use the term "behavior-based maximization criterion" to characterize the class of definitions of intelligence indicated in the previous paragraph.
So, sppose one has some particular behavior-based maximization criterion in mind. Then Marcus Hutter's work on the AIXI system, descrigives a software program that will be able to achieve intelligence according to the given criterion.
Now, there's a catch: this program may require infinite memory and an infinitely fast processor to do what it does. But he also gives a variant of AIXI which avoids this catch, by restricting attention to programs of bounded length L. Loosely speaking, the AIXItl variant will provably be as intelligent as any other computer program of length <= L, satisfying the maximization criterion, within a constant multiplicative factor and a constant additive factor.
Hutter's work draws on a long tradition of research into statistical
learning theory and algorithmic information theory, mostly notably
Solomonoff's early work on induction and Levin's work on computational
measure theory. At the present time, though, this work is more exciting
theoretically than pragmatically. The "constant factor" in his theorem may
be very large, so that in practice, AIXItl is not really going to be a good
way to create an AGI software program. In essence, what AIXItl is doing is
searching the space of all programs of length L, evaluating each one, and
finally choosing the best one and running it. The "constant factors"
involved deal with the overhead of trying every other possible program
before hitting on the best one!
If anyone knows of anything that should be added to this list, please let me know.