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I find very interesting the problem of existence of a machine $H$ which given as input any algorithm $P$ outputs whether $P$ halts or not. Alan Turing disproved the existence of such an $H$ machine in the following way:

"Assume $H$ exists. The input of $H$ is any algorithm $P$. The output of $H$ is $YES$ if $P$ halts and $NO$ if $P$ doesn't. Create another machine $N$ with input in the set $\{YES,NO\}$ and without output. The machine $N$ does this: if the input is $YES \to \text{loop for ever}$ otherwise, if input is $NO \to \text{halt}$. Let us call the composition of $H$ followed by $N$ as $X$. We will simply write $X(P) = N(H(P))$. Now $X$ has as input a machine and no output (it either halts or loops forever). What Turing does is to "feed" $X$ with itself ... and reason as follows: if $X(X)$ halts then $X(X) = N(H(X)) = N(YES) = \text{loops for ever}$ from which somehow the contradiction arises ..." but here one should assume (in order to obtain a contradiction) that $X(X) = X$ ...

Question

I do not know why such an assumption, $X = X(X)$ would be made. In fact, in my opinion, $X \neq X(X)$ for any input $P$ (that is all Turing is proving actually). Indeed assume $P$ is a machine which halts (runs for ever). Then $X(P)$ does not halt (does halt) when $X(X(P))$ does (does not). I do not understand therefore, why $H$ cannot exist! Where is my mistake ?

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  • $\begingroup$ Your transcription of the proof has several errors. It confuses $A_0$, $A_1$ and $A_2$ in several places. Would you like me to fix it? (I would start by not calling these $A_0$, $A_1$ and $A_2$, as that is quite unreadable.) $\endgroup$ Nov 3 '20 at 23:12
  • $\begingroup$ sure, please go ahead ... I will change back if I consider that the essence of the question was lost though! $\endgroup$
    – C Marius
    Nov 3 '20 at 23:12
  • $\begingroup$ In the meanwhile, you might find it helpful to watch this excellent video about Halting Problem. $\endgroup$ Nov 3 '20 at 23:13
  • $\begingroup$ I have seen the video already ... I think it glides over the core of my concerns expressed here in this question ... $\endgroup$
    – C Marius
    Nov 3 '20 at 23:15
  • $\begingroup$ @andrejBauer I do not think I confused $A_0, A_1, A_2$ anywhere though ... and $A$ stands for algorithm. Why you think is unreadable ? $\endgroup$
    – C Marius
    Nov 3 '20 at 23:17
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The proof you wrote in the question is faulty, because it uses the wrong definition of the Halting oracle. You tried to fix the proof by inserting the assumption $X(X) = X$, but that is not the way to do it. Here is the correct proof.

Definition: A halting oracle is a machine $H$ which takes as input a pair $(M,I)$ where $M$ is the (description of) a machine and $I$ an input. $H$ always halts and outputs:

  • "yes" if $M(I)$ halts,
  • "no" if $M(I)$ does not halt.

Theorem: A halting oracle does not exist.

Proof. Suppose $H$ exists. We shall derive a contradiction.

Let $N$ be a machine which does the following upon receiving an input $I$:

  • if $I$ is "yes" then $N$ loops forever,
  • otherwise $N$ halts.

Let $X$ be the machine which takes an input $I$ and then acts as $N(H(I, I))$. We now prove two contradictory facts:

  1. $X(X)$ does not halt.
  2. $X(X)$ halts.

To prove the first claim, suppose $X(X)$ halts. Then $H(X,X)$ outputs "yes", therefore $X(X) = N(H(X,X)) = N(\text{yes})$, but $N(\text{yes})$ does not halt. Therefore it is not true that $X(X)$ halts.

To prove the second claim, suppose $X(X)$ does not halt. Then $H(X,X)$ outputs "no", therefore $X(X) = N(H(X,X)) = N(\text{no})$, but $N(\text{no})$ halts. Therefore it is not true that $X(X)$ does not halt, so it halts. $\Box$.

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  • $\begingroup$ would you agree with defining the oracle $H$ as $H(M(I)) \in \{ YES, NO\}$ with obvious choice of yes and no ? Then $X(I) = N(H(I(I)))$ and $X(X) = N(H(X(X)))$ through definition. $\endgroup$
    – C Marius
    Nov 4 '20 at 0:16
  • $\begingroup$ I would not. The halting oracle takes two inputs: a machine and the input to be fed to the machine, and you need to explain what $H$ does withour referring to $H$ (because you are trying to define $H$). But you are suggesting that we should define what $H$ does on input $(M,I)$ by saying that it does $H(M(I))$ - this is not a valid definition because you referred to $H$. And also, $H(M(I))$ is wrong because $H$ takes two inputs, but "$M(I)$" is just a single input (namely the output of $M$ when run with intput $I$). $\endgroup$ Nov 4 '20 at 9:16
  • $\begingroup$ ok but why wouldn't you agree that for any halting oracle $H: \{machines\} \times \{ inputs\} \to \{Y,N\}$ you can consider a function $\bar{H}: \{ machines\} \to \{Y,N\}$. Since $M(I)$ is a machine still, we define $\bar{H}(M(I)) = H(M,I)$ (for those other machines $M^{0}$, which can not written as $M(I)$ consider $M^{0} = M^{0}(\emptyset)$ ) Now I wonder if $H(M, \emptyset)$ is defined actually ? $\endgroup$
    – C Marius
    Nov 4 '20 at 9:42
  • $\begingroup$ $H(M, \emptyset)$ is actually trivial to define $H(M,\emptyset) = YES$ if $M$ halts and $H(M, \emptyset) = NO$ if it doesn't. I have therefore "converted" your definition of oracle to mine, haven't I ? $\endgroup$
    – C Marius
    Nov 4 '20 at 10:01
  • $\begingroup$ When you write $H(M(I))$ that does not make sense. You must write $H(M,I)$. When you write $\emptyset$ as input, you are confusing the type of input and its value, so $H(M, \emptyset)$ makes no sense. What does it mean that the input to a machine is the empty set? It has to be something written on a tape. $\endgroup$ Nov 4 '20 at 10:43

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