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The original version of this question: When Linz Ĥ is applied to its own TM description and Ĥ has an embedded simulating halt decider does this decider correctly decide that its input does not halt?

Had to be simplified to this next question so that people would have some established basis to understand what a simulating halt decider is.

When we hypothesize that the halt decider embedded in Ĥ is simply a Universal Turing Machine (UTM) does this define a computation that never halts when Ĥ is applied to its own Turing machine description?

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Linz, Peter 1990. An Introduction to Formal Languages and Automata. Lexington/Toronto: D. C. Heath and Company. (318-320)

The following simplifies the syntax for the definition of the Linz Turing machine Ĥ, it is now a single machine with a single start state. The halt decider is embedded at state Ĥ.qx.

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


Figure 12.3 Turing Machine Ĥ

Does the Peter Linz Ĥ applied to its own Turing machine description have the following infinite cycle in its state transition graph when we hypothesize that the halt decider at Ĥ.qx is simply a UTM?

Ĥ.q0 copies its input then Ĥ.qx simulates this input with the copy then
Ĥ.q0 copies its input then Ĥ.qx simulates this input with the copy then
Ĥ.q0 copies its input then Ĥ.qx simulates this input with the copy then...
This is expressed in figure 12.4 as a cycle from qx to q0 to qx.


Figure 12.4 Turing Machine Ĥ

The follow on question is this:

If the answer to the original question is "yes", then would a simulating halt decider (initially acting as if it was a UTM) encounter an infinitely repeating pattern when the Linz Ĥ is applied to its own TM description?

If the answer to this question is yes, then does this provide a correct basis for this simulating halt decider at state Ĥ.qx to abort the simulation this input and report that its input does not halt?

Halting problem undecidability and infinitely nested simulation

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