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Every undecidable problem that I know of falls into one of the following categories:

  1. Problems that are undecidable because of diagonalization (indirect self-reference). These problems, like the halting problem, are undecidable because you could use a purported decider for the language to construct a TM whose behavior leads to a contradiction. You could also lump many undecidable problems about Kolmogorov complexity into this camp.

  2. Problems that are undecidable due to direct self-reference. For example, the universal language can be shown to be undecidable for the following reason: if it were decidable, then it would be possible to use Kleene's recursion theorem to build a TM that gets its own encoding, ask whether it will accept its own input, then does the opposite.

  3. Problems that are undecidable due to reductions from existing undecidable problems. Good examples here include the Post Correspondence Problem (reduction from the halting problem) and the Entscheidungsproblem.

When I teach computability theory to my students, many students pick up on this as well and often ask me if there are any problems we can prove are undecidable without ultimately tracing back to some kind of self-reference trickery. I can prove nonconstructively that there are infinitely many undecidable problems by a simple cardinality argument relating the number of TMs to the number of languages, but this doesn't give a specific example of an undecidable language.

Are there any languages known to be undecidable for reasons that aren't listed above? If so, what are they and what techniques were used to show their undecidability?

Raphael
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templatetypedef
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  • @EvilJS My understanding was that the undecidability proof there involved the ability to simulate TMs, though perhaps I'm mistaken? – templatetypedef Dec 26 '15 at 22:28
  • You can say Rice's theorem might not fit into any of these categories, but the proof of the theorem does. – Ryan Dougherty Dec 27 '15 at 02:07
  • So basically any proof of being undecidable preceding Gödel incompletness theorem would do the trick? What about axioms that are undecidable within ZFC? Manifolds homeomorphism? Semigroups? – Evil Dec 28 '15 at 19:54
  • @EvilJS If the proof technique used to show the undecidability of those results doesn't involve some sort of self-referential technique and actually involves some kind of alternative proof technique, that would be great. However, keep in mind that we're talking about undecidable problems here, so statements that are independent of the axioms of some set theory aren't in of themselves problems. There's a distinction between "this statement is independent of the axioms of this theory" and "this problem is undecidable." – templatetypedef Dec 28 '15 at 20:15
  • You are forming sentences of the form: "Statement P holds because of proof technique X". I'm not sure these make sense: statements hold or not (in classical logic), period. There are true statements with many proofs using different techniques. So rather than asking for "reasons of holding", are you not looking for different proof techniques? And/or problems all known proofs for which are not from the three-item list you give? – Raphael Dec 28 '15 at 20:42
  • I'm mostly interested in your last option - a problem that is undecidable and where every known proof does not involve any of the above techniques. – templatetypedef Dec 28 '15 at 20:46
  • But if proof was made preGödelTuringRiceChurch... goodness, and then proved in modern style, does it imply you are not interested in such? Because "every known proof" means, that it either is from different world (and axioms) or nobody translated it to such. – Evil Dec 28 '15 at 21:29
  • What would be really interesting is to identify problems that can be proven undecidable by one of the techniques, but not any of the others. As it stands, this just asks for a laundry list of problems that are easier to prove using technique X, or even just are customarily proven by technique X for random reasons, like most popular textbook. – vonbrand Dec 28 '15 at 22:02
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    @EvilJS That's a good point. Really, what I'm looking for here is whether there is some fundamentally different technique we can use. It would be nice, for example, if someone identified a problem as undecidable in a case where that problem has no known relation to TM self-reference or a Godeling-type argument. If the best we can do is "we figured this one out a long time ago, then realized that it's easier to prove it another way," that in a sense would be an answer - the three techniques above fundamentally account for all the proofs of undecidability we know of. – templatetypedef Dec 28 '15 at 22:22
  • @vonbrand My intent behind the question isn't to find a problem that's easiest to prove undecidable using any one of the above techniques. Rather, I'm hoping for something that's easiest to prove undecidable via none of the above techniques, or, ideally, that's only proven undecidable via a technique fundamentally different than the above three. – templatetypedef Dec 28 '15 at 22:23
  • another way of looking at this is that undecidability and self-reference are inherently intertwined. also the use of the word "trickery" seems subjective. they are just mathematical proofs. somewhat related, there are problems that are conjectured to be undecidable without any proof, eg collatz from number theory, etc. – vzn Dec 28 '15 at 22:56
  • @vzn I agree. One of the major reasons I'm asking this question is to see if there is any way to disentangle undecidability and self-reference. – templatetypedef Dec 28 '15 at 23:02
  • Sorry to barge in but shouldn't "Problems that are undecidable due to reductions from existing undecidable problems." read "Problems that are shown to be undecidable via reduction to known undecidable problems."? – David Tonhofer Dec 29 '15 at 11:31
  • Idle thoughts: You need a program that can go into an infinite loop where there is no way for a debugger to find out that this is indeed an infinite loop (i.e. the hope that the loop might eventually finish should spring eternal). Thus there must not be any meaningful distance-to-solution problem that can be evaluated on top of the loop and the "working data structure, if any" must be able to shrink or grow in non-deterministic fashion (are we talking deterministic or probablistic TMs)?. – David Tonhofer Dec 29 '15 at 12:15
  • The question of whether a value "near" a fractal set will eventually escape to infinity comes to mind (but the TM will work on it in IEEE 754 or UNUM notation, naturally, not in honest-to-god reals, it's a TM not an abstract state machine...) – David Tonhofer Dec 29 '15 at 12:15
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    The busy beaver function grows too fast for any program to compute. Concretely, you can define a function $f(n)$ as one plus the largest number computed by a program of length at most $n$. Does that count as diagonalization? – Yuval Filmus Dec 29 '15 at 20:29
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    @YuvalFilmus Perhaps I'm being too strict here, but that sounds like a diagonal-type argument to me: you're constructing a function that is defined to be different from all functions computed by TMs. – templatetypedef Dec 29 '15 at 20:48
  • maybe chaos theory and non-linear dynamics suits you as an answer, in the sense that although a dynamical system can have "known and deterministc laws" its long-term behaviour can be (computationaly) unpredictable (aka undecidable) – Nikos M. Dec 30 '15 at 21:56
  • the reason non-linear dynamical systems can be unpredictable, is due to Hopf bifurcations in the phase-space, which can be seen as another "reason for undecidability" – Nikos M. Dec 31 '15 at 10:22
  • also note that "self-reference" does not lead necesarily to "undecidability', again from a physics analogy, renormalisation techniques are exactly about self-reference, in the sense that a certain particle (e.g an electron) is charged and interferes with its own electric field, without re-normalisation this computation cannot be done (technicaly, it diverges to infinity), yet the electron is finite, re-normalisation solves that by re-defining parameters in-place – Nikos M. Dec 31 '15 at 10:28
  • maybe a related question can interest you (again some physics involved) – Nikos M. Dec 31 '15 at 10:39
  • I'm wondering if self-reference in the Gödel way can be formally defined. If not, the question may be invalid (i.e. Gödel's proof technically has no self-reference). – Albert Hendriks Jan 01 '16 at 05:58
  • @Albert, see William Lawvere, "Diagonal Arguments and Cartesian Closed Categories". – Kaveh Jan 03 '16 at 18:40
  • @Albert, Godel's theorem is essentially based a on fixed point theorem for arithmetic formulas (similar to Kleene's fixed-point theorem in computability): for any $\varphi(x,y)$ there is a $\psi(x)$ such that $\psi(x) \equiv \varphi(x,\ulcorner \varphi \urcorner)$. Now combine that with expressibility of provability of a recursively axiomatizable theory in first-order arithmetic. For details see e.g. Mendelson's book "Introduction to Mathematical Logic". – Kaveh Jan 03 '16 at 18:46
  • @templatetypedef Please edit your question to clarify what you are looking for. The current version allows misleading interpretations and has attracted ... philosophical answers. – Raphael Jan 04 '16 at 10:05

3 Answers3

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Yes, there are such proofs. They are based on the Low Basis Theorem.

See this answer to Are there any proofs the undecidability of the halting problem that does not depend on self-referencing or diagonalization? question on cstheory for more.

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Kaveh
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  • If anyone is interested in advanced techniques in computability theory then check out Robert I. Soare's books Recursively Enumerable Sets and Degrees and Computability Theory and Applications. – Kaveh Jan 03 '16 at 10:42
  • Correct me if I'm wrong, but doesn't the proof of the low basis theorem involve applying a functional to itself and asking whether it doesn't produce a value? If so, isn't this just a layer of indirection on top of a diagonal argument? – templatetypedef Jan 03 '16 at 18:58
  • @templatetypedef, I am not an expert but as far as I understand no. See e.g. page 109 in Soare's book. – Kaveh Jan 03 '16 at 19:02
  • @templatetypedef, ps1: there is some vagueness in the question about what we consider diagonalization. If we are not careful we may expand what we consider to be diagonalization every time we see something which was not. Take e.g. priority methods or any general method of constructing objects part by part in a way to avoid being equal to any object from a given class. – Kaveh Jan 03 '16 at 19:21
  • @templatetypedef, ps2: one possible way to make the distinction very formal (possibly too formal because it ignores the distinction between different proofs of the same fact essentially, i.e. is not refined enough to distinguish between different proof techniques) is to ask about the place of the low basis theorem or the priority method arguments inside reverse mathematics framework compared to the systems where we can formalize typical diagonalization arguments. – Kaveh Jan 03 '16 at 19:35
  • @templatetypedef, as far as I understand, the low basis theorem is beyond recursive mathematics RCA0 (or even WKL0). Simpson proves it in ACA0. – Kaveh Jan 03 '16 at 19:51
  • @Kaveh What is this black magic that allows you to get a Google Books URL in 54 bytes? :-) – David Richerby Jan 03 '16 at 20:27
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    @David, :) I open the page from the book I want to share, click on the share button on top, and remove the parameters except the id and pg from the link. – Kaveh Jan 03 '16 at 20:35
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This is a really interesting question and I also had this question before when I learn computability theory. Here I give an alternative angle to view this question. In the following paragraphs I assume that the formal system that I use is first-order logic.

If you can prove that a problem P is undecidable, then you have at least one “proof” showing that “P is undecidable”. Since the set of all possible proofs is countable, we can conclude that the set of undecidable problems that you can formally prove the undecidability is also countable. But the set of all undecidable problems is uncountable, which means most undecidable problems cannot be proved to be undecidable at all.

On the other hand, if you can show a countably infinite set of undecidable problems where each problem’s undecidability can be proved via self-reference, you can at least show that the cardinality of the set of “provably undecidable problems” and the cardinality of the set of “undecidable problems provable by self-reference” are the same. This might not be an interesting result, but if we can go one step further and prove something like “all/almost all provably undecidable problems could be proved to be undecidable by self-reference”, I think that will be interesting enough.

Regarding specific proof techniques, I don’t know non-self-reference techniques for proving undecidability but I can make an informal argument stating that those techniques may not exist for certain undecidable problems. In program analysis, which is a research area trying to design (incomplete) analyzers to inform programs’ non-trivial properties, according to Rice’s theorem, non-trivial extensional properties of programs are undecidable. For such undecidable problems, different program analyzers give different “precisions” (e.g. the number of programs that they can precisely analyze). This might imply that the undecidability indeed depends on the analyzer itself because for a specific program, whether its property could be precisely analyzed via a given analyzer depends on the analyzer itself. As a result, I feel that a proof showing the undecidability of program analysis (the non-existence of complete program analyzers) tends to somehow mention the analyzer itself, which constitutes a “self-reference”. Of course I’m talking about the prove-by-contradiction route “assuming that a complete analyzer exists”.

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this is not exactly an affirmative answer, but an attempt at something nearby to what is asked for via a creative angle. there are quite a few problems in physics now that are "far distant" from mathematical/ theoretical formulations of undecidability, and they seem increasingly "remote" from and "bear little resemblance to" the original formulations involving the halting problem etc.; of course they use the halting problem at the root but the chains of reasoning have become increasingly distant and also have a strong "applied" aspect/ nature. unfortunately there do not seem to be any great surveys in this area yet. a recent problem that was "surprisingly" proven undecidable in physics that has attracted a lot of attention:

The spectral gap—the energy difference between the ground state and first excited state of a system—is central to quantum many-body physics. Many challenging open problems, such as the Haldane conjecture, the question of the existence of gapped topological spin liquid phases, and the Yang–Mills gap conjecture, concern spectral gaps. These and other problems are particular cases of the general spectral gap problem: given the Hamiltonian of a quantum many-body system, is it gapped or gapless? Here we prove that this is an undecidable problem. Specifically, we construct families of quantum spin systems on a two-dimensional lattice with translationally invariant, nearest-neighbour interactions, for which the spectral gap problem is undecidable. This result extends to undecidability of other low-energy properties, such as the existence of algebraically decaying ground-state correlations.

what you seem to be observing in the question is that (informally) undecidability proofs all have a certain "self-referential" structure, and this has been formally proven in even more advanced mathematics, such that both the Turing halting problem and Godels theorem can be seen as instances of the same underlying phenomenon. see eg:

The halting theorem, Cantor's theorem (the non-isomorphism of a set and its powerset), and Goedel's incompleteness theorem are all instances of the Lawvere fixed point theorem, which says that for any cartesian closed category, if there is an epimorphic map e:A→(A⇒B) then every f:B→B has a fixed point.

there is also a long meditation on this theme of the (intrinsic?) interconnectedness of self-referentiality and undecidability in the books by Hofstadter. another area where undecidability results are common and were initially somewhat "surprising" is with fractal phenomena. the crosscutting appearance/ significance of undecidable phenomena across nature is nearly a recognized physical principle at this point, first observed by Wolfram as "principle of computational equivalence".

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vzn
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  • other "surprising/ applied" areas of undecidability: aperiodic tilings, eventual stabilization in conway game of Life (cellular automata) – vzn Dec 29 '15 at 22:47
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    My understanding is that the proofs that all of these problems are undecidable all boil down to reductions from the halting problem. Is that incorrect? – templatetypedef Dec 29 '15 at 23:45
  • the answer basically concedes that (all known undecidability results can be reduced to the halting problem). your question is nearly phrased as a conjecture, and am not aware of any conflicting knowledge to it, and see a lot of circumstantial evidence in favor of it. but the closest to a formal proof known is apparently the fixed-point formulations of undecidability (there does not seem to be other formal formulations of "self-referential".) another way of saying it all is that Turing completeness and undecidability are two views of essentially the same phenomenon. – vzn Dec 30 '15 at 16:28