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.

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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 sense.

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) Church Thesis. Bruno