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From: dbush <dbush.mobile@gmail.com>
Newsgroups: comp.theory
Subject: Re: DD specifies non-terminating behavior to HHH --- RECURSIVE CHAIN
 --- Saving Democracy
Date: Sat, 22 Feb 2025 14:40:44 -0500
Organization: A noiseless patient Spider
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On 2/22/2025 2:33 PM, olcott wrote:
> On 2/22/2025 1:17 PM, dbush wrote:
>> On 2/22/2025 1:54 PM, olcott wrote:
>>> On 2/22/2025 12:21 PM, dbush wrote:
>>>> On 2/22/2025 1:02 PM, olcott wrote:
> 
>>>>>>
>>>>> 01 int F(int i)
>>>>> 02 {
>>>>> 03      if (i<10) {
>>>>> 04          return 0;
>>>>> 05      } else {
>>>>> 06          return F(i+1);
>>>>> 07      }
>>>>> 08 }
>>>>> 09
>>>>> 10 int no_numbers_greater_than_10()
>>>>> 11 {
>>>>> 12      return F(0);
>>>>> 13 }
>>>>> 14
>>>>> 15 int main()
>>>>> 16 {
>>>>> 17   no_numbers_greater_than_10();
>>>>> 18   return 0;
>>>>> 19 }
>>>>
>>>> Actually, let's update main:
>>>>
>>>> int main()
>>>> {
>>>>     F((int)no_numbers_greater_than_10);
>>>>     return 0;
>>>> }
>>>>
>>>>>>
>>>>>> The function no_numbers_greater_than_10() checks if any natural 
>>>>>> number exists that is greater than 10.  It does this by checking 
>>>>>> all natural numbers one at a time.  If one such number exists it 
>>>>>> halts and return 0.   If no such number exists, it will run 
>>>>>> forever as no such number will satisfy the condition.
>>>>>>
>>>>>
>>>>> Your code is incomplete. I added main() with line numbers.
>>>>>
>>>>>> We can see that no_numbers_greater_than_10 correctly simulated by 
>>>>>> F cannot possibly terminate normal by reaching its own "return" 
>>>>>> instruction.  This means that F correctly reports that 
>>>>>> no_numbers_greater_than_10 is non-halting.  It further means, 
>>>>>> since no_numbers_greater_than_10  doesn't halt that there is no 
>>>>>> natural number greater than 10.
>>>>>>
>>>>>> Agreed?
>>>>>
>>>>> Here the execution trace that I see:
>>>>> 15, 16, 17, 10, 11, 12, 01, 02, 03, 04, 12, 18, 19
>>>>>
>>>>
>>>> Just as you say we're not talking about the direct execution of DD, 
>>>> we're also not talking about the direct execution of 
>>>> no_numbers_greater_than_10.  We're talking about 
>>>> no_numbers_greater_than_10 correctly simulated by F.
>>>>
>>>> It's a verified fact that no_numbers_greater_than_10 correctly 
>>>> simulated by F cannot possibly return so 
>>>> F(no_numbers_greater_than_10) is correct to report non-halting, 
>>>> which means that there is no natural number greater than 10.
>>>>
>>>> Agreed?
>>>>
>>>
>>> Leaving out main() made this difficult.
>>> We can assume that the address of no_numbers_greater_than_10 > 10.
>>> This will emulate no_numbers_greater_than_10 at incorrect byte offsets
>>> causing it to crash. This may or may not make F crash depending
>>> on how robust its emulator is.
>>>
>>
>> Let's make a small change so that wraparound is well defined:
>>
>> int F(uintptr_t i)
>> {
>>       if (i<10) {
>>           return 0;
>>       } else {
>>           return F(i+1);
>>       }
>> }
>>
>> This ensures that F((uintptr_t)no_numbers_greater_than_10) returns 0.
>>
>> This doesn't change the fact that no_numbers_greater_than_10 correctly
>> simulated by F cannot possibly return so F(no_numbers_greater_than_10)
>> is correct to report non-halting, which means that there is no natural
>> number greater than 10.
>>
>> Agreed?
> 
> i starts out as the address of
> no_numbers_greater_than_10
> Then causes the emulation to crash.
> 

If that address is greater than 10 then F returns 0 right away, 
otherwise it makes at most 10 recursive calls before returning 0, so 
there would be no crash.

So you agree that no_numbers_greater_than_10 correctly
simulated by F (i.e. if the body of the function F is replaced by an 
unconditional simulator as you said is correct) cannot possibly return?