Re: So you want APPROACHABLE oo, REAL oo, INTEGER oo, COUNTABLE oo, HIGHER oo, MEGA oo's

From: The Ghost In The Machine (ewill_at_sirius.athghost7038suus.net)
Date: 01/30/05 

Date: Sun, 30 Jan 2005 10:01:44 GMT

In sci.logic, |-|erc
<H@r.c>
 wrote
on Sun, 30 Jan 2005 17:50:16 +1000
<363i0gF4qgp42U1@individual.net>:
> "Barb Knox" <see@sig.below> wrote in
>> >> >MISTAKE : intentional assertion of an erronous concept, not spelling, typos
>> >> >etc.
>> >>
>> >> Goodness knows what you *indend*. I suppose you could be
>> >> meaning that the many mistakes in what you post are not
>> >> "really" mistakes because you didn't intend them to be mistakes...
>> >
>> >In natural language context, say a courtroom, you must show INTENT.
>> >Stop backpeddling and find a mistake of mine.
>> >
>> >Any proposition that is false, that I "asserted as if true".
>> >Intent is just to clarify that it wasn't quoted, I meant
>> >what was wrote AS YOU SEE IT.
>>
>> Ah, so the part you snipped was meant as a hypothetical QUOTE? --
>
>
> What? I'm not DEFINING mistake here, I'm saying it must be
>
> 1/ something I wrote
> 2/ that something was an *assertion* of that fact
>
> I've spent too long in prison because police stand up to the judge and
> say COOPER SAID THIS "and then they will all die".
>
> I get asked DID YOU SAY THAT?
> And then I go to prison because that's what happens to unemployed writers.
>
> A mistake that was MY MISTAKE when taken IN CONTEXT.
>
> e.g. Ghost said the sum on the board is 50.
>
> That is not MY mistake because even though I wrote
> "the sum on the board is 50" yes judge
>
> I didn't intend that as factual. No idiot in his right mind
> would fall for the sum of all numbers on the blackboard trick
> except Ghost

You're flattering me again. :-P

My mind tends toward the literal. Perhaps that's why I'm good
at math; when one says "a equals b to n digits", I assume
they actually *mean* "a equals b to n digits", as opposed
to simply "a equals b", "abs(a - b) < epsilon", or
"horse happy apple".

>
> WHO I WAS QUOTING. got it yet?
>
>
>>
>> >> >[Herc] : a UTM applied to any number always finished
>> >> > after time t, t e N
>> >> >
>> >> >[Mr Fixer] : that is wrong! All systems of computation
>> >> > equiv. to TMs have representations of
>> >> > "algorithms" that never halt.
>> >> >
>> >> >FIND ONE!
>>
>> And thus the referent of "FIND ONE" was meant to be some actual mistake, not
>> a non-halting algorithm?
>>
>> If so, I misinterpreted you (which your sloppy English does make rather easy
>> to do). But my misinterpretation was reasonable, since you have REPEATEDLY
>> made the mistake of constructing arrays like this:
>>
>> \ Input to the TM:
>> TM \ 0 1 2 3 ...
>> index +---------------
>> 0 | 1 1 0 1 ...
>> 1 | 0 0 1 1 ...
>> 2 | 0 1 0 0 ...
>> ... ... ... ... ...
>>
>> E.g, this is how you attempt to enumerate all computable binary sequences.
>> But this is a MISTAKE, since you make no allowance for the fact that
>> UTM(index,input) might not halt!
>>
>> This has been pointed out to you SEVERAL times, and you have never addressed
>> it satisfactorily: you say something like "... and let the value be 0 if
>> UTM(i,j) doesn't halt". But, D******, whether it halts or not is NOT
>> COMPUTABLE in the general case. So for all I know, the your hypothetical
>> dialogue with "Mr. Fixit", wherein you show ingorance of the meaning of the
>> unsolvability of the halting problem, might have actually occurred!
>>
>
>
> OK, now you've got the idea. You find an example, very good!
>
> Anyone care to prove this is an error?
>
> Herc
>

Is what an error?

The above array is merely a representation of a function
R : N x N -> sigma*, where sigma is an alphabet (the
star, of course, is from Kleene). If the machine halts,
the function R(machine, input) is well defined. If the
machine does not halt, the function R(machine, input) may
or may not be defined, depending on whether the machine
continually generates digits while moving its head to the
right (in which case one gets something slightly outside of
sigma*, as sigma* properly speaking contains all *finite*
strings only), or does silly stuff like:

state 1: on '0', write '1', move right, state 2
state 2: on '0', write '1', move left, state 1
state 1: on '1', write '0', move right, state 2
state 2: on '1', write '0', move left, state 1

which of course just jiggles the tape uselessly.

If one wants to restrict the domain of the function R to
inputs that generate well-defined outputs, that's fine;
one will get a sparse matrix (or, if one maps the inputs
to column indexes, one gets infinite strings with blanks
and/or filler characters).

And there will be blanks. R(machine, input) will generate
blanks for those inputs with a 0 or 1 in the first tape cell.

It is possible to modify the Turing machine by adding an
"output tape" that can only write and move right, if
it decides to write at all. A set of such modified machines is
isomorphic to the set of the original Turing machines (by redefining
the latter's alphabet) so this doesn't really buy one much apart
from convenience. In such a machine, R(machine, input) is
always well defined.

At this point I'm not sure what subproblem you wish to address. 

-- 
#191, ewill3@earthlink.net
It's still legal to go .sigless.

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