# Questions tagged [undecidability]

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

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### Is it decidable whether a Turing machine M will reach state q on input s?

Given a turing machine $M$, one of its states $q$ and an input word $w$, will $M$ ever reach $q$ on $w$? As we are not given anything about the word length, I assume that we have a finite length word....
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### Mapping reduction from $A_{TM}$ to $INFINITE_{TM}$ same as to $ALL_{TM}$?

I was trying to solve a problem with a mapping reduction from $A_{TM}$ to $INFINITE_{TM}$, and came across a solution that was 100% identical to another solution I saw for $A_{TM} \leq_M ALL_{TM}$. ...
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### A variation of the halting problem

Given an infinite set $S \subseteq \mathbb{N}$, define the language: $L_S = \{ \langle M \rangle : M$ is a deterministic TM that does not halt on $\epsilon$, or, $T_M \in S\}$ where $T_M$ is the ...
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### PCP undecidability

There is a popular proof for the undecidability of the PCP (Post correspondence problem), which is outlined here: https://en.wikipedia.org/wiki/Post_correspondence_problem I'll assume whoever will ...
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### M does not accept [M] | 'Correction' of proof possible?

The language $D=\{[M]|M([M])=0\}$ is not decidable because of the following argument: Suppose there was a $TM \space M_D$ that decides $D$. Then if we gave $M_D \space [M]$, there would be two ...
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### Can we enumerate finite sequences which have no halting continuation?

Note: this question has been cross-posted to Math.SE, after about a week here. I am trying to deepen my understanding of the relationship between the Halting Problem and Godel's Completeness Theorem (...
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### Prove ALL$_{\text{TM}}$ is undecidable reduction problem

Given ALL$_{\text{TM}}$ = { < M > | where M is a TM and L(M) = $Σ^*$ } show this is undecidable. I'm also told not to use Rice's theorem. I'm having difficulties with reduction type problems. How ...
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### A language which is neither r.e. nor co-r.e

First, consider $$L_\exists=\{\langle M\rangle \mid M \text{ is a Turing machine and accepts some input}\}$$ is RE. I tried to construct a Turing machine: ...
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### decidability about intersection of regular language and context free language

Is "Given a CFL L and a regular language R, is intersection of L and R an empty set?" decidable? What if we replace L with the complement of L? One of them is decidable and the other is not. For the ...
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### Why is $EQ_{cfg}$ not recognizable but is co-turing-recognizable

I've seen the proof that $EQ_{CFG}$ is not recognizable but its complement is, my problem is that in the proof that it's complement is recognizable, it says that we test every string in $\sum^*$ and ...
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### Turing machine on input w tries to move its head past the left end of the tape

Consider the language $$L = \{ \langle M,w \rangle \mid \text{M on input w tries to move its head past the left end of the tape}\}.$$ Prove whether L is decidable or not. I tried to prove ...
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### Reduction of Turing-machine language

How to show that the following language is undecidable using reduction on the halting problem? $L: = \{w \in \{0,1\}^* |$ TM $M$ with $w = \langle M \rangle$ does not accept any input $\}$ When TM ...
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### prove language is decidable

The question is: The language L contains $DFAs$ which can accept languages equal to $\Sigma$* prove this language is decidable. I'm new to the Decidability topic and I don't know where should I start ...
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### What are some problems that have higher degree of unsolvalbilty than $\Pi^0_2$-complete problems?

I'm looking for some problems that have higher degree of unsolvalbilty in term of arithmetical hierarchy that requires more than 2 quantifiers like $\Pi^0_3$ ,$\Pi^0_4$ etc.
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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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### Complement of equality problem of Turing machine is recognisable or not

Complement of equality problem of Turing machines is unrecognisable or not-recognizable but How?. As per my knowledge it is recognisable if you can decide its accept condition but not Reject and ...
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### CFG that generates $1^*$ is decidable

I read somewhere that the problem which asks whether or not a CFG $G$ generates $1^*$ is decidable. The proof goes like this: $1^* \cap L(G)$ is context free since it is the intersection of a regular ...
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### Is this string substitution problem decidable?

We have the following task: Take as input a finite set of string pairs. Each pair represents a substitution. Replace exactly one instance of the left with the right. A substitution can only be ...
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### Is it decidable when a TM M gets another as inputs and checks if it fullfiills certain property?

I was asking myself if it is not possible to decide the language where a TM M gets the Godel number of a TM M' as input and the checks if, let us say, the TM M' has a certain amount of transitions. My ...
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### Proving undecidability of a language with mapping reductions

I'm referring to questions like this one: Mapping reduction to show NeverHalt is undecidable I understand with Turing reductions, you have to use oracle calls of the unknown language you're trying to ...
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### Determine if a language is Decidable or semi decidable

Consider the language $L = \{\langle M \rangle: \text{$M$accepts at most two single-letter words}\}$, where $\langle M\rangle$ is the encoding of Turing machine $M$. We need to determine, without ...
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### An approximation variant of the halting problem

It always has been bugging me that we (humans) know pretty easily when most programs we write halt or not, but the halting problem is still undecidable. I have just thought of a variant approximation-...
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### Reduction of the diagonalization language to the universal language

I'm going through Jeffrey D. Ullman's Introduction to Automata Theory, Languages, and Computations. The author reduces an instance of the membership problem in $L_d$ (diagonalization language) to a ...
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### Are run time bounds in P decidable when the problem is promised that an input program must halt?

I'm solving Problem 11-10(b) in "what can be computed". 11.10 Consider the decision problem HALTSINSOMEPOLY (HISP), defined as follows. The input is a program P, and the solution is “yes” ...
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### A question on decidability

I have a homework question that is as follows: L(P) is a language of ASCII input strings for which a given program, P, returns "yes". Is the set of all input strings P decidable, such that P ...
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### Is the language of Turing Machine encodings decidable is this instance?

I have an exercise question as follows: L is a set of all Turing Machine encodings for which the Turing Machine halts after a number of steps less than or equal to the minimum value among |w| and 1000,...
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### Would a “TM starting with a blank tape will ever write a nonblank symbol anywhere before halting” be decidable?

Would a "TM starting with a blank tape will ever write a nonblank symbol anywhere before halting" be undecidable?
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### Please help me understand this proof of the undecidability of “Do two halting Turing machines accept the same language?”

Do two halting Turing machines accept the same language? Proof that it is undecidable(credit to another user on this website: "Tom van der Zanden"): Let M be an arbitrary Turing machine. Let ...
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### Is the halting problem decidable for TMs that do not write to the tape? [duplicate]

Is the halting problem decidable for TMs that do not write to the tape? Once a read only tape TM repeats a configuration, it will loop forever. Therefore, all we have to do to decide the above is ...
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### Undecidability of TMs recognizing a decidable language

The language $L = \{ \text{M} \mid \text{M is a TM and the set of words w such that M halts on w is decidable} \}$ is given. I need to prove that $L$ is NOT Turing recognizable. I've got a hint: it ...
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### Purpose of Acceptance Problem

I am confused about the purpose/statement of the Acceptance problem: $A_{TM} =\{\langle M\rangle\,s |$ Turing machine $M$ accepts $s\}$ It can be shown that $A_{TM}$ is uncomputable, so we know that, ...
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### Given an algorithm, is it possible to find all other equivalent algorithms for the same computable function in the same model

For any computable-function, there may be multiple different algorithms (possibly countably infinite). For example, sort has many different implementations/algorithms, that we know of or that we have ...
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### Are all Recursively Enumerable languages which are not Recursive also Undecidable?

Knowing that all Recursive languanges are Decidable and All Not R.E. Languages are Undecidable (correct me if I am wrong), Are all languages which are R.E. but not Recursive also Undecidable? R.E. ==&...
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### Undecidability of an Intersection

1)"Given a CFL L and a regular language R, is the intersection of L and R an empty set?" decidable? 2)What if we replace L with the complement of L? Either 1 or 2 is decidable and the other ...