Is there an alternative proof of the TM Halting Problem other than the "standard" one? [closed]

Question

I'm wondering if anyone is aware of a proof of the Halting Problem that is not just a permutation of the "standard" proof. Since there are so many formulations of this proof, rather than pick a specific one, I will sketch out the general outline that every proof I've seen follows (at least, I honestly don't remember one that doesn't follow this pattern):

1. Assume that there is a function halt() that can decide whether or not a given Turing Machine halts or not.
2. Construct a Turing Machine that embeds halt(), but "does the exact opposite".
3. ...
4. Proof by contradiction that because we have constructed a program that embeds halt(), but does the exact opposite, this proves that halt() can not exist.

I believe the above is "sufficiently accurate" for the purposes of this question. Please feel free to speak up if you feel it misses some nuance of the problem and / or proof.

My objection with this entire class of proofs is:

• Every formulation of the problem / proof begins by assuming that halt() exists and is capable of deciding whether or not a given Turning Machine halts or not.

At this point in the proof, there are really only two possibilities:

1. halt() does not, or can not, decide whether a Turing Machine halts or not.
2. halt() can, and does, decide whether or not a Turing Machine halts.

The problem is this: If we choose the first possibility, that is to say that for some unspecified reason halt() does not "work as advertised", the proof comes to an end. There is literally no point in proving halt() can not decide whether a Turing Machine halts or not if we handicap it from the start- we already know the answer.

I'm not aware of any proof that places qualifications on halt() at this point in the proof, therefore the only reasonable choice is the second possibility. This means that we axiomatically define halt() in order to "bring it in to existence". The problem with this is- halt() is axiomatically defined, and therefore is guaranteed to decide whether or a Turing Machine halts or not.

Since halt() is axiomatically defined as being able to decide whether a Turing Machine halts, any analysis that arrives at a result that contradicts this behavior can not, by definition, prove that halt() is undecidable. The axiomatic definition of halt() is so strong that any result that causes, or results in, a contradiction of halt() is proof that the hypothesis is invalid. Basically, just like the first possibility, there's no point in continuing on with the proof with this choice either because we have our answer- it is impossible to prove that the Halting Problem is undecidable using this technique because it is axiomatically guaranteed to be decidable.

Therefore, I personally consider this family of Halting Problem proofs to be invalid.

So, I'm wondering if anyone is aware of a proof of the Halting Problem that is "independent" of this style and family of proof. I have a strong preference for a proof that does not rely on "diagonalization" or "recursion" in any way. My preference is for something rooted in graph theory.

Which raises a different problem: So far, I have been unable to arrive at a zero-order approximation of a Halting Problem proof using other means. In fact, every back of the envelope attempt has completely failed to arrive at the same conclusion.

The strongest one I've managed to come up with is roughly: The $\delta$ transition table is finite, with a finite number of symbols. An appeal to König's lemma seems to be enough to justify that there is no infinite path, and therefore it must be possible to find a path from the start state to a halting state, if one exists. Not intended as proof, but a result from graph theory, whatever it is, would be much stronger than the current "Say something is true, then do the exact opposite, and then claim that by doing the exact opposite of what you first claimed, you've proved that it was never true to begin with" proof.

Does anyone:

1. Have a Halting Problem proof that is significantly different than the current "standard" proof? My preference is for a proof that gives the same result using completely different techniques and / or approach. So far, I have only been able to find derivatives of the "standard" proof.
2. See a problem with the analysis above?

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@Kaveh stated Note that undecidability of H is usually stated negatively. To be clear, from Turings On computable numbers, with an application to the Entscheidungsproblem paper that established the proof:

It may be thought that arguments which prove that the real numbers are not enumerable would also prove that the computable numbers and sequences cannot be enumerable. It might, for instance, be thought that the limit of a sequence of computable numbers must be computable. This is clearly only true if the sequence of computable numbers is defined by some rule.

Or we might apply the diagonal process. "If the computable sequences are enumerable, let $a_n$ be the $n$-th computable sequence, and let $\phi_n\left(m\right)$ be the $m$-th figure in $a_n$. Let $\beta$ be the sequence with $1 - \phi_n\left(n\right)$ as its $n$-th figure. Since $\beta$ is computable, there exists a number $K$ such that $1 - \phi_n\left(n\right) = \phi_K\left(n\right)$ all $n$. Putting $n = K$, we have $1 = 2\phi_K\left(K\right)$, i.e. $1$ is even. This is impossible. The computable sequences are therefore not enumerable".

The fallacy in this argument lies in the assumption that $\beta$ is computable. It would be true if we could enumerate the computable sequences by finite means, but the problem of enumerating computable sequences is equivalent to the problem of finding out whether a given number is the D.N of a circle-free machine, and we have no general process for doing this in a finite number of steps. In fact, by applying the diagonal process argument correctly, we can show that there cannot be any such general process.

The simplest and most direct proof of this is by showing that, if this general process exists, then there is a machine which computes $\beta$. This proof, although perfectly sound, has the disadvantage that it may leave the reader with a feeling that "there must be something wrong". The proof which I shall give has not this disadvantage, and gives a certain insight into the significance of the idea "circle-free". It depends not on constructing $\beta$, but on constructing $\beta'$, whose $n$-th figure is $\phi_n\left(n\right)$.

Let us suppose that there is such a process; that is to say, that we can invent a machine $\mathcal{D}$ which, when supplied with the S.D of any computing machine $\mathcal{M}$ will test this S.D and if $\mathcal{M}$ is circular will mark the S.D with the symbol "$u$" and if it is circle-free will mark it with "$s$". By combining the machines $\mathcal{D}$ and $\mathcal{U}$ we could construct a machine $\mathcal{M}$ to compute the sequence $\beta'$. The machine $\mathcal{D}$ may require a tape. We may suppose that it uses the $E$-squares beyond all symbols on $F$-squares, and that when it has reached its verdict all the rough work done by $\mathcal{D}$ is erased.

...

It is clear the original proof "assumes that there is a function halt() that can decide whether or not a given Turing Machine halts or not."

...

The machine $\mathcal{H}$ has its motion divided into sections. In the first $N - 1$ sections, among other things, the integers $1, 2, \ldots, N - 1$ have been written down and tested by the machine $\mathcal{D}$. A certain number, say $R\left(N - 1\right)$, of them have been found to be the D.N's of circle-free machines. In the $N$-th section the machine $\mathcal{D}$ tests the number $N$. If $N$ is satisfactory, i.e., if it is the D.N of a circle-free machine, then $R\left(N\right) = 1 + R\left(N - 1\right)$ and the first. $R\left(N\right)$ figures of the sequence of which a D.N is $N$ are calculated. The $R\left(N\right)$-th figure of this sequence is written down as one of the figures of the sequence $\beta'$ computed by $\mathcal{H}$. If $N$ is not satisfactory, then $R\left(N\right) = R\left(N - 1\right)$ and the machine goes on to the $\left(N + 1\right)$-th section of its motion.

From the construction of $\mathcal{H}$ we can see that $\mathcal{H}$ is circle-free. Each section of the motion of $\mathcal{H}$ comes to an end after a finite number of steps. For, by our assumption about $\mathcal{D}$, the decision as to whether $N$ is satisfactory is reached in a finite number of steps. If $N$ is not satisfactory, then the $N$-th section is finished. If $N$ is satisfactory, this means that the machine $\mathcal{M}\left(N\right)$ whose D.N is $N$ is circle-free, and therefore its $R\left(N\right)$-th figure can be calculated in a finite number of steps. When this figure has been calculated and written down as the $R\left(N\right)$-th figure of $\beta'$, the $N$-th section is finished. Hence $\mathcal{H}$ is circle-free.

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At this point, Turing has explicitly stated that $\mathcal{H}$ is "circle-free" and can be, and in fact is, constructed using a finite number of steps.

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Now let $K$ be the D.N of $\mathcal{H}$. What does $\mathcal{H}$ do in the $K$-th section of its motion? It must test whether $K$ is satisfactory, giving a verdict "$s$" or "$u$". Since $K$ is the D.N of $\mathcal{H}$ and since $\mathcal{H}$ is circle-free, the verdict cannot be "$u$". On the other hand the verdict cannot be "$s$". For if it were, then in the $K$-th section of its motion $\mathcal{H}$ would be bound to compute the first $R\left(K - 1\right) + 1 = R\left(K\right)$ figures of the sequence computed by the machine with $K$ as its D.N and to write down the $R\left(K\right)$-th as a figure of the sequence computed by $\mathcal{H}$. The computation of the first $R\left(K\right) - 1$ figures would be carried out all right, but the instructions for calculating the $R\left(K\right)$-th would amount to "calculate the first $R\left(K\right)$ figures computed by $\mathcal{H}$ and write down the $R\left(K\right)$-th". This $R\left(K\right)$-th figure would never be found. I.e., $\mathcal{H}$ is circular, contrary both to what we have found in the last paragraph and to the verdict "$s$". Thus both verdicts are impossible and we conclude that there can be no machine $\mathcal{D}$.

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And here is part of the problem- after Turing has just finished showing that $\mathcal{H}$ must be circle-free, and we are only constructing a machine made up of circle-free machines, the proof then immediately states that $\mathcal{H}$ can not be circle-free, even though we've shown that the machine we have constructed must be circle-free by definition.

The fact of the matter is this- the alleged issue that is being used to assert that the halting problem is undecidable has already been resolved by this point. If it were truly a problem that would cause the Turing Machine to be "circular" it never would have been written out as a circle-free machine. At this point we are executing the circle-free machines, so the alleged problem being discussed has already been filtered out along with all the other "circular" machines by halt(). The fallacy here is "denying the antecedent".

Answer 1

This was too long to post as a comment (in response to your comment), so I'm posting it as an answer.

Turing's proof of the Halting Problem is not mathematically fallacious. Here is the paragraph in which you explain why the proof bothers you:

Since halt() is axiomatically defined as being able to decide whether a Turing Machine halts, any analysis that arrives at a result that contradicts this behavior can not, by definition, prove that halt() is undecidable. The axiomatic definition of halt() is so strong that any result that causes, or results in, a contradiction of halt() is proof that the hypothesis is invalid. Basically, just like the first possibility, there's no point in continuing on with the proof with this choice either because we have our answer- it is impossible to prove that the Halting Problem is undecidable using this technique because it is axiomatically guaranteed to be decidable.

I think I can summarize your objection as follows: You are saying, "once we've contradicted the initial assumption that HALT exists, it is impossible to continue with a proof, because we've encountered an absurdity."

I'm not sure that it's quite right to say that the halting problem is axiomatically guaranteed to be decidable, but the point is that we've assumed that it is decidable, for no reason other than that we want to show that it isn't. I'm not sure what you think a proof by contradiction (or negation--I admit I use the two interchangeably, even if I shouldn't) is if not this.

This may be qualify as decidedly "non-research level" mathematics, but in the interest of giving you a metaphor that may help I'll say this anyway. Suppose I claim that I can build a spaceship that will fly to Alpha Centauri and back within ten seconds. You could say, "Well, let's assume that that's the case. Then you would have built a spaceship that is capable of traveling faster than the speed of light, which is impossible. Thus, no such spaceship can exist." This is (vaguely) similar logic to what is being used in the proof; we claim something can exist, then demonstrate an absurdity, and conclude that this indicates that it cannot exist.

If there's anything unclear about this, let me know and I'll try to re-explain. Also please see the link I gave in the comments for alternate proofs of the Halting Problem.

Answer 2

Although your reasoning is flawed, I believe it is educational to see why. First of all, I have a rule of thumb when I am trying to prove something that seems extraordinary or difficult: "Assume that your approach is wrong and try to find out why". Arguing against yourself is not an easy task, but it is very important.

One technique I use very often is trying to apply the same argument to different situations. In other words, I'm trying to find a counterexample. Assume that your reasoning is perfect. Then, apart from computability, you are disproving a good part of mathematic knowledge, which is based on contradiction proofs. For example, the statement that there is no natural number such that no other natural number is greater than it, uses contradiction. This result is much more intuitive than the undecidability of halting.

Perhaps the notion of contradiction could be better understood if you think of every person trying to prove a statement as a little God: Assume that you have a theory, i.e. a number of axioms and theorems. Assume also that you have a universe that instead of physical laws, obeys only this theory. Now , if you wanted to prove that statement S is false,you could work in the following way (an analogy to the physical world and a statement such as the conservation of energy is highly educational):

1. Split the original universe U into two parts, A and B, that are equal to U and obey its theory.Then make an addition that is unique to every part: Assume that in part A S holds, while in part B of the universe S is false. This is not the same as saying that S or its denial is an axiom in the respective part. It might be possible that a proof exists, however you are not aware of it (yet).

2. If S is false in both parts, then it must be also false in the whole universe. In the part B of universe that it is false (proven so by an unknown proof) you don't have to worry about it. If this argument does not sound very convincing, you can consider part B as small as you like: for example, if B consisted of only 2 atoms, you could easily verify the law of energy conservation.

3. Now, consider the part A. If you could show that A does not obey its own theory,that is the original theory and the theorem that S is true, it could not exist. However, you know that the original theory is true, therefore it must be that last theorem that does not allow part A to exist. Therefore, since part A does exist, S must be false.

4. Now you have that S is false in both parts of your universe, so it is false for the whole universe as well.

I would also highly recommend reading the following posts from Terence Tao's blog. I found them very instructive. He talks about contradiction and how to create objects that are so powerful, that contradict their own existence:

The “no self-defeating object” argument

The “no self-defeating object” argument, revisited

The “no self-defeating object” argument, and the vagueness paradox