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consider $g$ and $f$ and $h$ as

$ \begin{align*} g(m) &= \begin{cases} 1 & if\;program\;m\;halts\;on\;input\;m \\ 0 & otherwise \\ \end{cases} \end{align*} $

$ \begin{align*} f(m,n) &= \begin{cases} undefined & if\;m=n \\ 1 & if\;program\;m\;halts\;on\;input\;n \\ 0 & otherwise \\ \end{cases} \end{align*} $

$ \begin{align*} h(m,n) &= \begin{cases} 1 & if\;program\;m\;halts\;on\;input\;n \\ 0 & otherwise \\ \end{cases} \end{align*} $

we have a counter example for computability of $g$ and $h$ but we don't have any counter example for $f$ and even more interesting function $f_{all}$ $$f(m,n)=\dfrac{m-n}{m-n}h(m,n)$$ $$f_{all}(m)=1-sgn(\lim_{w \to \infty}\sum_{\substack{n=0 \\ n\neq m}}^{w}(1-f(n,m)))$$

we know that if $p$ is a universal diophantine polynomial then we can represent a non-computable solution to the halting problem as follows $$ h(m,n) = sgn(\lim_{w \to \infty}\sum_{x_1=1}^{w}...\sum_{x_u=1}^{w}\dfrac{1}{1 + wp^2(m,n,x_1,...,x_u)}) $$

now the question is: are $f$ and $f_{all}$ computable?

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It's easy to see that $f$ is not computable, using the padding lemma: there is a total computable function $t$ such that for all $x$, $t(x)>x$ but program number $x$ and program number $t(x)$ have the same behavior (= yield the same partial computable function).

This means that $f$ computes $g$, since we have $g(x)=f(x,t(x))$.

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    $\begingroup$ For instance, informally, $t(x)$ can be obtained from $x$ by inserting a no-op instructions. (I know that Turing machines don't have instructions, so make the corresponding translation to Turing machines.) $\endgroup$
    – D.W.
    Sep 24, 2022 at 20:28
  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – D.W.
    Oct 18, 2022 at 16:52
  • $\begingroup$ I will accept your answer if you change $g(x) = f(x,t(x))$ to $g(x) = f(t(x),x)$ $\endgroup$
    – raoof
    Nov 15, 2022 at 13:08

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