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Re: Poetential infinity

Source: http://sci.tech−archive.net/Archive/sci.logic/2006−01/msg00371.html



      • From: "Stephen Harris" <cyberguard1048−usenet@xxxxxxxxx>
      • Date: Wed, 25 Jan 2006 01:00:14 GMT


"Chris Menzel" <cmenzel@xxxxxxxxxxxxxxxxxxxx> wrote in message
news:slrndtdgb1.cpk.cmenzel@xxxxxxxxxxxxxxxxxxxx
> On 24 Jan 2006 14:19:07 −0800, Daryl McCullough
> <stevendaryl3016@xxxxxxxxx>
> said:
>> Chris Menzel says...
>>
>>>Well, I grant you that a tape with infinitely many nonblank cells is not
>>>relevant to the question of what function the machine calculates, but
>>>certainly we can talk about a TM computation on such a tape, i.e., what
>>>the TM does, step by step, when started at such and such a position on
>>>the tape.
>>
>> Oh, sure. We can certainly make sense of a TM with an "actually infinite"
>> tape (that's what computation with oracles is all about), but that's not
>> the usual setting for discussing computable functions.
>
> I think I said that! :−)
>

http://www.wolfson.ox.ac.uk/~floridi/ctt.htm
A method m qualifies as a procedure that effectively computes ¦ iff m
satisfies all the following four conditions:

1) m is finite in length and time

m is set out in terms of a finite number of discrete, exact and possibly
repeatable instructions, which, after a given time (after a given number of
steps), begin to produce the desired output. To understand the finite nature
of m in length and time recall that in a TM the set of instructions is
constituted by a finite series of quintuples (more precisely, we say that a
TM is a particular set of quintuples), while in an ordinary computer the set
of instructions is represented by a stored program, whose application is
performed through a fetch−execute cycle (obtaining and executing an
instruction). A consequence of (1) is the halting problem that we shall
analyse at the end of this section.

2) m is fully explicit and non−ambiguous


Re: Poetential infinity                                                        1
                                             Re: Poetential infinity


each instruction in m is expressed by means of a finite number of discrete
symbols belonging to a language L and is completely and uniquely
interpretable by any system capable of reading L.

3) m is faultless and infallible

m contains no error and, when carried out, always obtains the same desired
output in a finite number of steps.

4) m can be carried out by an idiot savant

m can (in practice or in principle) be carried out by a meticulous and
patient human being, without any insight, ingenuity or the help of any
instrument, by using only a potentially unlimited quantity of stationery and
time (it is better to specify "potentially unlimited" rather than "infinite"
in

order to clarify the fact that any computational procedure that necessarily
requires an actually infinite amount of space and time never ends and is not

effectively computable, see below). A consequence of (4) is that whatever a
UTM can compute is also computable in principle by a human being. I shall
return to this point in chapter five. At the moment, suffice to notice that,
to become acceptable, the converse of CTT requires some provisos, hidden by
the "in principle" clause, for the human being in question would have to be
immortal, infinitely patient and precise, and use the same kind of
stationery resources used by UTM. I suppose it is easier to imagine such a
Sisyphus in Hell than in a computer room, but in its most intuitive sense,
the one endorsed by Turing himself (see chapter five), the thesis is easily
acceptable as true by definition.




.



      • References:
             ♦ Poetential infinity
                    ◊ From: Bill Taylor
             ♦ Re: Poetential infinity
                    ◊ From: Stephen Harris
             ♦ Re: Poetential infinity
                    ◊ From: Chris Menzel
             ♦ Re: Poetential infinity
                    ◊ From: Daryl McCullough
             ♦ Re: Poetential infinity
                    ◊ From: Chris Menzel
             ♦ Re: Poetential infinity

Re: Poetential infinity                                                        2
                                           Re: Poetential infinity
                    ◊ From: Daryl McCullough
             ♦ Re: Poetential infinity
                    ◊ From: Chris Menzel

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Re: Poetential infinity                                              3