# Questions tagged [undecidability]

Questions about problems which cannot be solved by any Turing machine.

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### How to prove that the problem $\text{"If$L$is a context-free language, then, is$\overline{L}$also context-free?"}$ is undecidable?

Lately I came across a problem: $\text{"If$L$is a context-free language, then, is$\overline{L}$also context-free?"}$ And I need to comment on its decidability. Now I know that context free ...
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### Show that a language is not decidable by reducing from ATM

Let (ATM denotes the language $\{\langle M,w \rangle \mid \text{TM$M$accepts$w$}\}$) show that the language L={<M1,M2,w> | M1 and M2 both accept or reject w} is undecidable by reducing ATM ...
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### Are there decidable non-trivial properties of a LBA's accepted language?

The halting problem and therefore the acceptance problem are decidable for LBAs, but are the infinite extensions of these problems decidable? Given a LBA, can you decide whether there exists an input ...
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### Given a Turing Machine $M$, if I know $L(M)$ is finite, can I solve the halting problem?

Say I'm given an oracle that tells me whether or not $L(M)$, the set of words accepted by a Turing Machine $M$, is finite. By leveraging this oracle, can I solve the halting problem? That is, on an ...
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### A language is decidable iff some enumerator enumerates it in decreasing order [duplicate]

Show that a language is decidable iff some enumerator enumerates the language in decreasing order. What does it mean by enumerator enumerating in decreasing order? I am so confused about this concept....
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### Show that a language is decidable iff some enumerator enumerates the language in decreasing order

Show that a language is decidable iff some enumerator enumerates the language in decreasing order. I know a language is decidable iff some enumerator enumerates the language in the standard string ...
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### Undecidability and Unrecognizability of Language with two Turing Machines

I've been working on undecidability proofs and I found this question in the practice problems for the textbook "An Introduction to Automata Theory." I know that we start by contradicting the ...
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### Prove that the language Cats-Vs-Dogs is undecidable

Define Σ = {a, b, c, . . . , z} be the set of letters in the English alphabet. Prove that the following language is undecidable: Cats-VS-Dogs = {(M) | Either “cats” ∈ L(M) or “dogs” ∈ L(M), but not ...
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### Are there 3-colorable maps that can never be colored?

I just watched this explanation of zero-knowledge proofs with Avi Wigderson: https://www.youtube.com/watch?v=5ovdoxnfFV Key claims from the video: Every formal statement can be translated into a map ...
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### Turing machine that checks whether a given string is an output of a given machine and input

Is there a Turing machine such that, given a description $\langle M \rangle$ of a Turing machine $M$, an input $x$ and a string $y$, computes whether or not $y$ is the output of $M$ input $x$? My ...
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### Deteremine if Language is in $R$ or $RE$

$$L =\left \{ \langle M \rangle \mid \exists x\in \Sigma^* \left(\left | x \right |\leq 10000 \wedge H(M, x\right) \right \}$$ Where $H(M, x)$ denotes whether Turing machine $M$ halts on input $x$. My ...
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### halting problem vs watchdog

I have a theory that all finite state machines can be monitored by a second turing machine with infinite tape to determine if the state of the first machine was repeated thus reaching the conclusion ...
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### Characterization of computationally universal functions

Is it correct to state that $u$ is a universal function if and only if $$\forall f : \text{RE} \quad \exists g : \text{R} \quad \exists h : \text{R} \quad f = h \circ u \circ g$$ where RE is the set ...
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### A language is decidable iff it is Turing-recognizable and co-Turing-recognizable (WHY?)

I am trying to understand the proof for this theorem (theorem 4.22 of the book 'An introduction to the theory of computation'): ...
### Decidability of whether $w \in L(M_1) \setminus L(M_2)$
I'm studying for my finals and I came across this question from past exams: Is the following language decidable? $$L = \{ \langle M_1,M_2,w \rangle \mid w \in L(M_1) \setminus L(M_2) \}.$$ How can ...
Regarding the following languages $L_1$ and $L_2$, I want to prove that $L_1$ is decidable and $L_2$ is undecidable. I want to construct a turing machine which can decide $L_1$ and reduce the halting ...