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Questions tagged [semi-decidability]

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

A problem $L$ is decidable if there is an algorithm which, given an input $x$, halts and output whether $x \in L$ or not. In contrast, a problem $L$ is semi-decidable if there is an algorithm which, on input $x$, halts if $x \in L$, and runs into an infinite loop if $x \notin L$.

The quintessential example of a semi-decidable problem which is not decidable is the halting problem.

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Why are the total functions not enumerable?

We learned about the concept of enumerations of functions. In practice, they correspond to programming languages. In a passing remark, the professor mentioned that the class of all total functions (i.e. the functions that always terminate for every…
Raphael
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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 set must be countable since we can construct a…
Peter Olson
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undecidable problem and its negation is undecidable

A lot of "famous" undecidable problems are nonetheless at least semidecidable, with their complement being undecidable. One example above all can be the halting problem and its complement. However, can anybody give me an example in which both a…
Giulia Frascaria
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Why is the halting problem semi-decidable?

This is what is known about the halting problem and semi-decidability :- The 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 language is said to be Semi-decidable if there…
Zephyr
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Is it possible for a language and its complement to both be unrecognizable?

Given some unrecognizable language $L$, is it possible for its complement $\overline{L}$ to also be unrecognizable? If some other language $S$ and its complement $\overline{S}$ are both recognizable, then $S$ and $\overline{S}$ are decidable. If…
baffld
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Reduction from ATM to ATM-complement

Is there a reduction from ATM to ATM-complement? (ATM denotes the language $\{\langle M,w \rangle \mid \text{TM $M$ accepts $w$}\}$) I have been thinking about it too much and couldn't find the answer. I know that reduction from ATM-complement to…
Dvir
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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-recognizable language, then including the first…
templatetypedef
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How to prove that a language is not recursively enumerable

How does one prove that some arbitrary language $L$ is not recursively enumerable? I know I can prove that the language $L$ is recursively enumerable by constructing a Turing machine $M$ that accepts all words in the language (and the language would…
Smajl
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Is the set of TMs that does not reach most cells to the right computable?

Let $L_{NTF} = \{ \langle M \rangle \mid $ for every $x\in\Sigma^* $ the machine $M$ does not reach the $|x|+10$'th cell during its calculation on $x$. $ \}$. I would like to prove or disprove $L_{NTF} \in RE$. I know how to easily prove that…
Dan D-man
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Why is the class of recursively enumerable languages not closed under complementation?

I am having a hard time understanding closure properties of recrusively enumerable languages. I have read the explanation on this site but still unable to fully understand why they are not closed under complementation? Explanation also says, This…
c2h5oh
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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 total computable functions from N to N, such that…
agemO
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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 ${\displaystyle I}$ is infinite, but for every index ${\displaystyle…
shcolf
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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 it is in RE with a TM. I think its not in co-RE,…
Bar
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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 in stackexchange site. I have no problems in the reduction method. But, just trying to be…
Garrick
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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?
Garrick
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