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Questions tagged [semi-decidability]

Questions about which problems are semi-decidable, also known as recognizable or recursively enumerable.

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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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1answer
44 views

Is every decidable language recognizable by a Turing Machine space-bounded by some f(|w|)?

The negative answer to decidable = non-contracting grammar? suggests the following question: Is there a decidable language that can be recognized only by a space unrestricted Turing Machine (i.e. ...
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2answers
57 views

Are decidable set/languages EQUIVALENT to type 1 grammars (non-contracting)?

Suppose a Turing Machine (TM_G) that generates natural numbers following < or, equivalently, it generates words in lexicographical order. Then, that language/set is decidable. Because it is trivial ...
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1answer
31 views

Decidability of languages with dfa/turing-machines

For any alphabet and any natural number k, a language of strings at least k is decidable. So my question is having some alphabet (let's say (0,1)) and some number let's say k=5 then my language has ...
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73 views

How to prove (un)decidability

Let's say we have a string s , a code size limit of b bytes and a time limit t, the question is then whether or not it is possible to construct an algorithm that prints the string within the time ...
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1answer
30 views

TM with double-infinity tape semi-decides the same languages as classical TM

Show that a Turingmachine with tapes that are infinite in both directions semi-decides the same languages as a classical TM. Apart from the entry-word, the tapes are filled with empty spaces and the ...
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1answer
261 views

is there a constructive proof of the existence of a language which isn't recursive (without invoking infinities)?

My understanding is that a language cannot be decided if the language is actually infinite (not generated by any machine). However, actual infinites make me squirm. Is there any reason to believe in ...
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61 views

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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2answers
237 views

Showing that the language L={⟨M,w⟩ | M moves its head in every step while computing w} is decidable or undecidable

How would you go about showing that the language L={⟨M,w⟩ | M moves its head in every step while computing w} is decidable or undecidable? Intuitively speaking I think it is indeed undecidable because ...
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1answer
53 views

Understanding the union of an undecidable language with a finite or decidable language

I'm trying to prove that the language $L \cup A$ is undecidable, when the language $L$ is undecidable and the language $A$ is finite or decidable. This is confusing me because if $L$ were to be a semi-...
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28 views

Some questions regarding decidability and semi-decidability of $A/B = \{ w \text{ | }\exists z \in B, wz \in A\}$

I have found two interesting questions regarding the quotient of languages, described as: $A/B = \{ w \text{ | }\exists z \in B, wz \in A\}$ The first one is: Let $A$ and $B$ be regular languages, ...
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70 views

How this language belong to R?

Consider the following language $$L= \{ \langle M\rangle | \text{ $M$ is a TM, and $L(M)\in coRE$} \}$$ I don't understand why the language $L$ is in $R$, intuitively, I think this is not true. ...
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80 views

How to show a language is Partially Decidable?

I am trying to solve some questions on partial decidability of languages and I am getting confused in how to construct proper arguments through the idea of Universal Turing Machine. I am not posting ...
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1answer
25 views

Showing semidecidability without using diagonalization

All the methods I know which shows a given language $L$ is $RE$ but note $REC$ deep down boils down to the cantor's diagonalization arguement in one way or the other, and most commonly it boils down ...
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269 views

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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21 views

Can you build a solver for a language from a solver for its complement?

If you have a solver for an L in NP do you have enough information to build a solver for co-L in Co-NP? Meaning, is there a procedure that can take you from a solver for one to a solver for another?
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1answer
73 views

How to show ambiguous context-free grammars in Chomsky normal form is Turing recognizable?

So this question has two questions and i have to use the answer from 1 to answer question 2. Assuming that my answer for 1 is good. I need help with 2. ( Correct me if wrong please.) Question 1 : Show ...
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1answer
66 views

Can you build a solver from a verifier?

Given code to just an NP-verifier, where the certificate/witness is required to be of size polynomial in the instance, for a language L, can you, from that data alone, construct code for a solver, or ...
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55 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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0answers
40 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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0answers
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k-limited solution for PCP

So there's following problem, that has been bugging me for the last few days: A solution of a PCP $ i_{1},\dots,i_{n}$ with the cards $(x_{1} ,y_{1}),\dots,(x_{m}, y_{m})$ is considered as $k$-...
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0answers
43 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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1answer
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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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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
48 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
65 views

Decide if a string is a member of a language that represents $P$?

For some enumeration of the complexity class P (such as this as an example: How does an enumerator for machines for languages work?), for each string 𝑝 in the enumeration, does there exist some other ...
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1answer
102 views

How to prove semi-decidable = verifiable?

A language L is verifiable iff there is a two-place predicate R ⊆ Σ∗ × Σ∗ such that R is computable, and such that for all x ∈ Σ∗: x ∈ L ⇔ there exists y such that R(x, y) A language is semi-...
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1answer
65 views

Semi-decidability of the language $\overline{L_{\epsilon}}$

Firstly consider the problem: given $L_H = \{R(M)w : M \in TM_0, w\in L(M)\}$ where $R(M)$ are encoded transitions of $M \in TM_0$. Assume for contradiction $\overline{L_{H}}$ is semi-decidable, then ...
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1answer
115 views

“problematic” non-halting inputs for Turing machines

Let us start this question out, by defining for a Turing machine, the set of words it doesn't halt on. Define: $P(M)=\{w\in\Sigma^*|M$ doesn't halt on $w \}$ We know that the $HALT$ problem is $RE\...
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1answer
168 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
28 views

Is reaching in less lines semi-decidable?

Assuming we have two programs $p_1$ and $p_2$ and two line numbers $n_1$ and $n_2$. Does $p_1$ reach $n_1$ in less computational steps than $p_2$ reaches $n_2$? By reduction from Halting, this is ...
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2answers
270 views

Is determining if a Turing Machine stops for at least one entry decidable?

I can't find how to prove the decibility with a reduction. EDIT: I've tried the reduction from the halting problem and the aceptance problem. Stopping for at least one entry has infinite inputs (you ...
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1answer
67 views

Prove that a language is decidable

I need some help to prove that the language is decidable. $K$ = {$N$ : $N$ is a DFA (Sigma = {a, b, c}) and $L$($N$) contains at least one word in which there is no a}. It tried to make an algorithm ...
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1answer
106 views

Union of halting-like problem and non-halting-like problem

I came across the following problem: Define languages $L_0$ and $L_1$ as follows : $L_0=\{⟨M,w,0⟩∣M\text{ halts on }w\}$ $L_1=\{⟨M,w,1⟩∣M\text{ does not halt on }w\}$ Here $⟨M,w,i⟩$ is a triplet, ...
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A TM that doesn't decide Σ*, and a TM that doesn't decide the empty set?

I was wondering if it was possible to create a TM that semi-decides (but doesn't decide) either of the following two languages: $\emptyset$ $\Sigma^{*}$ I assume for 2, a one-state TM that just ...
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1answer
291 views

Proving a language is not Semidecidable

I have the language $L = \{ \langle M_1, M_2 \rangle : L(M_1) \subset L(M_2)\}$ and I'd like to prove that it is not Semidecidable. To do so, I need to use a reduction from $\neg H$. I cannot use Rice'...
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1answer
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Is it decidable “Given a TM M, whether M ever writes a non blank symbol when started on the empty tape.”

I came across below problem in this pdf: Given a TM M, whether M ever writes a non blank symbol when started on the empty tape. Solution given is as follows: Let the machine only writes blank ...
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3answers
488 views

Is it possible that the subtraction between two undecidable languages is regular?

If $L_1$ and $L_2$ are both non-decidable languages (Not decidable, so can be SD or $\lnot$SD), is it possible that $L_1-L_2$ is regular and $L_1-L_2\neq\phi$, where $\phi$ is the empty set? What's ...
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2answers
189 views

Is this language Recursively Enumerable or Not RE?

$L = \{\langle M, k\rangle : M\;\text{is a Turing Machine and } |\{w \in L(M) : w \in a^*b^*\}| \geq k \}$ My Interpretation of language is that $L$ is a language which contains Turing machine ...
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1answer
168 views

How can I show that a language is Turing-recognizable and decidable?

I was wondering how I can show that the language $\{a^n b^n c^n \mid n \geq 0 \}$ is Turing-recognizable. Also, if it is Turing-decidable?
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78 views

Decidability of the language of all deterministic LBA where all states are reachable

I have a exam task with 3 parts. (b) is no problem. I got a solution for (a) but the way (c) is asked makes me wonder if I even understood (a) (a) L := { < A > | A is a DFSA, where all states are ...
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1answer
132 views

Decidability of decision problems

Can somebody give intuition how to answer those questions? From one side I can say that most of them are undecidable because we can reduce the halting problem to them (or halting problem can appear ...
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1answer
1k views

Why is a subset of a undecidable language decidable?

I have problems with the understanding why a subset of a undecidable language is decidable. We've proved in the lecture that $HALT$$_T$$_M$$=${$<M,w>$|M is a TM and M halts on input w} is ...
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1answer
2k views

Which languages, decided by a turing machine are decidable?

How do I decide if a language is decidable and/or semi-decidable? I have theses languages: a) { < M > | L(M) ⊆ 0*} b) { < M > | L(M) contains at least one word of even length} c) {...
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1answer
121 views

Turing Machine equivalence in MinTM proof

The proof with contradiction that $MIN_{\mathrm{TM}}$ is not Turing-recognizable from Michael Sipser's textbook "Introduction to the Theory of Computation" (Theorem 6.7) is as follows: $C=$ "On ...
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1answer
398 views

Is the halting problem undecidable or unrecognizable? [duplicate]

Is the Halting problem in the class of undecidable problems, or it is just in the set of unrecognizable problems? I understand that if it is undecidable, then it is also unrecognizable. I have seen ...
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2answers
303 views

Is a language whose Turing Machine doesn't halt for some positive cases but for others does not recursive?

Say language $L$ is recursively enumerable, but not recursive. Say $a$ and $b$ are symbols of the alphabet and $w$ a word. Say we have the following language: $L' = \{ aw | w \in L \} \cup \{ bw | w \...
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1answer
66 views

Is $ L = \{ a^n\ |\ a^n \not\in L_n \} $ Turing recognizable (recursively enumerable)?

Say $ \Sigma = \{a\} $, $M_1, M_2, ... $ is an enumeration of all TMs that recognize languages over $\Sigma$ and $L_1, L_2, ... $ are respectively the languages that are recognized by those TMs. We ...
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1answer
194 views

Recognizably turing machine question (reject / loop)

The definition of proving recognizability using dove tailing is below. However I'm wondering if we can also prove loop or reject in the same way? Give a deterministic TM D that recognizes L such that ...
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1answer
86 views

Is the language of all TMs *not* accepting a given string, Enumerable?

Is the following language in RE? $$L = \{\langle M\rangle : M\text{ is a TM that does not accept }010\}$$ I could use Rice's Theorem with the property $P = \{L : 010\text{ is not in }L\}$ to show ...