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From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory
Subject: Re: Correcting the definition of the halting problem --- Computable
 functions
Date: Mon, 24 Mar 2025 21:07:35 -0500
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On 3/24/2025 8:46 PM, André G. Isaak wrote:
> On 2025-03-24 19:33, olcott wrote:
>> On 3/24/2025 7:00 PM, André G. Isaak wrote:
> 
>>> In the post you were responding to I pointed out that computable 
>>> functions are mathematical objects.
>>
>> https://en.wikipedia.org/wiki/Computable_function
>>
>> Computable functions implemented using models of computation
>> would seem to be more concrete than pure math functions.
> 
> Those are called computations or algorithms, not computable functions.
> 

https://en.wikipedia.org/wiki/Pure_function
Is another way to look at computable functions implemented
by some concrete model of computation.

> The halting problems asks whether there *is* an algorithm which can 
> compute the halting function, but the halting function itself is a 
> purely mathematical object which exists prior to, and independent of, 
> any such algorithm (if one existed).
> 

None-the-less it only has specific elements of its domain
as its entire basis. For Turing machines this always means
a finite string that (for example) encodes a specific
sequence of moves.

>> For example pure math functions don't have any specific
>> storage like a tape or machine registers.
> 
> No they don't. Why would they? A mathematical function is simply a 
> static mapping from elements of a domain to elements of a codomain.
> 
>> This also would seem to mean that they would require
>> some actual input.
>>
>>
>>> The above copypasta doesn't address this.
>>>
>>> I pointed out that the domain of a computable function needn't be a 
>>> string. The above copypasta doesn't address this.
>>>
>>
>> When implemented using an actual model of computation
>> some concrete form or input seems required.
>>
>>> I pointed out that there is no bijection natural numbers and strings, 
>>
>> finite strings of decimal digits: [0123456789]
>>
>>> but rather a one-to-many relation. The above copypasta doesn't 
>>> address this.
>>
>> "12579" would seem to have a bijective mapping to
>> a single natural number.
> 
> The natural number 12579 maps equally to the (decimal) string '012579', 
> '0012579',... so there is no bijection.
> 

The bijection is then to decimal digits without leading zeroes to 
Natural numbers.

>>>
>>> I pointed out that the exact same sort of one-to-many relation exists 
>>> between computations and strings. The above copypasta doesn't address 
>>> this.
>>>
>>
>> I pointed out above that the finite string of x86
>> machine code correctly emulated by EEE DOES
>> NOT MAP TO THE BEHAVIOR OF ITS DIRECT EXECUTION.
> 
> But I was not talking about EEE. I was talking about the halting 
> function. All you seem to be claiming above is that whatever EEE 
> computes, it isn't the halting function. Everyone already agrees to that.
> 
> André
> 

The math halting function is free to "abstract away" key
details that change everything. That is why I have never
been talking about the pure math and have always been
referring to its implementation in a model of computation.

A halt decider cannot exist because the halting problem is
defined incorrectly ignoring key details that change
everything.

To unequivocally see these key details we examine x86 code
such that every control flow instruction is implemented
within a directed graph of 100% specific state transitions.

---
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer