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From: =?UTF-8?B?QW5kcsOpIEcuIElzYWFr?= <agisaak@gm.invalid>
Newsgroups: comp.theory
Subject: Re: Correcting the definition of the halting problem --- Computable
 functions
Date: Mon, 24 Mar 2025 19:46:38 -0600
Organization: Christians and Atheists United Against Creeping Agnosticism
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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.

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).

> 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.

>>
>> 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

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