Re: Turing Machine computable functions apply finite string transformations to inputs

On 2025-05-06 15:03:52 +0000, olcott said:

On 5/6/2025 3:24 AM, Mikko wrote:

On 2025-05-06 02:32:01 +0000, olcott said:
 

On 5/5/2025 8:19 PM, Richard Damon wrote:

On 5/4/25 8:35 PM, olcott wrote:

On 5/4/2025 5:34 PM, Mr Flibble wrote:

On Sun, 04 May 2025 23:30:54 +0100, Richard Heathfield wrote:
 

On 04/05/2025 23:15, olcott wrote:

On 5/4/2025 2:21 PM, Richard Heathfield wrote:

On 04/05/2025 18:55, olcott wrote:

Changing my words then rebutting these changed words is dishonest.
 Functions computed by Turing Machines require INPUTS and produce
OUTPUTS DERIVED FROM THESE INPUTS.

 Counter-example: a Turing Machine can calculate pi without any input
whatsoever.
 As Mikko rightly said: a Turing machine does not need to require an
input.
 

IT IS NOT COMPUTING FUNCTION THEN

 Quoth Alan Turing:
 (viii) The limit of a computably convergent sequence is computable.
   From (viii) and TT— 4(1—i-|--i—...) we deduce that TT is computable.
 No input required.
 

IT IS NOT COMPUTING FUNCTION THEN IT IS NOT COMPUTING FUNCTION THEN IT
IS NOT COMPUTING FUNCTION THEN
 Computable functions are the basic objects of study in computability
theory. Computable functions are the formalized analogue of the
intuitive notion of algorithms, in the sense that a function is
computable if there exists an algorithm that can do the job of the
function, i.e. given an input of the function domain it can return the
corresponding output. https://en.wikipedia.org/wiki/ Computable_function

 That's a very second-rate summary of computability. Turing was far more
interested in whether a computation was possible than whether it needed
inputs. Do most computations need inputs? Most useful ones that we care
about, sure. But all? By no means.
 

*Computer science is ONLY concerned with computable functions*

 Computer science is concerned with the Halting Problem.
The Halting Problem is concerned with an incomputable function.
Therefore computer science is concerned with at least one incomputable
function.

 The function is neither computable nor incomputable because there is no
function at all, just a category error.
 /Flibble

 You can look at it that way or you can look
at it as simulating termination analyzer HHH(DD)
does correctly determine that DD cannot possibly
reach its own final state, thus is correctly
rejected as non-halting.
 

 Except that isn't the question that is being asked.
 In fact, that question has a trivial answer, as we can make an H0 that just aborts its emulation and returns 0 and it is correct by your definition,

 No that is stupidly wrong as I have said at least 100 times recently.
The termination analyzer must compute the mapping from the input
on the basis of the behavior that this input actually specifies.

 You often say so and you often say otherwise. More specifically, you
have said that is correct to call the actual input non-halting if a
hypothetical non-decider would correctly decide that a hypthetical
input would not halt.

 <MIT Professor Sipser agreed to ONLY these verbatim words 10/13/2022>
     If simulating halt decider H correctly simulates its
     input D until H correctly determines that its simulated D
     *would never stop running unless aborted* then
      H can abort its simulation of D and correctly report that D
     specifies a non-halting sequence of configurations.
</MIT Professor Sipser agreed to ONLY these verbatim words 10/13/2022>
 *would never stop running unless aborted*
looks at the actual input D and reports on the basis
of a hypothetical H/D pair where H does not abort.
 int DD()
{
   int Halt_Status = HHH(DD);
   if (Halt_Status)
     HERE: goto HERE;
   return Halt_Status;
}
 _DD()
[00002133] 55         push ebp      ; housekeeping
[00002134] 8bec       mov ebp,esp   ; housekeeping
[00002136] 51         push ecx      ; make space for local
[00002137] 6833210000 push 00002133 ; push DD
[0000213c] e882f4ffff call 000015c3 ; call HHH(DD)
[00002141] 83c404     add esp,+04
[00002144] 8945fc     mov [ebp-04],eax
[00002147] 837dfc00   cmp dword [ebp-04],+00
[0000214b] 7402       jz 0000214f
[0000214d] ebfe       jmp 0000214d
[0000214f] 8b45fc     mov eax,[ebp-04]
[00002152] 8be5       mov esp,ebp
[00002154] 5d         pop ebp
[00002155] c3         ret
Size in bytes:(0035) [00002155]
 DD emulated by HHH according to the rules of the
x86 language cannot possibly reach past its own
machine address [0000213c].

That is an example where you try to justify the claim that a termination
analyzer may correctly compute the mapping on the basis of a behaviour
that differs from what the input actually specifies.
Thank you for the confirmation.
--
Mikko