volatile/CT hypothesis.

[Steven Ryan]

In article <15422 at brl-adm.ARPA> dsill at nswc-oas.arpa (Dave Sill) writes:
>>Also, please do not assume the Church-Turing Hypothesis.
>
>Huh?  I think it's quite safe to assume the Church-Turing *Thesis*.
>And what does it have to do with optimization, anyway?
>
The reference was back to somebody saying AI will come up with the ultimate
optimiser. AI implicitly assumes CT.

For those not in the know, the Church-Turing(-Curry-...) Hypothesis in its
restricted form states that all formal systems (programming, type 0 grammars,
Turing machines, lambda calculus, operators, ...) are all equally powerful.
This is a hypothesis because this characterisations applies to all systems
existing and yet to be created. The most unrestricted form of the hypothesis
states any "computable function" can be expressed in any of these notations.

It is important to note that all the formal systems share common properties:
discrete and finite number of transistions, fixed and finite number of states,
and a discrete and finite store. Formal systems have a very aleph-null
character.

It is not safe to assume the unrestricted hypothesis because space-time may
be continuous and aleph-one. If so, there may exist some process characterised
by an arbritrarily long and complex path, that is continuous transistions,
infinite number of states, and an infinite store, which is all very aleph-one.
If such a process exists, it cannot be described by any aleph-null formal
system.

I do not claim such a process exists nor, unlike AI proponents, do I deny its
existence. As long as these questions remain unsettled, I feel it is unsafe
to declare the hypothesis resolved. If someone wishes to use it as working
hypothesis, fine, but just be honest.

By the way, if you want to see how complicated aleph-one sets can get, read
about fractals. I think, however, that fractals only cover a tiny chunk of
all sets. The set of sets which can only be characterised by a negation
remains larger than all other sets.

                                       Hafa an godne daege.
                                                    sm ryan

"Wow, and like every atom in my thumbnail might a complete universe...."
   - Pinto