Questions about Turing computability and recursion theory, including the halting problem and other unsolvable problems. Questions about the resources required to solving particular problems should be tagged (computational-complexity).
Questions tagged [computability]
2395 questions
167
votes
1 answer
What properties of busy beaver numbers are computable?
The busy beaver function $\text{BB}(n)$ describes the maximum number of steps that an $n$-state Turing machine can execute before it halts (assuming it halts at all). It is not a computable function because computing it allows you to solve the…
Qiaochu Yuan
- 419,620
83
votes
6 answers
Are there any examples of non-computable real numbers?
Is this true, that if we can describe any (real) number somehow, then it is computable?
For example, $\pi$ is computable although it is irrational, i.e. endless decimal fraction. It was just a luck, that there are some simple periodic formulas to…
Dims
- 1,149
29
votes
1 answer
How to interpret "computable real numbers are not countable, and are complete"?
On page 12 of this (controversial) polemic
http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf
Wildberger claims that
Even the "computable real numbers" are quite misunderstood. Most mathematicians reading this paper suffer from the…
Stephen
- 14,811
20
votes
3 answers
What is the difference between total recursive and primitive recursive functions
I am studying the theory of computation. Here are some terminologies that I am confused about.
Are total recursive function and primitive recursive function equivalent? I think they are equal because their domains are both total and they always…
Rambo
- 303
16
votes
1 answer
When is Chaitin's constant normal?
Chaitin's constant is not one constant, but depends on an effective prefix-free encoding $d$ of Turing machines as bit strings. Once such an encoding is chosen, the corresponding Chaitin's constant is
$$ \sum_{T\text{ halts on empty tape}}…
hmakholm left over Monica
- 286,031
15
votes
2 answers
Are there any examples of non "weakly computable" numbers?
When I first learned about computable numbers, I misunderstood the (informal) definition, thinking it was this: a number $x$ is computable if there exists a turing machine that outputs a sequence of numbers that converges to $x$. The real definition…
Ynir Paz
- 609
14
votes
1 answer
Are total recursive functions recursively enumerable?
In quite some literature I found that primitive recursive functions are recursively enumerable, but total recursive ones are not. To what set do total recursive functions belong?
I am asking this since I learned that even the halting problem is in…
zpavlinovic
- 381
- 2
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12
votes
3 answers
Must a function that 'preserves r.e.-ness' be computable itself?
Does there exist a non-recursive function (say, from naturals to naturals) such that the inverse of every r.e. set is r.e.?
If yes, how to construct one?
If no, how to prove that?
Any References?
Anonymous
- 123
9
votes
3 answers
what is actually a "definable" real number?
Wikipedia states that a definable (real) number is a number "that can be uniquely specified by its description". I naïvely thought that this was the same as "a number which may be computed", but elsewhere in math.SE I found out that it is not the…
mau
- 9,774
9
votes
4 answers
Example of a number that is not the limit of a computable sequence
Let's define a real number as computable iff there's an algorithm that can generate a sequence with the number as its limit (turing machine or any of the equivalent programming models).
Not all real numbers fall into this category. In particular,…
John
- 203
8
votes
0 answers
Mathematical intro to Turing machines
Is there a good mathematically rigorous introduction to computability theory based on Turing machines? I have looked at some CS books but found them quite unsatisfying for a mathematician (too wordy in the chatty parts, too vague when it comes to…
user138530
8
votes
1 answer
How many busy beavers with the same number of states
Can the number of busy beavers with n states be computed, or would it be necessary to analyze all the machines to count them ?
Peter
- 84,454
7
votes
1 answer
Asymptotic bound to all computable functions lower than the Busy Beaver function
The busy beaver function $BB$ asymptotically bounds any computable function. It is easy to show that there are lower bounds, for example, $log(BB)$.
Is there a function $f$ that asymptotically bounds all computable functions, so that $BB$ is not…
Dani
- 339
7
votes
2 answers
Intuitive meaning of the concept “computable”
My question is a follow-up question to this one: How to show that a function is computable?
The original question was:
Is the following function $$g(x) = \begin{cases} 1 & \mbox{if } \phi_x(x) \downarrow \mbox{or } x \geq 1 \\ 0 & \mbox{otherwise }…
Xaver
- 586
7
votes
3 answers
Fast-growing noncomputable functions
A famous 1962 paper by Tibor Radó shows that the "busy beaver" function $h$ (which computes the maximal number of steps for which a halting Turing machine with $n$ states can run for) satisfies the property
(A) for every computable total function…
Gro-Tsen
- 5,541