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I had this question in my exam. but my answer is wrong(I didn't receive explanations why...) $$f(\langle M\rangle,1^n)=\left \{ \texttt{the lexicographically smallest } x\in\left \{ 0,1 \right \}^n \cap L(M) \texttt{ if } n>100\texttt{ and }L(M)\cap \left \{ 0,1 \right \}^n \neq\varnothing \texttt{, otherwise undefined}\right \}$$

I answered it is computable.
for input $(\langle M\rangle,1^{n})$ when $n \geq101$
I run the machine on all possible inputs in $\Sigma^{n}$ and output the first result when conditions are
met.

I was wrong and apparently the language is not computable. what did I miss?

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    $\begingroup$ How do you run $M$ on all possible inputs in $\Sigma^n$? You can't do it one at a time, otherwise one execution of $M$ could not halt. If you do it in parallel and one execution with input $x$ halts and accepts, how do you know that no other execution with input $x' <x$ will accept? $\endgroup$ – Steven Feb 11 at 14:41
  • $\begingroup$ @Steven Ok I understand your point. I meant in parallel(when reading the input). following your explanantion I can't confirm if some x is the minimal lexigoraphical because possibly there may be an x' in a later iteration that is smaller than x which I assumed is the smallest?thus this is not computable? $\endgroup$ – user6394019 Feb 11 at 14:49
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    $\begingroup$ Yes, you can't be sure that $x$ is the smallest such input. My comment is just intuitive (non formal) argument that shows you why your approach fails. For a formal proof that $f$ is not computable see my answer. $\endgroup$ – Steven Feb 11 at 14:53
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Since for a given $n$, there are only finitely many strings in $\Sigma^n$, saying "run $M$ on all words from $\Sigma^n$" does make sense. But we need to be somewhat careful about the details here.

  1. We could run $M$ on all $w \in \Sigma^*$ in parallel. If we do that, then (assuming that $M$ halts on at least one of them, which we may do here), there is a first word for which we learn that $M$ halts on it. But "first" here refers to our particular simulation, and to how long $M$ takes on the various words. This will generally not be the lexicographically first word; so this is not doing the job.

  2. We could run $M$ on the words $w \in \Sigma^*$ in lexicographic order. But now, if $M$ doesn't halt on $0^n$, we never proceed beyond this particular simulation.

To prove that the function is indeed non-computable, mirror the proof of Rice' theorem.

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Consider this version of the halting problem: given $T$, decide whether $T(\varepsilon)$ halts (where $\varepsilon$ denotes the empty string).

Let $\overline{0}$ (resp. $\overline{1}$) be the string containing $101$ zeros (resp. ones) and suppose towards a contradiction that your function is computable. Then, given $T$ you can construct a Turing machine $M_T$ that behaves as follows on input $x$:

  • If $x\neq \overline{0}$ then $M_T(x)$ accepts;
  • Otherwise $M_T(x)$ simulates $T(\varepsilon)$, and when (if) $T(\epsilon)$ halts $M_T(x)$ accepts.

It is easy to see that $f(\langle M_T \rangle, \overline{1}) = \overline{0}$ if $T(\varepsilon)$ halts and $f(\langle M_T \rangle, \overline{1}) = \underbrace{00\dots0}_{100 \mbox{ times}}1$ if $T(\varepsilon)$ does not halt.

Since $f$ is computable by hypothesis, we can decide the halting problem by computing $f(\langle M_T \rangle, \overline{1})$ and accepting iff $f(\langle M_T \rangle, \overline{1}) = \overline{0}$. This is a contradiction.

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