XPost: comp.theory, sci.logic, sci.math

Halting problem undecidability and infinitely nested simulation (V3)

We define Linz H to base its halt status decision on the behavior of its
pure simulation of N steps of its input. N is either the number of steps
that it takes for its simulated input to reach its final state or the
number of steps required for H to match an infinite behavior pattern
proving that its simulated input would never reach its own final state
in any finite number of steps. In this case H aborts the simulation of
this input and transitions to H.qn.

Simulating halt deciders never determine whether or not their input
stops running. Every input to a simulating halt decider always stops
running either because it reached its final state or its simulation was aborted. Simulating halt deciders determine whether or not the Linz
criteria can possibly be met … the Turing machine will halt whenever it enters a final state. (Linz:1990:234)

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.
A copy of Linz H is embedded at Ĥ.qx.

Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qy ∞
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn

Because it is known that the UTM simulation of a machine is
computationally equivalent to the direct execution of this same machine
H can always form its halt status decision on the basis of what the
behavior of the UTM simulation of its inputs would be.

When Ĥ applied to ⟨Ĥ⟩ has embedded_H simulate ⟨Ĥ⟩ ⟨Ĥ⟩ these steps would
keep repeating:
Ĥ copies its input ⟨Ĥ⟩ to ⟨Ĥ⟩ then embedded_H simulates ⟨Ĥ⟩ ⟨Ĥ⟩...

This shows that the simulated input to embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ would never reach its final state conclusively proving that this simulated input
never halts. This enables embedded_H to abort the simulation of its
input and correctly transition to Ĥ.qn.

It is the case that if embedded_H recognizes an infinitely repeating
pattern in the behavior of its simulated input: ⟨Ĥ⟩ applied to ⟨Ĥ⟩ such
that this correctly simulated input cannot possibly reach its own final
state then this is complete proof that this simulated input never halts.

Because a halt decider is a decider embedded_H is only accountable for computing the mapping from ⟨Ĥ⟩ ⟨Ĥ⟩ to Ĥ.qy or Ĥ.qn on the basis of the
behavior specified by these inputs. embedded_H is not accountable for
the behavior of the computation that it is contained within: Ĥ applied
to ⟨Ĥ⟩ because this is not an actual input to embedded_H.




Halting problem undecidability and infinitely nested simulation (V3)

https://www.researchgate.net/publication/358009319_Halting_problem_undecidability_and_infinitely_nested_simulation_V3



--
Copyright 2021 Pete Olcott

Talent hits a target no one else can hit;
Genius hits a target no one else can see.
Arthur Schopenhauer

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