92

Consider the following Turing machine $M$: it searches over valid ZFC proofs, in lexicographic order, and if it finds a proof that $M$ halts, then it halts.

If we fix a particular model of Turing machine (say single-tape Turing machine), and if we fix an algorithm to verify that a given string is a valid ZFC proof of the fact that $M$ halts, this should constitute an unambiguous description of a Turing machine $M$.

(Standard arguments in computability theory, i.e., Kleene's recursion theorem, allows $M$ to compute functions of its own description).

Does $M$ halt?

I find this question puzzling because there's no apparent logical contradiction either way. There could be a proof, in which case it will halt. If there is no proof, then it doesn't halt. What would the answer "depend" on? $M$ either halts or doesn't halt, but could its behavior be independent of ZFC?

I should note that a closely related Turing machine $M'$ can be used to give a simple proof of Godel's incompleteness theorem. It's much more "rebellious" in its behavior, where if it finds a proof that it halts, it doesn't halt, and if it finds a proof that it doesn't halt, it halts. It follows that there cannot be a proof of its halting or non-halting in ZFC (unless ZFC is inconsistent).

However $M$ is just earnestly trying to figure out its fate. Which is it?

Henry Yuen
  • 1,909
  • 13
    I'm not sure you have really described $M$ unambiguously. Doesn't it have to "know how it is coded"? – Sam Hopkins Aug 04 '22 at 00:04
  • 5
    @SamHopkins It seems at least to me to be unambiguous, since it is clear that there exists at least one $M$ that meets the informal description of enumerating and checking ZFC proofs, and Kleene's recursion theorem guarantees that it can encode the statement "M halts" in ZFC language – Joe Bebel Aug 04 '22 at 00:12
  • 3
    This is a hilariously interesting question! :) :) – paul garrett Aug 04 '22 at 02:18
  • 4
    @SamHopkins Here's a simple way to construct $M$. As it stands, $M$ is a zero-input program. Define $M_1$ to be a 1-input program, with the following behavior: if the input is $P$, then $M_1(P)$ interprets $P$ as a 1-input program and searches for a proof that $P(P)$ halts, and halts when it does. Then OP's $M$ is $M_1(M_1)$ (that is, $M_1$ run on itself). Similarly, in my answer, we can construct $N_1$ on similar lines, and let $N$ be $N_1(N_1)$. I leave actually writing the pseudocode (for p in proofs etc) to someone else. – Akiva Weinberger Aug 04 '22 at 13:05
  • A bounded version of this problem is considered here: https://www.lesswrong.com/posts/TNfx89dh5KkcKrvho/ai-cooperation-in-practice. It's bounded in the sense that M only searches proofs up to some maximum length. They also consider the case with two programs that each search for a proof the other halts. This is eventually used to construct agents which achieve mutual cooperation in the Prisoner's Dilemma, while still defecting against exploitable opponents. https://arxiv.org/abs/1401.5577 – Oscar Cunningham Aug 04 '22 at 14:58
  • If anyone wishes to republish this problem somewhere, they should call this machine $C$ for "compliant," to distinguish from what OP calls $M'$ and what I think should be called $R$ for "rebellious." – Akiva Weinberger Aug 31 '23 at 05:27

2 Answers2

78

It is a very nice question. The answer is yes, the machine will find a proof of its own halting nature, and it will halt when it does so.

I claim this is a consequence of Löb's theorem. Let $M$ be a Turing machine such as you describe. Note that it is not quite correct to say "the" Turing machine that does what you say, since there will be infinitely many different machines $M$ that search for proofs that they themselves halt. It may not be clear initially that they all have the same behavior, but let me show that indeed they do all halt.

Let $\psi$ be the assertion "$M$ halts." Thus, we can prove in ZFC that if $\psi$ is provable, then it is true, since $M$ would discover the proof. Thus, ZFC proves $\text{Pr}_{ZFC}(\ulcorner\psi\urcorner)\to\psi$. But this is exactly the situation that Löb's theorem is about, and it tells us that we can prove $\psi$ directly in ZFC. So we can prove in ZFC that $M$ halts, as I claimed. It follows that we can prove in PA and much less that $M$ halts, since once we have the actual ZFC proof that it halts, then we can prove in a very weak theory that the actual Turing machine computation halts in whatever specific number of steps it would take to verify the finding of it.

That argument uses the ZFC version of Löb's theorem, but we can get by with the standard PA version, even though M is searching for proofs in ZFC. The reason is that in PA we can prove that $\text{Pr}_{PA}(\ulcorner\psi\urcorner)\to\psi$, since if PA proves that $M$ halts, then we can prove that ZFC will prove it as well, and so $M$ will halt. Thus, we need only the standard PA version of Löb's theorem to see that PA proves that $M$ halts.

Incidentally, regarding the negated version and the proof of the incompleteness theorem you mention at the end of the post, these ideas are also the basis of the universal algorithm. See my paper The modal logic of arithmetic potentialism and the universal algorithm.

Joel David Hamkins
  • 224,022
  • 7
    I was just about to mention Löb's theorem in a comment! – Steven Stadnicki Aug 04 '22 at 00:38
  • 6
    I wrote an answer that expands out the proof of Löb. – Akiva Weinberger Aug 04 '22 at 01:45
  • Thanks--it seems very nice. – Joel David Hamkins Aug 04 '22 at 01:49
  • 4
    And thanks to @JoelDavidHamkins for writing this explanation! Exhilarating ideas. :) – paul garrett Aug 04 '22 at 02:20
  • 2
    I think Joel's was the first so I will accept that, but I wish I could accept both answers! – Henry Yuen Aug 04 '22 at 11:03
  • What's the weakest you can go? PA− plus Σ1-induction? – user21820 Aug 05 '22 at 09:46
  • Hmm. Consider a variant, a machine $M'$ that searches for either of two proofs: a proof that $M'$ halts at a tape position where symbol $\alpha$ is written (assertion $\psi_\alpha$), or a proof that $M'$ halts at a tape position where symbol $\beta$ is written ($\psi_\beta$); and whichever proof it finds first, it writes out that symbol and halts. Your argument would seem to imply that we can prove both $\psi_\alpha$ and $\psi_\beta$ directly in ZFC, since either one, if provable, is true. But both can't be true. What am I missing? – mjqxxxx Aug 05 '22 at 20:43
  • You can't prove that these statements, if provable, are true, since what you would also need to know in each case is that the statement is provable before the other one is, in order for it to be true. For example, if the theory is inconsistent, then both proofs are there, but only one of them is found first, and so only that one is true. – Joel David Hamkins Aug 05 '22 at 21:18
  • 1
    @mjqxxxx Yeah, neither situation actually has the Löb condition (provable => true). The only reason $M$ (as given in the OP) has the Löb condition, is that it's only looking for one thing. If you give me a proof that $M$ halts, then $M$ would eventually find it... unless it finds an earlier thing that makes it stop searching proofs - but we don't care because it leads to the same conclusion ($M$ halting) anyway. (continued) – Akiva Weinberger Aug 07 '22 at 03:47
  • 1
    On the other hand, if you give me a proof that $M'$ halts at $\beta$, then $M'$ would eventually find it... unless it finds an earlier thing that makes it stop searching, which is an issue now because that earlier thing might be a proof that $M'$ halts at $\alpha$. (If ZFC is consistent then both proofs can't exist simultaneously, but we can't assume ZFC is consistent because we need this argument to work within ZFC.) – Akiva Weinberger Aug 07 '22 at 03:50
  • 3
    Another example: Suppose we write a machine $L$ that, if it finds proof that $L$ halts then it halts (and stops searching), and if it finds a proof that $L$ doesn't halt then it enters an infinite loop (and stops searching). Naïvely, we might think (since Con(ZFC) poisons our brains) that $L$ has the same behavior as $M$, so it halts. But since $L$ is searching for two things, our proof does not work. (I think that the fate of $L$ depends on implementation (unlike $M$ which halts no matter how you write it).) – Akiva Weinberger Aug 07 '22 at 03:53
78

Build a second machine $N$. $N$ searches for a proof in ZFC of, "if $N$ halts then $M$ halts". If it finds one, it halts.

ZFC can argue as follows. "Suppose $N$ halts. Then it found a proof that if $N$ halts then $M$ halts. This, combined with the trace of $N$ halting, would be a ZFC proof that $M$ halts. Thus $M$ finds this and halts."

That paragraph is a ZFC proof that if $N$ halts then $M$ halts… which $N$ finds, so $N$ halts. Thus, by the same logic as above, there is a proof that $M$ halts, which it will find and then halt on.

Isn't this hilarious? We prove this fact about our self-referential machine by constructing a second self-referential machine. This is basically how the proof of Löb's theorem works (see Joel David Hamkins's answer).

Akiva Weinberger
  • 888
  • Several edits because the details are pretty tricky. (My original version - which I think you can't find on the edit log because I fixed it quick enough - didn't actually use the behavior of $M$ in the proof… whoops.) – Akiva Weinberger Aug 04 '22 at 01:53
  • 9
    Yes, hilarious, indeed! Thanks for writing this out! – paul garrett Aug 04 '22 at 02:20
  • 2
    I suppose, when I wrote that "$M$ finds this [proof that $M$ halts] so it halts", technically $M$ could have already found an earlier proof and halted already. (Similar for when I wrote a similar thing about $N$.) But it doesn't change the argument – Akiva Weinberger Aug 04 '22 at 02:50
  • 5
    Beautiful -- this is a lot more understandable to me than "modal logic" (what shows up on Lob's theorem's Wiki page). – Henry Yuen Aug 04 '22 at 11:04
  • 2
    @HenryYuen Wikipedia does it by constructing a sentence $\Psi$ that satisfies $\vdash\Psi\leftrightarrow(\Box\Psi\rightarrow P)$ (where $P$ is "$M$ halts"). In words, that means it's a sentence $\Psi$ that's true iff, if $\Psi$ has a proof then $P$ holds. In other words, $\Psi$ is the sentence "If this sentence is provable then $P$ holds" (that is, $\Psi$ is the sentence "If this sentence is provable then $M$ halts"). Compare this to what I use, which is that $N$ halts iff, if there's a proof that $N$ halts then $M$ halts. – Akiva Weinberger Aug 04 '22 at 11:28
  • 5
    @HenryYuen Wikipedia's proof is more general because it applies not just to the provability operator (what Hamkins calls $\text{Pr}{ZFC}(\ulcorner\cdot\urcorner)$, which takes a ZFC formula as input and produces a ZFC formula as output), but to _anything that behaves like the provability operator. It's also easier to verify, as it's all syntactic rules. The downside is it's harder to see what's going on. (I suppose this is a metaphor for formalization in general.) – Akiva Weinberger Aug 04 '22 at 12:03
  • The wackiness of the proof of Löb's theorem was noted by Nik Weaver in an answer to another MO question. – Timothy Chow Aug 04 '22 at 19:37
  • @TimothyChow Note that that answer resolves it using Gödel's second incompleteness theorem. I think if you "unzip" the proof of the second incompleteness theorem (phrased using Turing machines), you essentially get the argument as I wrote it. – Akiva Weinberger Aug 04 '22 at 20:12
  • Why is it the case that such an $N$ exists? – mathlander Dec 26 '22 at 18:38
  • 2
    @mathlander This is a consequence of Kleene's recursion theorem (pronounced /ˈkleɪni/ KLAY-nee), believe it or not), which essentially states that we may assume programs can access their own source code. For a specific construction, see this comment I wrote above. – Akiva Weinberger Dec 26 '22 at 19:15