Questions tagged [computable-analysis]
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32 questions
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Can a path in Kleene's $\mathcal{O}$ enumerate all of the computable reals via uniform diagonalization?
It's a well-known fact that there are computable diagonalization functions on Baire space $\mathbb{B} = \mathbb{N}^\mathbb{N}$ (i.e., functions which take a sequence $(r_i)_{i\in \mathbb{N}}$ of ...
- 12.4k
3
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87
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Is the probability distribution of a graphon given as a graph limit computable?
Let $G_i$ be a sequence of finite graphs that is Cauchy in the space of graphons. That is, for every $\epsilon \in \mathbb Q_+$ there is a $N \in \mathbb N$ such that $$\forall n, m > N. \delta_\...
- 6,353
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229
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Why does Weihrauch reducibility make use of multi-functions?
This is probably a kinda dumb question, but why is Weihrauch reducibility defined in terms of multi-functions (i.e. why isn't it just the degree structure of regular functions under that reducibility)?...
- 3,029
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Integration in polynomial time
The work of Friedman and Ko and
Müller guarantee the polynomial time computability of the integrals of analytic functions inside the circle of convergence. But do algorithms have practical value? Is ...
- 81
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Reference for a proof of Ceitin's theorem (Borel computable = Markov computable)?
Ceitin's theorem says that if a function $\mathbb R_c \to \mathbb R_c$ is Markov computable, then it is Borel computable (or TTE-computable). I find this theorem on Klaus Weihrauch's Computable ...
- 1,755
5
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1
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118
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Understanding the definition of a (computably / continuously) “transparent” function
The following definitions of a “transparent function” are essentially taken from references [1] (where it is called a “jump operator”), [2] and [3], except that the variation “primitively recursively ...
- 32.4k
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3
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392
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Intuition behind Kleene's “second algebra” $\mathcal{K}_2$
The “second Kleene algebra” $\mathcal{K}_2$ is defined, e.g. here on nLab, or in section 1.4.3 of van Oosten's book Realizability: an Introduction to its Categorical Side (2008), or as example 3.4 of ...
- 32.4k
2
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1
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145
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Computability of fillability of unit cube in $\mathbb{R}^n$ by $k$ $\varepsilon$-balls
Let $\mathbb{N}$ denote the set of positive integers. We define a relation $R \subseteq \mathbb{N}^4$ in the following way:
$(p,q,n,s)\in R$ if and only if there is $S\subseteq [0,1]^n$ with $|S| = s$...
- 48.8k
3
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1
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172
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LLPO as constructivity/computability for dense subsets
LLPO is the statement $\forall x \in \mathbb R. x \leq 0 \vee x \geq 0.$ The statement should be understood as a fragment of the Law of Excluded Middle, rather than a statement about the ordering of ...
- 4,943
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Is the one-point compactification of $\mathbb{N}$ computably countable?
The one-point compactification $\mathbb{N}_\infty$ of $\mathbb{N}$ is obtained from the discrete space $\mathbb{N}$ by adjoining a limit point $\infty$. It may be identified with the subspace of ...
- 48.8k
8
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Are representations in computable analysis the equivalent to countably-generated condensed sets?
This is the first in a pair of questions. For the other see here.
Dustin Clausen and Peter Scholze have a theory of condensed sets, which is a slightly different take on topology. For most cases, ...
- 6,287
6
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1
answer
421
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Computing the complex roots of a monic polynomial
The map from monic complex polynomials to the unordered tuples of their roots (each appearing according to its multiplicity) is computable. This seems to have been known for a long time, and with ...
- 4,667
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306
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Coding third-order objects via second-order ones
As is well-known, the language of second-order arithmetic only has variables for natural numbers and sets of natural numbers. Higher-order objects, like functions on $\mathbb{R}$, have to be ...
- 4,359
2
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Classification and computation of the entanglements of pairs of planar curves
Let $C$ is the set of continuous $f:[0;1]\to \mathbb R^2$ with $\|f\|=\max_t\|f(t)\|$. For $f,g\in C$ let $(f,g)\in E$ iff $\{f(0),f(1)\}\cap {\rm Range}(g)=\emptyset$ and $\{g(0),g(1)\}\cap {\rm ...
4
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1
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Is the Hilbert space-filling curve bijective over computable numbers?
The Hilbert curve is a continuous space-filling curve that maps $I$ to $I^n$ where $I$ denotes the unit interval [1]. Like all other space-filling curves, it is not one-to-one. I am wondering if the ...
1
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1
answer
164
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Partial computability results on integrals over open intervals
It's well known in Type 2 Effectivity that integration over a compact interval is computable. So what about integration over an open interval? What rigorous computability results exist?
My thoughts ...
- 4,943
20
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2
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2k
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Is it possible to constructively prove that every quaternion has a square root?
Is it possible to constructively prove that every $q \in \mathbb H$ has some $r$ such that $r^2 = q$? The difficulty here is that $q$ might be a negative scalar, in which case there might be "too many"...
- 4,943
2
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103
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Buridan's principle in computable analysis
In (Lamport, 2012), Lamport proposes the principle
A discrete decision based upon an input
having a continuous range of values cannot be made within a bounded length of time.
I think it could be ...
13
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1
answer
364
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Is there a computable homeomorphism between two different Cartesian powers of the computable real numbers?
It's well know that it is surprisingly difficult to prove that $\mathbb{R}^n$ and $\mathbb{R}^m$ are not homeomorphic for $n\neq m$. Commonly proofs go through Brouwer's fixed point theorem, which is '...
- 12.4k
3
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186
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Banach spaces: A ball being a subset of the interior of the union of two balls
Let $X$ be a separable reflexive Banach space and let $A$, $B_1$, and $B_2$ be three closed balls in $X$. Is there a `handy' necessary and sufficient condition for checking whether $A \subseteq (B_1 \...
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221
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Are these two definitions of arithmetical hierarchy of real numbers equivalent?
Zheng and Weihrauch (http://www-sst.informatik.tu-cottbus.de/~wwwti/zheng/publications/1999/mfcs99.pdf) define a real number $x$ to be $\Sigma_n$ if and only if there is a computable function $f:\...
- 2,130
4
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1
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Is the following product-like space a Polish space?
Let $\mathcal{M}_1(\mathbb R)$ denote the space of Borel probability measures on $\mathbb R$. The space is a Polish space (a space which admits a complete, separable, metric) using, say the Levy-...
- 6,287
3
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1
answer
446
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floating point representation via the perspective of TTE/computable analysis
Floating point numbers are not compatible with the usual theory of type 2 theory of effectivity (TTE), and not even the real-RAM model; there are functions that are computable in one model but not ...
- 459
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2
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2k
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Is this property equivalent to Lusin's property (N) for continuous functions?
A function $F:[0,1]\rightarrow\mathbb{R}$ satisfies Lusin's (N) property if for every measure zero set $A\subseteq [0,1]$, $F(A)$ has measure zero. (This includes the assertion that $F(A)$ is ...
- 1,601
17
votes
1
answer
874
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An NP-hard $n$ fold integral
We are given rational numbers $[c_1, c_2, \ldots, c_n]$ and $v$ from the interval $[0,1]$.
Consider the $n$-fold integral
$$
J = \int_{\theta_1 \in I_1, \theta_2 \in I_2 \ldots, \theta_n \in I_n} d\...
- 627
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Can nonstandard analysis be used to prove results in constructive or computable analysis?
Nonstandard analysis is a useful tool which can be used to prove a number of results in analysis.
Question
Can it also be used to prove results in computable or constructive analysis?
If so, what are ...
- 6,287
7
votes
2
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417
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tennenbaum phenomena for the reals?
Let $\mathfrak{M} = \langle R, +,\times,> \rangle$ be such that $R$ is the set of real numbers and $\mathfrak{M} \models RA^1$ (the first-order axioms for the reals). Do we have characterisations ...
- 71
11
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1
answer
420
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The complexity of the leading fractional bit of a power of a rational number
On a mailing list (math-fun) that I subscribe to Dan Asimov asked what's the most efficient way to calculate the leading decimal digits (say 10 of them) of $(p/q)^n \bmod 1$ where $p$ and $q$ are ...
- 4,636
5
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Simple example of a sequence without computable modulus of convergence
Can anyone give a simple example of a sequence that converges, but there's no computable function that gives $N$ as a function of $\epsilon$, i.e., the modulus of convergence is not computable?
In ...
- 702
2
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0
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332
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Computable distribution on [0,1] with C-infinity distribution function
Does anyone know of an easily-describable distribution on $[0,1]$ with a density $p$ (with respect to Lebesgue measure) that satisfies the following properties:
$p$ is $C^\infty$
$p(0) = a$, $p(1) = ...
- 2,221
5
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5
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Uncomputability of the identity relation on computable real numbers
Let $f_{=}$ be a function from $\mathbb{R}^{2}$ be defined as follows:
(1) if $x = y$ then $f_{=}(x,y) = 1$;
(2) $f_{x,y} = 0$ otherwise.
I would like to have a proof for / a reference to a textbook ...
- 61
13
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3
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Differentiability of computable functions
Call a computable function a total function $\mathbb{R} \to \mathbb{R}$, for which there exists a Turing machine outputting arbitrary close approximation to $f(x)$ given arbitrary close approximation ...
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