Questions tagged [undecidability]

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

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22 views

Not understanding this way of proving undecidability of the termination problem

I am reading some slides on Algorithm to understand why termination is an undecidable problem. The slides say the following: – Assume termination(P) always terminates and returns true iff P always ...
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1answer
57 views

Cannot understand reductions from the halting problem and its complement

When I was going through the reductions from $HP$ and $\overline{HP}$ in this handout, I do not understand how everywhere the following claim is made: $$⟨M,x⟩ \in \overline{HP} ⇒ \text{M does not halt ...
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1answer
62 views

Is this Language decidable?

As the title says; is this language decidable and how do you prove it? $$L =\{\langle M\rangle \mid M \text{ is a Turing Machine and there is an input that } M \text{ halts on} \} $$
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1answer
77 views

Algorithmic problem of regular, context-free, and recursively enumerable languages

Consider a language $L_1$ that is recursively enumerate, $L_2$ that is regular, and $L_3$ that is context-free. Are the following problems algorithmically decidable? Is $L_1 \cap L_2 = L_3$? Is $L_1 \...
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1answer
49 views

Is $\{w~|~\forall x \in T(M_v):|w|>|x|~\}$ decidable?

I want to ask if $\{w|\forall x\in T(M_v):|w|>|x|\}$ is decidable if v is a Index of a random but fixed Turing Machine with $|T(M_v)|<\infty$. My idea: It is co-semi-decidable since as soon as i ...
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1answer
30 views

Union of every language within group of decidable languages is also decidable?

So I was trying to solve following exercise: Let $(L_{i})_{i \in \mathbb{N}}$ be a family of decidable languages - this means that every $L_{i}$ is decidable. Then $\cup_{i \in \mathbb{N}}L_{i} $ is ...
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1answer
31 views

Why is the following language undecidable?

I'm currently learning for my exams this semester and tried to solve some old exams from the last years. The question is to show, that L ist undecidable. $L=\{w|T(M_w)\neq\emptyset \land \forall x \in ...
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1answer
51 views

Is the problem that determines whenever the word member $\in$ L(M) decidable or not?

Given a Turing machine M on alphabet {m,e,b,r} we're asked to determine if member $\in$ L(M). You must realize that M is not one specific machine and can be any turing Machine with the same alphabet. ...
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1answer
34 views

Undecidability of the language of all Turing Machines with repeat strings as their language

Show that the language consisting of all Turing machines whose language consists of strings that can be broken up into two consecutive and equal strings is undecidable. I would prefer if reduction was ...
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0answers
40 views

Undecidability of the language of PDAs that accept some ww

I'm trying to solve problem 5.33 from Sipser's Introduction to the Theory of Computation, "Consider the problem of determining whether a PDA accepts some string of the form $\{ww|w\in \{0,1\}^∗\}$...
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1answer
70 views

How to prove the language of all Turing Machines that accept an undecidable language is undecidable?

I want to prove that $L=\{\langle M \rangle |L(M)\text{ is undecidable}\}$ is undecidable I am not sure about this. This is my try : Suppose L is decidable. Let $E$ be the decider from $L$. Let $A$ be ...
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1answer
20 views

Is the Post Correspondence Problem with more than two rows harder than the standard two-row variant?

The standard Post Correspondence Problem concerns tiles with two rows of symbols, and whether a tile arrangement can be made so that the sequence of the top symbols of the tiles is equal to the bottom ...
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5answers
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Proof of the undecidability of compiler code optimization

While reading Compilers by Alfred Aho, I came across this statement: The problem of generating the optimal target code from a source program is undecidable in general. The Wikipedia entry on ...
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30 views

Decidability for $ \exists w´, w´´\in L:$ so that |w´´| - |w´| is prime

I tried to decide wheter the given Language $ L = \{ \langle M \rangle | M \space is \space TM \space and \space \exists \space w´,w´´\in L(M):|w´´|-|w´| \space is \space prime \} $ is recursive or ...
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2answers
69 views

Is checking if regular languages are equivalent decidable? [duplicate]

Is this problem algorithmically decidable? L1 and L2 are both regular languages with alphabet $\Sigma$. Does L1 = L2? I think that it is decidable because you can write regular expressions for each ...
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2answers
49 views

Proof: is the language $L_1=\{\langle M\rangle\mid\emptyset \subseteq L(M)\}$ (un)-decidable?

I want to show that $L_1 = \{\langle M\rangle \mid \emptyset \subseteq L(M)\}$ is decidable/undecidable - without rice theorem (just for the case that I can apply it). Every language contain the $\...
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1answer
37 views

What undecidable language $B$ is reducible to its complement?

I encountered a problem which asks to give an example of an undecidable language $B$ such that $B \leq_m \overline{B}$... However, I could find it hard to construct an example ... my difficulty is ...
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1answer
29 views

Is the languague L={<M>, M accepts a finite amount of words} decdidable?

Is $L=\{<M> | L(M) \ is \ finite\} $ decidable ? M is a TM. I think its relative simple to proof with the theorem of rice. But I am interested in a solution which does not use the Rice theorem. ...
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1answer
38 views

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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1answer
43 views

Why is this language Turing recognizable and not not-Turing recognizable

I read that the following language is r.e. but not not-Turing recognizable $L$: On input $M$ (where $M$ is a Turing Machine), $M$ accepts at least 20 inputs I am not sure why it is not not-Turing ...
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1answer
39 views

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: $1^* \cap G$ is context free since it is the intersection of a regular ...
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0answers
36 views

How to show that these two disjoint sets $A$ and $B$ exist

I came across this problem which asks to show the existence of two disjoint Turing-recognizable sets $A$ and $B$ such that no decidable set $C$ can separate them... In this case, a set $C$ is said to ...
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0answers
25 views

Is the emptiness of intersection of two CFLs decidable? [duplicate]

Consider $L = \{\langle L_1, L_2\rangle\mid L_1, L_2 \in \text{CFL} \text{ and } L_1 \cap L_2 = \emptyset \}$. How to prove that $L \notin R $? $L_1, L_2$ encoded in chomsky-normal-form.
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Is the Languague which contains all TMs which write the blank symbol at firs by the given input w decidable?

Consider the problem of determining whether a Turing machine M on an input w writes the blank symbol at first. Is this decidable ?
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1answer
24 views

Help me understand this Turing-machine Problem concerning $A_{TM}$

I'm a Physics/C.S. student and have been struggling with this particular problem for a few days now. So the task is as following: Consider the following languages: $\hspace{20pt} L_1 = \{\langle M \...
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1answer
38 views

Show that for every language there exists a harder language

I came across this problem that I could not figure out... For every language $A$, there is supposed to be a language $B$ such that: $$ A \leq_T B $$ but: $$ B \not \leq_T A $$ If it is $A \leq_TB$ and ...
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1answer
35 views

Is $L_2:=${$<M>$|$L(M)=\overline{A_TM}$} (un-)decidable?

I have to prove that the language $L_2:=${$<M>$|$L(M)=\overline{A_TM}$} is (un-)decidable. In a previous assignment we proved that $L_1:=${$<M>$|$L(M)=A_TM$} is undecidable. I would say ...
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1answer
252 views

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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38 views

difference between two uncomputable functions

for $L1$ regular language, $L2$ some language, $L1$ \ $L2$ is regular, decidable. yet, the next transition function might not even be computable: $F′=${$\,q:δ(q,w)∈F \, for\, some\, w∈ L2\,$}$.$ $F$ ...
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1answer
30 views

Proving undecidability for a language which contains string with certain syntax

Lets say we have the following problem: $$\mathcal{L}_1 = \{\langle \mathcal{M} \rangle | \mathcal{M}\ \text{is a Turing machine and $\mathcal{L}(\mathcal{M})$ contains a string with exactly three ...
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2answers
80 views

Is a 'discrete language' well-defined?

Are the following well-defined formal languages (in these cases: subsets of {0,1}*) ? An argument w is a member of L under the following rules... Example1: If more than half of w's digits are 1's --...
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0answers
20 views

TM for Language With a Specific Cardinality

I'm curious about how to build TM that decide and recognize languages defined by cardinality. For example, with the language $L_1$ = $\{w \in \{0,1\}^* | |w| = 1\}$ this is the language with a single ...
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1answer
111 views

How to decide whether a language is decidable when not involving turing machines?

For instance, consider L = {k : the binary expansion of sqrt(2) contains k consecutive 1s}. Obviously Rice Theorem would not work. I also tried the method of how it is to PCP undecidable but still no ...
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1answer
49 views

Is $nHALT$ undecidable even if $M$ halts on input $w$ in finite steps

If we have the language $nHALT=\{<M,w,n>;$ $M$ halts on input $w$ in less than $n$ steps$\}$ Is this language also undecidable in the same way that $HALT$ is undecidable? And if so, $nHALT\...
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1answer
19 views

Decidability of a language and inclusion between two other languages

I have this assignement that asks to say if the following statement is true or false, and possibly justifying the answer: "Let L₁, L₂ be decidable languages. For every language L s.t. L₁ ⊆ L ⊆ L₂, L ...
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1answer
98 views

Check if language is decidable

I would like to determine if the following language is decidable or not. L = { w $\in$ $\Sigma^*$ | $T(M_w)$ is recognized by a Turing machine with at most 42 states}. I know that every finite ...
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1answer
26 views

Decide if a language has a word of a given size

Suppose that $L$ is some language over the alphabet $\Sigma$. I was asked to show that the following languages is decidable: $$L' = \{w \in \Sigma^* | \text{ there exists a word } w'\in L \text{ ...
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0answers
19 views

Many-one reductions between the set of true sentences and a particular arithmetical set

Never used this site before so not sure the best way to get help. However, I've been looking at many-one reductions in relations to sentences in logic. Let TH(N) = {ϕ : ϕ is a first order sentence ...
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1answer
220 views

Decidability of Turing machines that never move their heads past any input string

$L_1 = \{ \langle M, w\rangle : M \text{ is a TM that never moves its head past the input string } w \}$ $L_2 = \{ \langle M\rangle : M\text{ is a TM that never moves its head past any input string} ...
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1answer
49 views

Decidability of equality, and soundness of expressions involving elementary arithmetic and exponentials

Let's have expressions that are composed of elements of $\mathbb N$ and a limited set of binary operations {$+,\times,-,/$} and functions {$\exp, \ln$}. The expressions are always well-formed and form ...
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35 views

Is the undecidability of a given problem undecidable?

Given an input problem P, can you construct an algorithm A to compute whether or not P is decidable or undecidable? In other words, is the undecidabiliy of a problem undecidable? My initial guess is ...
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6answers
2k views

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 ...
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1answer
36 views

About computable sets

Let TOT be the set of all Turing Machines that halt on all inputs. Find a computable set B of ordered triples such that: TOT = {e : ($\forall$x)($\exists$y)[(e, x, y) $\in$ B] This definition means ...
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2answers
49 views

How could you “solve” the halting problem if, hypothetically, the busy beaver numbers were “small”?

I read that if BB(n) did not grow faster than all computable sequences of integers, you could solve the halting problem and contradict Turing's theorem. I'm trying to figure out how you could ...
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3answers
96 views

What does “Every CFL is decidable” exactly mean?

I am trying to prove the fact that every CFL is decidable, however I can't come to terms with what the statement exactly means. I know that generation of a particular string by a given CFG is a ...
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1answer
31 views

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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1answer
94 views

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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1answer
68 views

Turing machines moving left at least once

Is the following language decidable? $$ L = \{ \langle M,w \rangle \mid \text{$M$ moves its head left at least once when run on $w$}\}. $$ I feel like this is a decidable language. But I don't know ...
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1answer
64 views

Is there a language that cannot be polynomially reduced to?

Is there a language A that cannot be polynomially reduced to by some language B? Or is it always possible to reduce a language B to A?
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2answers
79 views

Difference between regular grammar and CFG in generating computation histories and $\Sigma^*$

I would like to ask for intuition to understand the difference between a CFG generating $\Sigma^*$ and a regular grammar generating $\Sigma^*$.. I got the examples here from Sipser. Let $ALL_{CFG}$ ...

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