Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V52 (HP refutation Sufficiency Specs)

From	Ben Bacarisse <ben.usenet@bsb.me.uk>
Newsgroups	comp.lang.c
Subject	Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V52 (HP refutation Sufficiency Specs)
Date	2020-09-07 02:05 +0100
Organization	A noiseless patient Spider
Message-ID	<871rjebdml.fsf@bsb.me.uk> (permalink)
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Malcolm McLean <malcolm.arthur.mclean@gmail.com> writes:

> On Sunday, 6 September 2020 at 21:14:19 UTC+1, Ben Bacarisse wrote:
>> 
>> > But it seems that most of the code 
>> > isn't actually run, it's emulated under supervision. That might allow it to 
>> > know its own recursion depth, which would be away round the proof.
>> What do you mean by "a way round the proof"?
>>
> A cheat, basically. We pass an extra parameter to "H" to show it's being
> called recursively. You then have to hide what you are doing, somehow,
> to not make it too obvious.

OK.  I'd call that not a way round the proof.  It's just another
program not deciding halting.

But then I want to ask what a recursive call has got to do with it?
What form of the proof are you considering?  Are you ignoring the
encoding?  There are some pretty sketchy arguments out there.

>> > There's a better argument against the halt decider than the paradox 
>> > proof. Consider a Turing machine which answers one comuptational 
>> > question with a true/false answer, like "is the 100th digit of PI greater 
>> > than 5?". Then if the answer is true, it halts, if false, it goes into an 
>> > infinite loop. 
>> > Now our halt decider either has some faster way of computing the 100th 
>> > digit of PI, or it has not. If it has, we replace our Turing machine with the 
>> > faster method, and re-run the argument. So now, our halt decider has 
>> > no faster way of computing the 100th digit of PI. So it must run the 
>> > machine under test until the flag is calculated. Then it can decide whether 
>> > the machine halts or not.
>> I think you are using the term "decider" in much the same way that PO 
>> does. The usual definition relates to infinite sets, not specific 
>> programs. A halt decider (in the standard sense) is program that 
>> computes the characteristic function (the "is or is not a member" 
>> function) for a some set. I'm not sure how your argument extends to the 
>> standard meaning of the term. Such a halt decider obviously can't have 
>> special logic for specific problems, so there would be nothing to 
>> replace. 
>>
> The halt decider is a general purpose hat decider, ot "H", if you like, but
> we don't know how it works. Implicit in my description is the assumption
> that it has some logic to distingush an empty infinite loop from a
> "halt" instruction. The program under test calcuates a bit on some basis
> or other, then either goes into the infinite loop or halts. Once the halt decider
> knows the status of the bit, it can make its decision. The question is, "how
> can it know the status of the bit?". Let's assume for now that it can prove
> that the calculation of the bit itself doesn't halt. It's still got to know the status
> of the bit. Now there might be many operations in the machine under test 
> that don't contribute to the bit value, or can be optimised in some way. But
> if that's the case, we abandon that example, and use the optimised version 
> as our example. "H" could special case a lot of computations, we don't know
> exactly how it works, but we can produce a version of the test machine that
> it can't optimise any more.

This does not seem to be a proof.  It appears to be an explanation of
why the obvious strategies won't work.  Can the argument be put in a
more formal way?

I think the direct diagonal argument is by far the simplest.  It is not
a proof by contradiction.

>> From a rather dim memory, this is reminiscent of Chaitin's proof, but 
>> that uses algorithmic complexity than about speed, and it is a proof of 
>> the nonexistence of the usual universal halt decider.
>>
> It's not a proof of the non-existence of "H".

Oh sorry.  I thought that was what you were talking about.  When you
said "There's a better argument against the halt decider than the paradox
proof" it seemed for all the world that you thought so too.

> It's an argument about the usefulness
> of "H". H has to run the machine under test, for the general case.

I'd have something to say about this if you were presenting an
alternative argument for the halting theorem, but apparently you are
not.

As you know, what everyone in these threads seems to call H does not
exist.  Can I urge everyone to pick another letter when talking about
machine/programs that really do exist?  Talk about a halting analyser
called A or a partial decider P, but not H.  I know the name does not
really matter in formal text, but it confuses so many people who come
across posts out of context.

(PO makes matters worse.  He keeps talking about "Linz's H" when he not talking
about a universal halting decider.)

<cut>
>> > Obviously, because the  analyser is another Turing machine. When put like this, 
>> > the tautology becomes obvious.
>>
>> Not to me, unfortunately. What tautology becomes obvious? 
>> 
> A Turing machine can't be more efficient than the most efficient Turing machine.
> That' all we're really saying. We're not saying that a halt decider is impossible,
> just that it can't work other than by the brute force algorithm of running the
> machine under test and checking for a loop. (You could now extend the argument
> to the loop and the loop checker itself, but I'm not actually going
> there).

I can't get on with this kind of argument.  You are describing what
something that does not exist must be doing.  PO did this some years
ago.  It seems to me to be a nonsense argument.

-- 
Ben.

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                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V52 (HP refutation Sufficiency Specs) olcott <NoOne@NoWhere.com> - 2020-09-06 15:51 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V52 (HP refutation Sufficiency Specs) olcott <NoOne@NoWhere.com> - 2020-09-06 16:10 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V52 (HP refutation Sufficiency Specs) Malcolm McLean <malcolm.arthur.mclean@gmail.com> - 2020-09-06 15:11 -0700
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V52 (HP refutation Sufficiency Specs) Richard Damon <Richard@Damon-Family.org> - 2020-09-06 21:33 -0400
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V52 (HP refutation Sufficiency Specs) olcott <NoOne@NoWhere.com> - 2020-09-06 20:40 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V52 (HP refutation Sufficiency Specs) Richard Damon <Richard@Damon-Family.org> - 2020-09-06 22:30 -0400
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V52 (HP refutation Sufficiency Specs) olcott <NoOne@NoWhere.com> - 2020-09-06 21:45 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V52 (HP refutation Sufficiency Specs) André G. Isaak <agisaak@gm.invalid> - 2020-09-05 15:29 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V52 (HP refutation Sufficiency Specs) André G. Isaak <agisaak@gm.invalid> - 2020-09-05 15:52 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V52 (HP refutation Sufficiency Specs) André G. Isaak <agisaak@gm.invalid> - 2020-09-05 15:28 -0600
          Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V52 (HP refutation Sufficiency Specs) olcott <NoOne@NoWhere.com> - 2020-09-05 00:40 -0500
      Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V52 (HP refutation) Malcolm McLean <malcolm.arthur.mclean@gmail.com> - 2020-09-03 12:32 -0700
  Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V52 (Machine equivalence) adultfriendfindercom08@gmail.com - 2020-09-04 04:51 -0700

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