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Questions tagged [semi-decidability]

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Does undecidability violate Turing completeness? Shouldn't “complete” include “decidability”?

Does undecidability violate Turing completeness? Shouldn't "complete" include "decidability"? That is, if one has a language that's Turing complete, but expresses infinite computation (i.e. may not ...
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1answer
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How do I construct a NTM that accepts the language consisting of the coding of turing machines that halt on one input?

I currently have a problem with the following question: Let $L = \{ \langle M \rangle \mid \exists w: \text{$M$ halts for $w$ in at most $|w|^3$ steps} \}$. Construct an NTM (non-deterministic Turing ...
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1answer
54 views

Is Rice-Shapiro theorem bidirectional?

Rice-Shapiro theorem states that version A Let $\Gamma$ be a set of computably enumerable sets, and $I = \{e : W_e \in \Gamma\}$ its index set in some admissible enumeration of c.e sets. If $I$...
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1answer
24 views

Can the complement of an unrecognizable language be a recognizable language?

I know that complement of a language that is recursively enumerable, but not recursive, is definitely not recursively enumerable (or unrecognizable). So my question is what can be said about the ...
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2answers
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Why are not all recursive languages undecidable?

I learned that recursive language are decidable; correct me if I am wrong. However, I have found some arguments that seem to contradict this. These may or may not be correct; please let me know. If ...
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0answers
25 views

A set that is not recursively enumerable and not (K'≤ A)

Is there a set A such that it's not recursively enumerable and not(K'≤ A) ? where K' is complement of K= {n| φ n (n) halts} Thanks!
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How to know if a lanugae is undecidable or semi-decidable

I recently learnt about undecidable languages and semi-decidable languages. But I am still quite confused on how I can determine if a language is semi-decidable. Is there any standard theorem or axiom ...
3
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1answer
47 views

Efficient algorithm to determine if a lambda calculus term is equivalent to one without a given free variable

Consider the following problem: given a lambda calculus term $t$ and free variable $v$ determine whether $\phi(t,v)$, where $\phi(t,v) := \exists t'. t' \equiv t \land v \notin FV(t')$. This problem ...
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1answer
35 views

Need help understanding what co-recursively enumerable means

Lets say I have a set: $ L = \{\langle G \rangle | L(G) = \sum^{\star}\}$ and the question asks if it is co-RE. I know that if something is co-RE, it halts on every input not in L but may or may not ...
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1answer
35 views

Is the set of context free grammars that generate all words in co-RE?

Is $\{\langle G \rangle | L(G) = \sum^{\star}\}$ in co-RE? $\langle G \rangle$ is the encoding of a context free grammar. My intuition is that this is false.
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Whether language of all turing machines is decidable or undecidable or semi-decidable?

I recently came across this language: $L=\{<TM>| \text{TM accepts recursively enumerable languages}\}$ It was asked in the question to find out whether language L is decidable or undecidable. ...
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0answers
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Non-Turing-recognizable Language [duplicate]

I have been stuck on this problem for a while: Show that $L=\{\langle M \rangle : L(M) \text{ contains an even number of strings} \}$ is not Turing-recognizable. I know that by Rice Theorem, this ...
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1answer
36 views

Prove/disprove that the class of decidable (resp. partially decidable) languages is closed under symmetric difference

Prove/disprove that the class of decidable (resp. partially decidable) languages is closed under symmetric difference. A symmetric difference of sets A and B is the set (A \ B) ∪ (B \ A). I know that ...
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1answer
17 views

Proving a set is semi-decidable

Let $S = \{ ⟨M,q⟩ | (\exists x) M $ reaches state $q$ when running $M$ on $x$$\}$, where ⟨M,q⟩ is coded TM M and state q. To prove that $S$ is semi-decidable, I've tried to use the equivalence: ...
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1answer
343 views

Determining if given languages are regular or recursively enumerable

I came across following problem: Suppose $L_1$ and $L_2$ are two languages, $M$ is a Turing machine $L_1 =\{M|M$ accepts at most 2016 strings$\}$ $L_2=\{M|M$ accepts at least 2016 strings$\}$ ...
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Proof that total computable functions are not enumerable

In an answer to this question, a sketch of the proof that total computable functions are not enumerable is made: Because of diagonalization. If $(f_e:e \in N)$ was a computable enumeration of all ...
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1answer
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When does an extendible 1:1 p.c. function have a 1:1 computable extension?

A partial computable function $\varphi_e$, defined on a c.e. set $W_e$, is called extendible if there exists some computable function $f$ which extends $\varphi_e$, i.e. $\varphi_e(W_e) = f(W_e)$. My ...
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1answer
54 views

If Q1 and Q2 are countably enumerable, then is Q1\Q2 countably enumerable?

If Q1 and Q2 are countably enumerable, then is Q1\Q2 countably enumerable? I am reading a text where they claim that this is not the case and ask the reader to come up with a counter example. Can ...
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1answer
170 views

Why is the intersection of these two Languages Recursively Enumerable, not Recursive?

I am only several days exposed to computational theory, so my understanding is quite slim: in a question, it says that for a regular language L1 and a recursively enumerable but not recursive language ...
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1answer
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Show that $L = L_\phi \cup L_{\{\sum^*\}} \notin RE$ with Rice theorem

Show that $L = L_\phi \cup L_{\{\sum^*\}} \notin RE$ with Rice theorem. Well I did show that with reduction, by using $HP'$. Simply by creating a function from $f(\langle M \rangle, x) = (M')$ Thus,...
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1answer
359 views

What is the difference between undecidable language and Turing Recognizable language?

I was wondering what is the difference difference undecidable language and Turing recognizable language. I've seen in some cases where they ask: Prove that the language $ A_{TM} = \{ \ <M,w> | \...
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0answers
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How to prove that for a decidable problem the problem and the compliment of the problem are semi-decidable?

Given a decidable problem, how would I go about proofing that the problem and the complement of the problem have to be semi-decidable?
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1answer
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Closure of Turing-recognizable languages under homomorphism

I've proven that the Turing-recognizable languages are closed under concatenation and I need to show that they are closed under homomorphism. But what's really the difference? Doesn't closure under ...
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1answer
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Does the language of TM's that repeat a configuration infinite times semi-decidable or not?

Let us define the following languages: $$ {L_1 = \{\langle M\rangle : M \ \text{is a TM and $\exists w\in \Sigma^*$ s.t $M(w)$ repeats a configuration infinite times}\}} $$ $$ L_2 = \{\langle M\...
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0answers
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Decidability of intersection of two languages of same type

Given two context-sensitive languages, $L_1$ and $L_2$ is the problem of "whether $L_1 \cap L_2$ also belongs to CSL" decidable? I have the same question for the case when $L_1$ and $L_2$ belongs to ...
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0answers
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Give an example of a language where both L and ¬L is not semidecidable? [duplicate]

I know ¬H is not semidecidable so I was thinking of creating a language that combines both H and ¬H. Therefore L would be undecidable for ¬H and ¬L would be undecidable for H. Is this a proper ...
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2answers
45 views

Function whose domain is strictly included in its range

Hi I'm doing some exercices on reductions and one of them is like this: My problem rather than doing the reduction is understanding it. For the positive case I want that if Mx(x) never stops then p'...
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0answers
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Is the language below not decidable , if yes , it is then R.E ?

I am given a Turing machine M as input and i have to find out if this language below decidable and if not is it then in this case recursively enumerable . $$L=\\\{<M> \mid \text{ is ...
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1answer
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Prove that an infinite set is semidecidible

I have been asked to prove that: Being C an infinite set. Prove that C is semidecidable if and only if exists a total computable function that is injective and whose image is C. I've read the ...
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1answer
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Completeness problem of TM

$L = \{ \langle M \rangle \mid L(M) = \Sigma^∗ \}$ Is above problem R.E ? I found an explanation in one of the websites and I have doubt in few lines of paragraph. The explanation was Now, given a ...
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1answer
333 views

If a problem is “not semi-decidable” and “not decidable” can we say it is “undecidable”?

I was under impression that when a Language (or problem) is not semi-decidable and not decidable then we can say it's undecidable and I think it makes sense also based on diagram. However, in my ...
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0answers
61 views

Decidability of {(M,w); M terminates on input w and tape of M is empty after computation}

I am currently trying to prove whether the above language is decidable, partially decidable or fully undecidable. I am certain that this language is partially decidable and reducible to the halting ...
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108 views

Turing machine accepts two different strings

I am having hard time to proving this problem $C=\{\langle M \rangle \mid M \text{ is a Turing Machine , } L(M) \text { only contains two different strings}\}$ some ideas that i have tried are : i ...
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0answers
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Prove whether this language is (partially) decidable [duplicate]

I'm currently working on a few turing machine exercises and I can't understand how I can prove whether the below is at least partially decidable: $\{M \mid L(M) = \{x \mid |x| = 10\}\;\}$ where $|x|$ ...
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2answers
417 views

Showing that the language $L = \{\langle M, w \rangle\ |\ M$ moves left at least three times while computing $w \}$ is decidable or undecidable

How would you go about showing that the language $L = \{\langle M, w \rangle\ |\ M$ moves left at least three times while computing $w \}$ is decidable or undecidable? Intuitively my thoughts are ...
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2answers
86 views

Enumerable disjoint subsets whose union is equal to the union of the sets

I'm given that two sets, $A$ and $B$ are enumerable. I have to show that there exist subsets $A \supset C$ and $B \supset D$ ($C$ and $D$ also enumerable) such that $C$ and $D$ are disjoint and $A\cup ...
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3answers
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Are there any countable sets that are not computably enumerable?

A set is countable if it has a bijection with the natural numbers, and is computably enumerable (c.e.) if there exists an algorithm that enumerates its members. Any non-finite computably enumerable ...
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1answer
241 views

The Church-Turing-Thesis in proofs

Currently I'm trying to understand a proof of the statement: "A language is semi-decidable if and only if some enumerator enumerates it." that we did in my lecture. One direction of the proof goes ...
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0answers
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The set of words accepted by TMs simultaneously is finite is non semi-decidable

Consider $L = \{(M_1,M_2):\text{the set of words accepted by both TM at the same time is finite}\}$. I want to determine if this language is decidable, semi-decidable or not semi-decidable. My ...
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1answer
92 views

Semi decidability proof

I'm studying for my theory of computation exam and came accross the following question: Construct an appropriate Turing machine for the following language and prove or disprove it's semi-decidability:...
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1answer
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Why is the halting problem semi-decidable?

This is what is know about halting problem and semi-decidability :- Halting problem says that for a given input x and a machine H, we can't say whether the machine H halts or not on input x. A ...
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1answer
113 views

Unrecognizable languages relationship to NP-hard languages

I would like to know if there exists an NP-hard language which is also a member of co-RE\R? I think it depends if P=NP or not, but i'm not sure. Can I simply assume NP-hard is in R? Can you direct ...
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1answer
300 views

Is it decidable whether a Turing Machine will visit every non-end state from input x?

L = {w#x | w,x ∈{0,1}∗ and Turing Machine Mw with input x visits every non-end state at least once} I believe this problem is undecidable. My proof would consist of me reducing L to a Halting ...
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1answer
248 views

Deciding on decidability of a problem and reducing it to halting problem if not decidable

Given are two following languages: $L_1 = \{ w\#x \hspace{1mm} | \hspace{1mm} w,x \in \{0, 1\}^* \text{ where } M_w \text{ with input } x \text{ visits each of its states at least once} \}$ $L_2 = ...
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1answer
424 views

which of the following languages are Recursively Enumerable?

Which of the following languages are recursively enumerable? A={⟨M⟩∣ TM M accepts at most 2 distinct inputs} B={⟨M⟩∣ TM M accepts more than 2 distinct inputs} For first language I think that we can ...
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1answer
80 views

Recognizability of $\left\{ \left\langle M\right\rangle |M\text{ is a TM and }A_{TM}\leq\mathcal{L}\left(M\right)\right\} $

Determine whether the following language is decidable, recognizable but not decidable, co-recognizable but not decidable or neither recognizable nor co-recognizable. Prove your answer. $$L=\left\{ \...
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Why is $L=\{\langle M \rangle \mid |L(M)| \geq k\}$ not recognizable?

Here $M$ denotes a turing machine. By set theory, $L = \overline{E_{TM}} \cap \overline{L_0} \cap \overline{L_1}$ where $L_i=\{\langle M \rangle \mid |L(M)|=i\}$. And I know that $\overline{E_{TM}}$ ...
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1answer
337 views

Are recursively enumerable languages closed under shuffle?

For languages $L_1, L_2$ over some alphabet $\Sigma$, we define $$ \textit{Shuffle}(L_1, L_2) = \{a_1b_1a_2b_2 \cdots a_nb_n : n \geq 1 \wedge a_1, \ldots , a_n \in L_1 \setminus \{ \varepsilon \} \...
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2answers
117 views

Show that RE is closed against right-quotient

I have a problem that I have no idea how to approach. I've been looking at using mapping reductions, but I can't find a way to apply it. Assume some alphabet $\Sigma$ and two languages $A, B \...
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1answer
173 views

cardinality of recursive/r.e/not r.e languages? [duplicate]

I was just looking into properties of languages and wondered about the cardinality of them are all recursive languages countable or can they also be uncountable (can u have a recursive language which ...