# Questions tagged [semi-decidability]

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### 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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### 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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### 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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### 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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### 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 ...
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### How can I prove that there is a decidable language which is not in P?

Generally, I want to use the diagonal argument to prove it. I tried to define a language $A$ which is constructed by a Turing machine $D$: It will only take a input which has a form of a ...
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### Semidecidability/Decidability for strings seperated by a non alphabet symbol

I am trying to prove the following: Let $\Sigma$ be an alphabet not containing the symbol "$;$", and suppose that $L \subseteq \Sigma^*$; $\Sigma^*$ is recursively enumerable. If this is the case ...
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### Which Turing machine problems are Decidable?

Let $M_0$, $M_1$, $M_2$,..., be an effective enumeration of all Turing machines. Which of the following problems is (are) decidable ? Given a natural number $N$, does $M_N$ starting with an empty ...
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### Complexity of the language of all TMs $M$ such that $L(M)$ is decidable

Let $$R = \{\langle M \rangle \mid L(M) \text{ is decidable}\}.$$ Is $R$ recursively enumerable or co-recursively enumerable?
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### Is undecidability of TMs' properties a statistical statement?

We know (by Rice's theorem) that is it not possible to decide a non-trivial property of a given TM. We could say therefore that we cannot be sure at 100 percent that a given TM has a certain non-...
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### Given language consisting Turing machines is decidable or not?

$$L=\{M\mid \text{there exist }x,y\in\Sigma^* \text{ s.t. }x \in L(M)\text{ and } y \notin L(M)\}\,.$$ I think it's not recursively enumerable because this language reduces to complement of the ...
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### Showing an equivalent definition of $CO-RE$

Let $L_1$ be a language. I would like t prove that $L_1\in CO-RE$ if and only if exists a language $L_2$ s.t. $L_1=\{ u\,\, |\,\, \forall_{v\in\Sigma^*} : <u,v>\in L_2 \}$ and $L_2\in R$. I'm ...
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### Partial recursive function with no total recursive extension

We define a partial recursive function $f:\{0,1\}^* \longrightarrow \{0,1\}^*$ to be semi-good if we can define a total recursive function $g:\{0,1\}^* \longrightarrow \{0,1\}^*$ from $f$, such for ...
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### Definition of an immune set

I'm reading a theorem about existence of a simple set. The definition of an immune set can be found from here A set ${\displaystyle I\subseteq \mathbb {N} }$ is called immune if ${\displaystyle I}$ ...
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### how to prove that the property 'doesn't halt for some input' is not semidecidable

I am taking a computer theory class and one of the exercises is to prove that the property "doesn't halt for some input" is not semidecidable. This property is the negation of the property "halts for ...
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### Language of TMs that accept some x in less than 50 steps. Is it in co-RE?

L = {M | M is a TM and there exists an input that the TM M accepts in less than 50 steps} I need to find a minimal class it belongs to between R/ RE/ co-RE/ not in RE∪co-RE. I managed to show that ...
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### Decidability of $\{p|∃y : \operatorname{Dom}(φ p ) ⊇ \operatorname{Dom}(φ y )\}$

I need to classify the set $$\{p|∃y : \operatorname{Dom}(φ p ) ⊇ \operatorname{Dom}(φ y )\}$$ as decidable, semidecidable or not semidecidable. I don't know how to start. Any ideas? Dom (φp) it's ...
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### Is the given language decidable for turing machine?

I am going through undecidability of TM and found this question $L=\left \{ \left \langle M \right \rangle |M\ is\ TM \ and \ number\ of\ strings\ in\ the\ language\ \ is\ prime\right \}$ I think it ...
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### Is the Rice Theorem applicable for these problems?

I have 1 problem :--> L = { < M > | TM halts on no inputs } I have solved the above problems by reductions given in the book and even there are many links ...
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### Does it matter for this function if the set we check membership of is finite?

I have the following problem. Let $\Phi$ be an admissible numbering of the single-parameter partially-recursive functions. That is, $\Phi(i, x) = f_i(x)$ with $f_i$ the $i$th partially-recusive ...
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### Dovetailing in Turing Machines?

I was going through TM here and encountered a term "Dovetailing". What is exactly the dovetailing in Turing Machines? Can anyone mind to provide a good explaination with some examples?
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### Is the given language decidable?

L = { < M > | M is a turing machine and } Obviously, the language which L(M) is polynomially reducible to, is context free and hence recursive, so it is a decidable language . Now, L(M) is ...
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### Is the set of pairs of TMs at least one of which accepts the empty word semi-deciable?

L = { < M1,M2 > | M1,M2 are TM's and Ɛ ∈ L(M1) ∪ L(M2) } Where Ɛ = Epsilon I know that this language is undecidable, but why it is semidecidable too. What i have tried is => Using Rice's ...
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### Is this language semi-decidable?

Let $M_w$ be the DTM encoded by the binary string $w$ and let $$L=\{w\#x\,|\,\text{all states are reached when running }M_w\text{ on }x\}.$$ I've already proved that this language is undecidable (the ...
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### Are there any RE-complete languages w.r.t. polynomial reduction?

I need to decide if there exists $L\in RE$ so that for every $L'\in RE$ we have $L' \leqslant_p L$, meaning a polynomial-time reduction. I've tried to use $L=A_{TM}$ (the accepting problem), but got ...
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### How to argue about semi-decidability of a problem?

I have the following problem, and I try to prove that it is semi-decidable, but I have a hard time arguing about it. I know that if a problem $\mathcal{P}$ is semi-decidable, then we can build a ...
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### Undecidable language and Turing Machines

I am reviewing some old papers for a final tomorrow, and there is a question that I'm not sure about. If a language A is Turing-Recognizable and Undecidable, what can be said of the Turing-Machine ...
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### Is English Recursively enumerable? [closed]

The title says it all. I've tried digging up debate on this issue to see a proof one way or the other but it doesn't look like anyone is able to say whether or not it is. Clearly there are recursive ...
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### Showing that $H'$ is not semi-decidable

I have an introductory class in computability theory and I'm currently working on my first exercises. I'm wondering if I'm on the right track with proving undecidable languages. Could you please have ...
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### A non-mechanical way to get an infinite decidable subset of a Turing-recognizable language?

There's a famous theorem that every infinite Turing-recognizable language has an infinite decidable subset. The standard proof of this result works by constructing an enumerator for the Turing-...
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### set of Kolmogorov-random strings is co-re

given RC = {x : C(x) ≥ |x|} is a set of Kolmogorov-random strings. How can I show that RC is co-re I have been reading this paper What Can be Efficiently Reduced to the Kolmogorov-Random Strings?...
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### Mapping reducibility vs. Turing reducibility

Let $A$ and $B$ be two languages. If $A \le_{m} B$ ( reducible by mapping ) then I know that if $B$ is decidable so is $A$ and if $B$ is recognizable so is $A$. And if $A \le_{T} B$, then if $B$ is ...
I know that $L=\{ \langle M \rangle \mid |L(M)| < \infty \}$ is not decidable (by Rice's theorem or using reduction, I followed it from $L$ not being decidable ). But is $L$ recognizable? What I ...
What would be the general approach for that? And what are the things that generally overlooked while proving such things? For example, I have a Language, L ={e:$L(M_e)$ such that it accepts only 'a ...