XPost: comp.theory, sci.logic   
   From: F.Zwarts@HetNet.nl   
      
   Op 28.jul.2025 om 15:36 schreef olcott:   
   > On 7/28/2025 4:13 AM, Fred. Zwarts wrote:   
   >> Op 27.jul.2025 om 01:28 schreef olcott:   
   >>> On 7/26/2025 5:49 PM, olcott wrote:   
   >>>> On 7/26/2025 2:58 PM, olcott wrote:   
   >>>>> On 7/26/2025 2:52 PM, Mr Flibble wrote:   
   >>>>>> On Sat, 26 Jul 2025 14:26:27 -0500, olcott wrote:   
   >>>>>>   
   >>>>>>> On 7/26/2025 1:30 PM, Alan Mackenzie wrote:   
   >>>>>>>> In comp.theory olcott  wrote:   
   >>>>>>>>   
   >>>>>>>>> The error of all of the halting problem proofs is that they   
   >>>>>>>>> require a   
   >>>>>>>>> Turing machine halt decider to report on the behavior of a   
   >>>>>>>>> directly   
   >>>>>>>>> executed Turing machine.   
   >>>>>>>>   
   >>>>>>>>> It is common knowledge that no Turing machine decider can take   
   >>>>>>>>> another   
   >>>>>>>>> directly executing Turing machine as an input, thus the above   
   >>>>>>>>> requirement is not precisely correct.   
   >>>>>>>>   
   >>>>>>>>> When we correct the error of this incorrect requirement it   
   >>>>>>>>> becomes a   
   >>>>>>>>> Turing machine decider indirectly reports on the behavior of a   
   >>>>>>>>> directly executing Turing machine through the proxy of a finite   
   >>>>>>>>> string   
   >>>>>>>>> description of this machine.   
   >>>>>>>>   
   >>>>>>>>> Now I have proven and corrected the error of all of the halting   
   >>>>>>>>> problem proofs.   
   >>>>>>>>   
   >>>>>>>> No you haven't, the subject matter is too far beyond your   
   >>>>>>>> intellectual   
   >>>>>>>> capacity.   
   >>>>>>>>   
   >>>>>>>>   
   >>>>>>> It only seems to you that I lack understanding because you are so   
   >>>>>>> sure   
   >>>>>>> that I must be wrong that you make sure to totally ignore the subtle   
   >>>>>>> nuances of meaning that proves I am correct.   
   >>>>>>>   
   >>>>>>> No Turing machine based (at least partial) halt decider can possibly   
   >>>>>>> *directly* report on the behavior of any directly executing Turing   
   >>>>>>> machine.  The best that any of them can possibly do is indirectly   
   >>>>>>> report   
   >>>>>>> on this behavior through the proxy of a finite string machine   
   >>>>>>> description.   
   >>>>>>   
   >>>>>> Partial decidability is not a hard problem.   
   >>>>>>   
   >>>>>> /Flibble   
   >>>>>   
   >>>>> My point is that all of the halting problem proofs   
   >>>>> are wrong when they require a Turing machine decider   
   >>>>> H to report on the behavior of machine M on input i   
   >>>>> because machine M is not in the domain of any Turing   
   >>>>> machine decider. Only finite strings such as ⟨M⟩ the   
   >>>>> Turing machine description of machine M are its   
   >>>>> domain.   
   >>>>>   
   >>>>   
   >>>> Definition of Turing Machine Ĥ   
   >>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.∞,   
   >>>>    if Ĥ applied to ⟨Ĥ⟩ halts, and        // incorrect   
   requirement   
   >>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn   
   >>>>    if Ĥ applied to ⟨Ĥ⟩ does not halt.    // incorrect   
   requirement   
   >>>>   
   >>>> (a) Ĥ copies its input ⟨Ĥ⟩   
   >>>> (b) Ĥ invokes embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩   
   >>>> (c) embedded_H simulates ⟨Ĥ⟩ ⟨Ĥ⟩   
   >>>> (d) simulated ⟨Ĥ⟩ copies its input ⟨Ĥ⟩   
   >>>> (e) simulated ⟨Ĥ⟩ invokes simulated embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩   
   >>>> (f) simulated embedded_H simulates ⟨Ĥ⟩ ⟨Ĥ⟩ ...   
   >>>>   
   >>>> The fact that the correctly simulated input   
   >>>> specifies recursive simulation prevents the   
   >>>> simulated ⟨Ĥ⟩ from ever reaching its simulated   
   >>>> final halt state of ⟨Ĥ.qn⟩, thus specifies non-termination.   
   >>>>   
   >>>> This is not contradicted by the fact that   
   >>>> Ĥ applied to ⟨Ĥ⟩ halts because Ĥ is outside of   
   >>>> the domain of every Turing machine computed function.   
   >>>>   
   >>>   
   >>> In the atypical case where the behavior of the simulation   
   >>> of an input to a potential halt decider disagrees with the   
   >>> behavior of the direct execution of the underlying machine   
   >>> (because this input calls this same simulating decider) it   
   >>> is the behavior of the input that rules because deciders   
   >>> compute the mapping *FROM* their inputs.   
   >>   
   >> But the input specifies halting behaviour,   
   > It never was the actual input that specifies non-halting   
   > behavior.   
      
   Indeed. But HHH must decide on the actual input that specifies halting   
   behaviour. Not on another hypothetical other input that specifies   
   non-halting behaviour.   
   Sum(2,3) must calculate the sum of the actual input, not of hypothetical   
   other inputs.   
      
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