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]
2536 questions
169
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
- 468,795
90
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
- 1,219
71
votes
3 answers
Recognizable vs Decidable
What is difference between "recognizable" and "decidable" in context of Turing machines?
metdos
- 1,007
61
votes
2 answers
Are there any Turing-undecidable problems whose undecidability is independent of the Halting problem?
To be more specific, does there exist a decision problem $P$ such that
given an oracle machine solving $P$, the Halting problem remains undecidable, and
given an oracle machine solving the Halting problem, $P$ remains undecidable?
David Zhang
- 9,125
- 2
- 43
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47
votes
1 answer
Computability viewpoint of Godel/Rosser's incompleteness theorem
How would the Godel/Rosser incompleteness theorems look like from a computability viewpoint?
Often people present the incompleteness theorems as concerning arithmetic, but some people such as Scott Aaronson have expressed the opinion that the…
user21820
- 60,745
44
votes
7 answers
In what sense does a number "exist" if it is proven to be uncomputable?
Uncomputable functions: Intro
The last month I have been going down the rabbit hole of googology (mathematical study of large numbers) in my free time. I am still trying to wrap my head around the seeming paradox of the existence of natural numbers…
Andreas Tsevas
- 2,639
- 6
- 15
43
votes
5 answers
Why do we believe the Church-Turing Thesis?
The Church-Turing Thesis, which says that the Turing Machine model is at least as powerful as any computer that can be built in practice, seems to be pretty unquestioningly accepted in my exposure to computer science. Why? Do we have any more…
GMB
- 4,256
39
votes
3 answers
Can someone explain the Y Combinator?
The Y combinator is a concept in functional programming, borrowed from the lambda calculus. It is a fixed-point combinator. A fixed point combinator $G$ is a higher-order function (a functional, in mathematical language) that, given a function $f$,…
Chris Taylor
- 29,755
39
votes
6 answers
Example of uncomputable but definable number
Every computable number is definable. However, the converse is not true.
What is an example of a real number that is definable but that is NOT computable? I guess if it is there, we can "define" (describe) it, can't we?
islamfaisal
- 598
33
votes
2 answers
Is there a "true" value of BB(745)?
I was reading this reddit thread and I got confused by one part. I always thought that there is always a "true" value of BB(n), even though it might not be provable or findable.
There is a 745-state Turing Machine that halts iff ZFC is inconsistent.…
David Lui
- 6,899
32
votes
6 answers
Is it possible to solve any Euclidean geometry problem using a computer?
By "problem", I mean a high-school type geometry problem.
If no, is there other set of axioms that allows that?
If yes, are there any software that does that?
I did a search, but was not able to find a single program that allows that. It is strange…
Artium
- 1,018
30
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
- 15,346
29
votes
5 answers
Are some real numbers "uncomputable"?
Is there an algorithm to calculate any real number. I mean given $a \in \mathbb{R}$ is there an algorithm to calculate $a$ at any degree of accuracy ?
I read somewhere (I cannot find the paper) that the answer is no, because $\mathbb{R}$ is not a…
Ricky Bobby
- 723
28
votes
2 answers
Are transcendental numbers computable?
Wikipedia states: "The computable numbers include many of the specific real numbers which appear in practice, including all real algebraic numbers, as well as e, π, and many other transcendental numbers."
I remember my professor saying incomputable…
joker
- 449
27
votes
0 answers
What functions from $\Bbb N\to\Bbb N$ can be made from $+$, $-$, $\times$, $\div$, exponentiation, and $\lfloor\cdot\rfloor$?
What functions from $\Bbb N\to\Bbb N$ can be made from $+$, $-$, $\times$, $\div$, exponentiation, and $\lfloor\cdot\rfloor$? Call this class of functions $\mathcal Flex$ (for "floor and exponentiation").
The $\rm mod$ function…
Akiva Weinberger
- 25,412