On Mon, Jul 7, 2008 at 5:44 PM, Linas Vepstas <[EMAIL PROTECTED]> wrote:
> 2008/7/2 Hector Zenil <[EMAIL PROTECTED]>:
>
>> Hypercomputational models basically pretend to take advantage from
>> either infinite time or infinite space (including models such as
>> infinite resources, Zeno machines or the Omega-rule, real computation,
>> etc.), from the continuum. Depending of the density of that space/time
>> continuum one can think of several models taking advantage at several
>> levels of the arithmetical hierarchy.
>
> Lest various readers of this thread be confused, lets be careful
> to make a distinction between physical infinity and the multitude
> of mathematical infinities.  Whether or not one beleives that
> some model of of physical quantum computer depends on
> what one might believe about physics at the plank length --
> which would be speculative.
>
> Next, using words like "infinite space" is misleading: the
> surface of a sphere has an infinite number of points; yet,
> upon hearing the words "infinite space", its unusual to think
> of a sphere as an example.

If a sphere has infinite number of points then it is an infinite
bounded space and you can theoretically build a hypercomputer on it.

Any level in the arithmetical hierarchy can correspond to a level in
the physical world. It depends on the cardinality of the physical
space, from the continuum to any power of it, and whether its dense
and how dense.

> Note that a single qubit is a
> 2-sphere, and so, in the same way that there are an infinite
> number of points in the surface of a sphere, so to are there
> an infinite number of states in a qubit.
>
> Does any given quantum computer make use of the infinite
> number of states in one qubit? That depends ... but the point
> is that one can ponder hypercomputation in a finite volume;
> this is crucial; don't let words like "infinite space" accidentally
> imply "infinite volume" - it does not.
>
>> But even if there is infinite
>> space or time another issue is how to verify a hypercomputation. One
>> would need another hypercomputer to verify the first and then trust in
>> one.
>
> Sure ... but we can express many theorems about Turing
> machines, and know them to be true, without "verifying"
> them on some other computer which we need to magically
> trust. Similarly, one can make plenty of true statements
> about hypercomputation without actually having to build
> one and verifying its operation.  But perhaps I missed
> some subtle point.

The big difference is that in principle you can verify the output of
any digital computer by hand in finite time, however you cannot verify
any non-Turing output from a hypercomputer by hand, not even in
principle unless having infinite time.

>
> --linas
>
>
> -------------------------------------------
> agi
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-- 
Hector Zenil    http://zenil.mathrix.org


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agi
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