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From: olcott <polcott2@gmail.com>
Newsgroups: comp.theory,sci.logic
Subject: We finally know exactly how H1(D,D) derives a different result than
 H(D,D)
Date: Thu, 7 Mar 2024 15:05:45 -0600
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H1(D,D) maps its input + its own machine address 00001422 to its output.
  H(D,D) maps its input + its own machine address 00001522 to its output.
Thus both H1 and H are computable functions of their input.

Turing machines don't even have the idea of their own machine
address so this exact same thing cannot be Turing computable.

Olcott machines entirely anchored in Turing machine notions
can compute the equivalent of H1(D,D) and H(D,D).

Because Olcott machines are essentially nothing more than
conventional UTM's combined with Conventional Turing machine
descriptions their essence is already fully understood.

The input to Olcott machines can simply be the conventional
space delimited Turing Machine input followed by four spaces.

This is followed by the machine description of the machine
that the UTM is simulating followed by four more spaces.

When this input is ignored Olcott machines compute the
exact same set as Turing machines.

Unlike Turing machines, Olcott machines have the basis to
determine that they have been called with copies of their
own TMD.

Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqy ∞ // Ĥ applied to ⟨Ĥ⟩ halts
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqn   // Ĥ applied to ⟨Ĥ⟩ does not halt

With Olcott machines Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ <Ĥ> and H ⟨Ĥ⟩ ⟨Ĥ⟩ <H> do
not have the same inputs thus can compute different outputs
when they do not ignore their own TMD.


-- 
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer