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I have some problems to understand computability and hope you can help me. In the lecture we had following problem:

Consider the three partial functions $f,g,h\colon N \to N$, where $f$ is computable and $g$ is not computable. The following two statements are correct:

  1. If $h(x) = g(f(x))$, then $h$ could be computable.

  2. If $g(x) = h(f(x))$, then $h$ is not computable.

Well, 2 is clear I think, because we know that $g$ is not computable and $f$ is computable, so $h$ has to be not computable, otherwise $g$ would be computable.

But 1 is a big problem for me. If $f$ is computable and $g$ is not computable, how can $h$ be computable?

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Suppose that $f(x) = 0$. Then $g(f(x))$ is computable for any function $g$, computable or not. This explains 1.

As for 2, the composition of two computable functions is computable, so if both $f$ and $h$ are computable, so is $g$. Since $f$ is computable and $g$ isn't, the only conclusion is that $h$ is not computable.

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  • $\begingroup$ So what the professor said is wrong? (2) h has to be computable. mmh $\endgroup$ – Lisa Feb 5 at 16:14
  • $\begingroup$ Sorry, I got the notation wrong. $\endgroup$ – Yuval Filmus Feb 5 at 19:31
  • $\begingroup$ No problem :=) But i don't understand 1. if $f(x)=0$ then we get $g(0)$. How can this be computable if $g$ by definition isn't? $\endgroup$ – Lisa Feb 5 at 19:47
  • $\begingroup$ If $f(x) = 0$ for all $x$, then $g(f(x)) = g(0)$ for all $x$. Now $g(0)$ is just some natural number, say 555. Are you claiming that the function $h(x) = 555$ isn't computable? $\endgroup$ – Yuval Filmus Feb 5 at 20:13

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