Stephen Paul King wrote (in the FOR list):
The notion of "intelligence" that you mention below seems very close to
the notion of "expressiveness" that Peter Wegner develops in several of his
papers. How do we balance the notion of the universality of computation
against this notion? It seems to me that the notion of universality implies,
at least for Church-Turing machines, that they are all equally expressive
since, if we neglect the number of steps the machines take, any one
universal computer can perform exactly the same computation as any other.
It has been shown by Putnam that there is no "perfect" universal learning
machine, that is, machine capable to identify in a finite time any total
(everywhere defined on N) function.
If you allow a learning machine to change its mind infinitely often (that
is to change his explanation (program) when he get more big sample of the
function it tries to identify) AND if you allow a finite but unbounded
number of mistakes in the explanation, then, at least in principle, there
is a universal learning machine.
As a side note, I have read a paper discussing the computational theory
of Malament-Hogarth machines in which it was pointed out that there does not
exist a universality property for them. Would the notion, of intelligence,
that you seem to imply below be more applicable to such rather than machines
defined by the Church-Turing thesis?
I don't think so. Malament-Hogart Machines are abstract *computer* having
some infinite capacities (if I remember correctly). Learning machine are
just any computer programmed to generate explanation (in the form of
computer program) when they are given data (sequence of input/output).
Of course such machine are "stream-interactive" in a Peter Wegner related
Have you considered more abstract notions of computation that are not
limited to those expressible by "physical systems"? For example, could there
exist a notion of computation that would involve functions C -> C, where C
is the "space" of complex numbers, analogous to the notion of Church-Turing
computations as functions N -> N?
Blum Shub and Smale have generalize the notion of computer by
computer on a ring (like R or C). They have prove in this setting
that the Mandelbrot set is undecidable (answering a conjecture by myself
and Penrose). From this you can look at the Mandelbrot set as a sort of
compactified projection of a universal dovetailer.
Blum & Al. approach to computability gives interesting bridge between
numerical analysis and classical computability. I remember also having read
a Los Alamos "quant-phys" suggesting to found quantum computing on
a similar ring-generalization. All those approach subsumes the (classical)