Questions about problems which cannot be solved by any Turing machine.
Questions tagged [undecidability]
919 questions
148
votes
4 answers
How can it be decidable whether $\pi$ has some sequence of digits?
We were given the following exercise.
Let
$\qquad \displaystyle f(n) = \begin{cases} 1 & 0^n \text{ occurs in the decimal representation of } \pi \\ 0 & \text{else}\end{cases}$
Prove that $f$ is computable.
How is this possible? As far as I know,…
Raphael
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52
votes
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What makes type inference for dependent types undecidable?
I have seen it mentioned that dependent type systems are not inferable, but are checkable. I was wondering if there is a simple explanation of why that is so, and whether or not there is there a limit of "dependency" where types can be indexed by…
Victor
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48
votes
2 answers
Perplexed by Rice's theorem
Summary: According to Rice's theorem, everything is impossible. And yet, I do this supposedly impossible stuff all the time!
Of course, Rice's theorem doesn't simply say "everything is impossible". It says something rather more specific: "Every…
MathematicalOrchid
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47
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2 answers
How to show that a function is not computable? How to show a language is not computably enumerable?
I know that there exists a Turing Machine, if a function is computable. Then how to show that the function is not computable or there aren't any Turing Machine for that. Is there anything like a Pumping lemma?
Similarly, how can we show a language…
user5507
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33
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7 answers
Is there a more intuitive proof of the halting problem's undecidability than diagonalization?
I understand the proof of the undecidability of the halting problem (given for example in Papadimitriou's textbook), based on diagonalization.
While the proof is convincing (I understand each step of it), it is not intuitive to me in the sense that…
user118967
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33
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1 answer
Rice's theorem for non-semantic properties
Rice's theorem tell us that the only semantic properties of Turing Machines (i.e. the properties of the function computed by the machine) that we can decide are the two trivial properties (i.e. always true and always false).
But there are other…
Kaveh
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29
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3 answers
Are there any specific problems known to be undecidable for reasons other than diagonalization, self-reference, or reducibility?
Every undecidable problem that I know of falls into one of the following categories:
Problems that are undecidable because of diagonalization (indirect self-reference). These problems, like the halting problem, are undecidable because you could use…
templatetypedef
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28
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1 answer
What are the strongest known type systems for which inference is decidable?
It's well known that Hindley–Milner type inference (the simply-typed $\lambda$-calculus with polymorphism) has decidable type inference: you can reconstruct principle types for any programs without any annotations.
Adding Haskell-style typeclasses…
Joey Eremondi
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27
votes
5 answers
Why isn't this undecidable problem in NP?
Clearly there aren't any undecidable problems in NP. However, according to Wikipedia:
NP is the set of all decision problems for which the instances where the answer is "yes" have [.. proofs that are] verifiable in polynomial time by a…
BlueRaja - Danny Pflughoeft
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24
votes
2 answers
Is there a "natural" undecidable language?
Is there any "natural" language which is undecidable?
by "natural" I mean a language defined directly by properties of strings, and not via machines and their equivalent. In other words, if the language looks like
$$ L = \{ \langle M \rangle \mid…
Ran G.
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23
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4 answers
Do undecidable languages exist in constructivist logic?
Constructivist logic is a system which removes the Law of the Excluded Middle, as well as
Double Negation, as axioms. It's described on Wikipedia here and here. In particular, the system doesn't allow for proof by contradiction.
I'm wondering, is…
Joey Eremondi
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22
votes
6 answers
Halting problem theory vs. practice
It is often asserted that the halting problem is undecidable. And proving it is indeed trivial.
But that only applies to an arbitrary program.
Has there been any study regarding classes of programs humans usually make?
It can sometimes be easy to…
Jack Fleming
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21
votes
3 answers
Is the unsolvability of the N-Body Problem equivalent to the Halting Problem
There is no general analytic solution to the n-body problem that can produce an analytic function which can be used to give an n-body system's state at arbitrary time t with exact precision. However, there are some special cases of n-body systems…
Shufflepants
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21
votes
1 answer
Ratio of decidable problems
Consider decision problems stated in some “reasonable” formal language. Let's say formulae in higher-order Peano arithmetic with one free variable as a frame of reference, but I'm equally interested in other models of computation: Diophantine…
Gilles 'SO- stop being evil'
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20
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Regular expressions with backreferences over unary alphabet
Setting:
regular expressions with backreferences
unary language (1-symbol alphabet)
Is the following problem decidable in this setting:
Given a regular expression with backreferences, does it define a regular language?
For example, (aa+)\1…
Jukka Suomela
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