# Sketch of Solution to the Halting Problem

Would a Halting Problem decidability decider refute Rice’s Theorem?
If every self-referential Turing machine description Halting Problem counter-example could be recognized and rejected such that halting decidability would be decidable (in these cases) would this refute Rice’s Theorem?

Halting decidability decider could possibly make halting always decidable:
A Turing machine H halt decider attempts to build a formal mathematical proof from its inputs to its own final states of H.halts and H.loops. If it finds that no such proof exists, then it has decided halting undecidability and rejects its inputs as erroneous, otherwise one of the two proofs that it built has decided halting on these inputs.

An Introduction to Formal Languages and Automata (1990) by Peter Linz Pages 318-319 show the definition of the two Turing machines that are being analyzed: The top of page 320 shows that Ĥ is being used self-referentially.

Halting Problem Proof from Finite Strings to Final States, (2018) by Pete Olcott This paper begins its first two pages with a slight adaptation of the notational conventions of Linz. It labels the second q0 state in Ĥ as Ĥ.qx. It includes non final state of Ĥ.qy. It numbers each copy of the input Turing machine descriptions. These adaptations to the Linz notation make it possible to perform a hypothetical execution trace of H(Ĥ, Ĥ).

This execution trace shows that no formal proof exists from H (Ĥ, Ĥ) to final states H.halts or H.loops. No proof exists because the input pair (Ĥ, Ĥ) specifies infinite recursion long before it reaches its paradoxical states of (qy) or ((qn)).

The Refutation of Peter Linz Halting Problem Proof is complete.
This refutation applies to the other conventional (self-referential) Halting
Problem proofs. This algorithm was completed December 13th 2018 7:00PM

Peter Linz Halting Problem Proof (1993) with Turing Machines H and Ĥ.

Every detail of the encoding of virtual machines implementing the Peter
Linz H deciding halting for the Peter Linz input pair: (Ĥ, Ĥ) is complete.

The only step remaining is the C++ encoding of the UTM that executes
these virtual machine descriptions.

When this last step is complete I will provide the full execution trace
of H actually deciding halting for input pair: (Ĥ, Ĥ).

The only reason that this is possible is a key undiscovered detail that
no one noticed for more than eight decades.