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Questions tagged [computability]

Questions related to computability theory, a.k.a. recursion theory

The theory of computability studies which functions can be computed using which machine models.

Computability theory (also known as recursion theory) is a branch of mathematical logic and computer science. It is mainly concerned with computability.

When some resource is restricted (such as time or space), then the tag is more appropriate.

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Is busy beaver the fastest growing function known to man?

I just had this interesting question. What is the fastest growing function known to man? Is it busy beaver? We know functions such as $x^2$, but this function grows slower than $2^x$, which in turn grows slower than $x!$, which in turn grows slower…
bodacydo
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In what sense is the Mandelbrot set "computable"?

The Mandelbrot set is a beautiful creature in Mathematics. There are a lot of beautiful images of this set created with high precision, so obviously this set is "computable" in some sense. However, what concerns me is the fact that it is not even…
Earth Engine
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Can a computable function converge to an uncomputable number?

Does there exist a computable function $f:\mathbb{N}\rightarrow \mathbb{Q}$ such that: For all $t\in\mathbb{N}: 0\le f(t) < X$ $\lim\limits_{t\rightarrow\infty} f(t) = X$ Where $X$ is an uncomputable real number. The only reference to this…
cjnash
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Easy to state open problems in computability theory

I was searching for interesting and easy to state open problems in computability (understandable by undergraduate students taking their first course in computability) to give examples of open problems (and obviously I want the students to be able to…
Kaveh
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Turing complete and computational power

In a lecture a professor mentioned that modern computers don't have as much computational power as a Turing machine because they don't have infinite memory, and since no computer can have infinite memory the Turing machine is therefore unattainable…
JustAnotherSoul
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Is smn theorem the same concept as currying?

I am studying the s-m-n theorem and the concept reminded me of currying. From wikipedia article about s-m-n theorem: the theorem says that for a given programming language and positive integers m and n, there exists a particular algorithm that…
emanek
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Can a RAM calculate its own Gödel number?

You can get the Gödel number of a RAM by making it a list of commands and making this list an integer. So, what I thought is something like "The RAM that would return its own Gödel number (say, $x$) would have to have the information $x$ in it, so…
palsch
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Given computable function, what are conditions for computability of inverse function?

If $f:\mathbb{N}\rightarrow\mathbb{N}$ is computable and has an inverse, under what conditions is $f^{-1}$ also computable? I couldn't find that in a textbook, and googling gets some vague suggestions about bijective, but I couldn't find a clearly…
John Forkosh
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Sub language is not Turing-recognizable, or could it be?

Let A and B be languages with A ⊆ B, and B is Turing-recognizable. Can A be not Turing-recognizable? If so, is there any example?
gfe
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Are "computable problem" and "computable function" the same thing?

I'm confused by the use of the expressions "computable problem" and "computable function" in the context of computability theory. Are they refer to the same thing or are there differences?
Sanyo Mn
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Collection of real world non computable programming tasks

I'm quite new to computability theory and would be interested in some examples of realistic programming tasks you could encounter when working as a C++ programmer that are not computable.
Graf Aal
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Clarification of Theorem 1.11 in the book Computational Complexity by Arora/Barak

The book Computational Complexity: A Modern Approach by Arora/Barak provides the following: Theorem 1.10 There exists a function $UC: \{0,1\}^* \to \{0,1\}$ that is not computable by any TM. Theorem 1.11 HALT(The halting problem) is not computable…
lgidwani
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Are these sets of indices also index sets?

An index set is a set of all indices of some family of computably enumerable sets. It is known that the empty set is an index set and that $K = \{e \mid e \in W_e\}$ is not an index set. The following was an exam question I didn't manage to…
girlonbeach
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How can primitive recursion be a special case of minimization?

In several posts on StackExchange and elsewhere, I have seen claims that you only need to show you can construct constants, successor, projection, composition, and minimization to prove a language Turing-complete, since primitive recursion is…
Circuit Craft
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What is the complement of a language?

If given any language L, how do I find the complement of said language? I lack the basic understanding required to determine if a language is co-recognizable. I understand that a language, $L$, is co-recognizable if the complement of that language,…
Jakir
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