Re: The halting problem as defined is a category error

Am Wed, 23 Jul 2025 07:22:55 -0500 schrieb olcott:

On 7/23/2025 2:34 AM, Mikko wrote:

On 2025-07-22 13:56:36 +0000, olcott said:

On 7/22/2025 5:51 AM, Mikko wrote:

On 2025-07-21 14:07:27 +0000, olcott said:

The category error is a type mismatch error where a Turing Machine
decider is required to report on the behavior of a directly executed
machine yet cannot take a directly executed machine as an input.

>
That is not a category error. A category error is a word or phrase of
some category in a context that requires a word or phrase of a
different category.

>
The category error is the mistake of assuming that a directly
executing Turing machine is in the category of input to a Turing
machine halt decider.

 
That error is not present in the halting problem. It is also not
present in https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf
which is the prototype of proofs that you falsely claim to have
refuted.
 

I am either going to go by the Linz proof or my own code

Your decision. Anyway the hypothetical halting decider obviously
only takes descriptions of TMs as input, with the OUTput required
to be what the universal simulation or direct execution *of that
input* (namely, DDD *not* calling a UTM) does.

It can be easily corrected by changing the requirement to report on
the behavior that its finite string input specifies.

 
Then the word "speifies" needs a definition. The clearest formulation
is to ask about an Universal Turing machine. Then the input to the
decider is the same as the input tu the Universal Turing machine.
 

When an input calls its own decider in recursive emulation this must be
modeled. The best way to determine the behavior that the input specifies
is for DD to be emulated by HHH according to the semantics of the x86
language. That does cause recursive emulation.

No, the best way to determine what the input "specifies" is to just run
it or use a UTM - unless you actually mean the behaviour of HHH simulating
DDD, in which case it is always tautologically correct.

*I have conclusively proven that these behaviors diverge*

 
No, you have not. A Turing machine has only one behaviour and the
halting problem requires that the input to the halting decider
describes that behaviour. If you inteprete the input differently then
either the input or the interpretaion is wrong.

--
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.