As part of a reduction I am trying to come up with a computable function that will fill in the holes of another function. Suppose $A$ is the set of all $n$ such that $\Phi(x,n)$ halts for all $x \in \mathbb{N}$, and suppose $B$ is the set of all $n$ such that $\Phi(x,n)$ halts for almost all $x \in \mathbb{N}$ ($\mathbb{N}$ minus finitely many natural numbers). I'm trying to show that $A \leq_m B$, i.e. $a \in A \Leftrightarrow f(a) \in B$ for some $f$.

What I have so far is: Suppose $f(x) = S^1_1(x,p)$ for some program encoded by $p$. let $f(a) \in B \Leftrightarrow \Phi(x,f(a))$ halts for almost all $x \Leftrightarrow \Phi(x,S^1_1(a,p))$ halts for almost all $x \Leftrightarrow \Phi(x,a,p)$ halts for almost all $x \Leftrightarrow$ (somehow by def of p) $\Phi(x,a)$ halts for all $x \Leftrightarrow a \in A$.

I'm stuck on what the program P should be to complete this proof. I think that it should "fill the holes" from almost all to all natural numbers. Any hints?

Note: $\Phi$ is the universal $L$ program and $S^1_1$ is from the S-n-m theorem (or parameter theorem).

  • $\begingroup$ Now that's a SEO title! $\endgroup$ – Raphael Mar 4 '16 at 6:14
  • $\begingroup$ Not sure what SEO means. Search Engine Optimized was the first thing that came up on google. $\endgroup$ – McAngus Mar 4 '16 at 6:17
  • $\begingroup$ Hint: One end of the equivalence chain needs to be $a \in A$. That informs your choice of $p$! See also our reference question. $\endgroup$ – Raphael Mar 4 '16 at 6:17
  • $\begingroup$ Hmm, I still don't see what to do after reading through that. Because halting is not computable, how could I determine (in the program p) if the input $x$ is something that we should run with $\Phi(x,n)$ or if we should choose a different number to make it halt on all inputs $x$? $\endgroup$ – McAngus Mar 4 '16 at 6:34

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