On 7/16/2021 8:34 AM, Ben Bacarisse wrote:
> Mr Flibble <fli...@reddwarf.jmc> writes:
>> All extant halting problem proofs appear to be predicated on a
>> misunderstanding of the following contradiction:
> I don't think you've read any actual proofs, let along all of them. Why
> you would even say such a thing?
>> Suppose T[R] is a Boolean function taking a routine
>> (or program) R with no formal or free variables as its
>> argument and that for all R, T[R] — True if R terminates
>> if run and that T[R] = False if R does not terminate. Consider
>> the routine P defined as follows
>> rec routine P
>> §L :if T[P] goto L
>> Return §
>> If T[P] = True the routine P will loop, and it will
>> only terminate if T[P] = False. In each case T[P] has
>> exactly the wrong value, and this contradiction shows
>> that the function T cannot exist.
>> [Strachey 1965]
>> T is indeed unable to decide P but for the wrong reason: T[P] is
> T[P] is not recursive. Maybe you don't understand what the CPL means?
> Further, this argument must fail for any of the actual proofs that are
> based on Turing machine because TMs have not functions, not calls and no
Peter Linz Ĥ applied to the Turing machine description of itself: ⟨Ĥ⟩
Ĥ.q0 wM ⊢* Ĥ.qx wM wM ⊢* Ĥ.qy ∞
if M applied to wM halts, and
Ĥ.q0 wM ⊢* Ĥ.qx wM wM ⊢* Ĥ.qn
if M applied to wM does not halt
When we hypothesize that the halt decider embedded in Ĥ is simply a UTM
then it seems that when the Peter Linz Ĥ is applied to its own Turing
machine description ⟨Ĥ⟩ this specifies a computation that never halts.
Ĥ0.q0 copies its input ⟨Ĥ1⟩ to ⟨Ĥx⟩ then Ĥ0.qx simulates this input with
the copy then
Ĥ1.q0 copies its input ⟨Ĥ2⟩ to ⟨Ĥy⟩ then Ĥ1.qx simulates this input with
the copy then
Ĥ2.q0 copies its input ⟨Ĥ3⟩ to ⟨Ĥz⟩ then Ĥ2.qx simulates this input with
the copy then ...
This is expressed in figure 12.4 as a cycle from qx to q0 to qx.
Within the hypothesis that the internal halt decider embedded within Ĥ
simulates its input Ĥ applied to its own Turing machine description ⟨Ĥ⟩
derives infinitely nested simulation, unless this simulation is aborted.
When the pure simulation of a machine on its input never halts we know
that the execution of this machine on its input never halts.
The ⟨Ĥ⟩ ⟨Ĥ⟩ input to the embedded simulating halt decider at Ĥ.qx is
pure simulation that never halts.
∴ Sound Deductive Conclusion
The embedded simulating halt decider at Ĥ.qx correctly decides its
input: ⟨Ĥ⟩ ⟨Ĥ⟩ is a computation that never halts.
Ĥ.q0 ⟨Ĥ⟩ specifies an infinite chain of invocations that is terminated
at its third invocation. The first invocation of Ĥ.qx ⟨Ĥ⟩, ⟨Ĥ⟩ is the
first element of an infinite chain of invocations.
It is common knowledge that when any invocation of an infinite chain of
invocations is terminated that the whole chain terminates. That the
first element of this infinite chain terminates after its third element
has been terminated does not entail that this first element is an actual
The above is more clear when you can see the cycle in the state
transition diagram of Ĥ(⟨Ĥ⟩) provided in this paper:
Copyright 2021 Pete Olcott
"Great spirits have always encountered violent opposition from mediocre