Re: Correcting the definition of the halting problem --- Computable functions

On 3/24/2025 6:05 PM, André G. Isaak wrote:

On 2025-03-24 17:04, olcott wrote:

On 3/24/2025 5:49 PM, André G. Isaak wrote:

On 2025-03-24 16:43, olcott wrote:
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Computable functions don't have inputs. They have domains. Turing machines have inputs.p
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Maybe when pure math objects. In every model of
computation they seem to always have inputs.
https://en.wikipedia.org/wiki/Computable_function

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Computable functions *are* pure math objects. You seem to want to conflate them with C functions, but that is not the case.
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The crucial point is that the domains of computable functions are *not* restricted to strings, even if the inputs to Turing Machines are.
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While the inputs to TMs are restricted to strings, there is no such such restriction on computable functions.

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The vast majority of computable functions of interest do *not* have strings as their domains, yet they remain computable functions (a simple example would be the parity function which maps NATURAL NUMBERS (not strings) to yes/no values.)
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Since there is a bijection between natural numbers
and strings of decimal digits your qualification
seems vacuous.

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There is not a bijection between natural numbers and strings. There is a one-to-many mapping from natural numbers to strings, just as there is a one-to-many mapping from computations (i.e. turing machine/input string pairs, i.e. actual Turing machines directly running on their inputs) to strings.
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André
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_III()
[00002172] 55         push ebp      ; housekeeping
[00002173] 8bec       mov  ebp,esp  ; housekeeping
[00002175] 6872210000 push 00002172 ; push III
[0000217a] e853f4ffff call 000015d2 ; call EEE(III)
[0000217f] 83c404     add  esp,+04
[00002182] 5d         pop  ebp
[00002183] c3         ret
Size in bytes:(0018) [00002183]
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When III is emulated by pure emulator EEE for any finite
number of steps of emulation according to the semantics
of the x86 language it never reaches its own "ret"
instruction final halt state THUS DOES NOT HALT.
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When III is directly executed calls an EEE instance
that only emulates finite number of steps then this
directly executed III always reaches its own "ret"
instruction final halt state THUS HALTS.

 And that has what, exactly, to do with the post you are allegedly responding to?
 André
 

THE INPUT FINITE STRING DOES SPECIFY RECURSIVE EMULATION.
The behavior specified by the finite string input to a
computable function implemented on a model of computation
does differ from the direct execution of this same finite
string because the direct execution avoids the pathological
self-reference that causes the recursive emulation.
THE INPUT FINITE STRING DOES SPECIFY RECURSIVE EMULATION.
--
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer