Questions tagged [computing-over-reals]

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Relation between BSS and Turing models

$P_\mathbb R$ is the set of languages decidable in polynomial time over the real $BSS$ machine defined in https://en.wikipedia.org/wiki/Blum%E2%80%93Shub%E2%80%93Smale_machine. Let $0-1-P_\mathbb R=\{...
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Is the Fermat-Weber problem $\mathsf{NP}$-hard?

Given a set of $n$ points in a Euclidean space, the Fermat-Weber problem asks to find a center that minimizes the sum of distances of points to that center. There are iterative algorithms known for ...
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6 votes
1 answer
144 views

Complexity of approximating a real function using queries

Consider the following computational problem, where $I$ is the real interval $[-1,1]$: There is a monotonically-increasing function $f: I\to I$. You are allowed to access it only through queries of ...
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5 votes
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Complexity of a specific class of definite integrals

INTRODUCTION: From the answer to this question I learned that deciding whether a definite integral is $0$ or not can be NP-complete, as the following integral representation of the Number Partition ...
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A complexity-class of problems that cannot be solved in finite time

Is there a complexity class for problems such as the following? Problem FindSum(s), where $s\in[0,1]$ is a real parameter: Input: $g: [0,1]\to [0,1]$, a continuous monotone-increasing bijective ...
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1 answer
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How can theoretical modelling be converted into viable product? [closed]

Many a times modelling is just for the sake of modelling like exploring some properties of materials which does not help in converting the theoretical modelling into a viable product that can serve ...
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3 votes
1 answer
897 views

Examples of Fat-Shattering Dimension

What are some good examples for analysis of a class's Fat-Shattering dimension? By (Alon et al) I know that the Fat-Shattering Dimension characterizes the learnability of real-valued function classes ...
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1 answer
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Characterisation of computability of partial functions from infinite words into finite words by functions with prefix-free domain

The following is taken from K. Weihrauch, Computable Analysis, page 21. The notation $f : \subseteq A \to B$ means a partial function. By $\Sigma^{\omega}$ and $\Sigma^{\ast}$ we denote the set of ...
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12 votes
2 answers
737 views

Euclidean TSP in NP and square root complexity

In this lecture notes by Ola Svensson: http://theory.epfl.ch/osven/courses/Approx13/Notes/lecture4-5.pdf, it is said that we don't know if Euclidean TSP is in NP: The reason being that we do not ...
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  • 363
2 votes
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136 views

Are there analogues of Specker sequences for other complexity classes?

Consider the standard definition of computable real numbers: a real number $r$ is computable just in case $r$ is the limit of a sequence $(a_i)_{i \in \mathbb{N}}$ such that (1) the function $i \...
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3 votes
1 answer
277 views

Zero of a multivariate cubic equation

Given a multivariate cubic equation $f(\vec{x})$ over reals where all coefficients of $f$ are integers, and a hyper-rectangle $B=\bigwedge_i x_i\in[a_i, b_i]$ where $a_i, b_i$ are constant integers, ...
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NP Intermediate problems over Reals

While studying ${\bf NP}$ complete problems we have from Ladners' theorem - if ${\bf P}$ $\neq$ ${\bf NP}$-there are ${\bf NP}$ problems not in the class ${\bf P}$ nor ${\bf NP}$-complete. Ladners' ...
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19 votes
3 answers
612 views

Computation of reals: floating point vs TTE vs domain theory vs etc

Currently, computation of reals in most popular languages is still done via floating point operations. On the other hand, theories like type two effectivity (TTE) and domain theory have long promised ...
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2 votes
0 answers
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Is the nonnegativeness of a polynomial hard for $\mathsf{NP}_\mathbb{R}$?

It is clear that the following problem is in $\mathsf{NP}_\mathbb{R}$. Input: a list $P$ of triplets $(a,s,t)$ where $s$ and $t$ are nonnegative integers. Output: is there an $x\in \mathbb{R}$ such ...
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1 vote
1 answer
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Is there any research on approximation of reals with computable numbers

I was wondering if there is any research in the field of Diophantine Approximation using the computable numbers. It seems to be a good fit, a dense countable set with a variety of different potential ...
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Computing log of sum of positive integers

As input, we are give $k$-bit approximation (after the decimal point) of $\log(a)$ and $\log(b)$ for positive integers $a$ and $b$, i.e, we are given $\alpha$ and $\beta$ (as binary strings) as input ...
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11 votes
2 answers
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How to judge the definition of computational complexity of reals is natural or suitable?

As we know, definition of computational complexity of algorithm is almost without controversy, but the definition of computational complexity of reals or the computation models over reals is not in ...
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10 votes
1 answer
327 views

Berman-Hartmanis Isomorphism for NP$_{\mathbb{R}}$?

Using the real-RAM/BSS model, we have the class NP$_{\mathbb{R}}$, (where a BSS is the Blum-Shub-Smale model of a computer with operations over reals). We have NP$_{\mathbb{R}}$ complete problems. ...
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8 votes
1 answer
459 views

The exponential function over algebraic numbers

Given an algebraic number $\alpha$, I am interested in finding an approximation of $\Re(e^\alpha)$ up to a given precision, where $\Re()$ refers to the real part of the complex number. Formally, I ...
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16 votes
1 answer
960 views

To what extent can the mathematics of Reals be applied to Computable Reals?

Is there a general theorem that would state, with proper sanitization, that most known results regarding the use of real numbers can actually be used when considering only computable reals? Or is ...
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7 votes
1 answer
281 views

What complexity issues are there in considering quantum algorithms with infinite gate-sets?

Short Version Suppose that you want to consider a model of quantum computation in which the gates used in the circuits may depend on the input size. Are there pitfalls to avoid when defining the ...
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14 votes
1 answer
303 views

NP completeness over reals

I am studying the BSS model of computation recently (cf. for instance Complexity and Real Computation; Blum, Cucker, Shub, Smale.) For reals $R$, it is shown that, given a system of polynomials $f_1,\...
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2 votes
0 answers
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Is this question $NP_R$ hard?

Consider $n$ variables $x_1, \cdots, x_n$ and $f=\sum a_i x_1^{d_{i1}}\cdots x_n^{d_{in}}$ such that for each $i$, $d_{i1}+\cdots+d_{in}=d$ for some fixed $d$ and $a_i\geq 0$. I am interested in the ...
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7 votes
1 answer
394 views

Uniformity vs. nonuniformity in algebraic complexity theory

I understand that the study of Boolean circuits and nonuniform complexity classes was introduced to (hopefully) prove separation of uniform complexity classes. In this sense, nonuniform computational ...
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1 vote
2 answers
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Foundational textbook(s) for Complexity and Computability on Real Numbers

It would be extremely helpful if someone can suggest foundational textbooks on Recursive Analysis (Computability over Reals) which explains connections between Computability and the Topological ...
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6 votes
3 answers
1k views

Computing Functions with Dynamical Systems

I was trying to make a set of differential equations "compute" some given function just like a Turing Machine does. Essentially, a given Turing Machine with an initial configuration (which includes ...
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21 votes
5 answers
5k views

Is it possible to test if a computable number is rational or integer?

Is it possible to algorithmically test if a computable number is rational or integer? In other words, would it be possible for a library that implements computable numbers to provide the functions <...
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  • 345
6 votes
2 answers
356 views

Oracle complexity of a problem in the Counting Hierarchy

In "On The Complexity of Numerical Analysis" (SIAM J. Comp. Vol. 38, 2009), Allender et al. introduce the problem of PosSLP and show that its complexity lies in the counting hierarchy, and more ...
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1 vote
1 answer
293 views

Concerning decidability of a problem on real numbers [closed]

This question is an outgrowth of a certain maths problem I've been thinking about. Suppose you use an oracle to represent a real number. The oracle is of the following form: you give it an integer ...
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18 votes
2 answers
7k views

Complexity of computing the discrete Fourier transform?

What is the complexity (on the standard integer RAM) of computing the standard discrete Fourier transform of a vector of $n$ integers? The classical algorithm for fast Fourier transforms, ...
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2 votes
0 answers
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mean/integral, except where positive differences between values and "mean" are weighted differently from negative differences?

Given a curve $f(x)$ (for $x \in [0,1]$), and a line $y=a$, let $U$ be the total area below $f$ and above $a$, and let $L$ be the total area above $f$ and below $a$. If $L=U$, this means that $a =\...
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10 votes
1 answer
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Reference for the undefinability of modulus of continuity functional in PCF?

Can someone point me to the reference for the non-definability of the modulus of continuity functional in PCF? $\newcommand{\N}{\mathbb{N}}$ $\newcommand{\bool}{\mathsf{bool}}$ Andrej Bauer has ...
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6 votes
2 answers
499 views

Iterative algorithms in algebraic complexity (Blum-Shub-Smale-Model)

I know the Blum-Shub-Smale model. It is claimed to provide a theoretical framework for algorithms in real and complex algebra and analysis. A very general question: Most algorithms compromise of ...
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1 vote
1 answer
157 views

Using a Polynomial Time Algorithm for Upper Bound Recognition to Show Polynomial Time for Evaluation?

Let's say I had an optimization problem $$ \min_{x \in D} f(x) $$ Where $D \subset \mathbb{R}^n$ and $f:\space D \rightarrow \mathbb{R}$, and the minimum is said to exist. Imagine I had a ...
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40 votes
2 answers
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Sum-of-square-roots-hard problems?

The sum of square roots problem asks, given two sequences $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_n$ of positive integers, whether the sum $\sum_i \sqrt{a_i}$ less than, equal to, or greater ...
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22 votes
1 answer
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Complexity of computing shortest paths in the plane with polygonal obstacles

Suppose we are given several disjoint simple polygons in the plane, and two points $s$ and $t$ outside every polygon. The Euclidean shortest path problem is to compute the Euclidean shortest path ...
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16 votes
2 answers
4k views

Time complexity of Bellman-Held-Karp algorithm for TSP, take 2

A recent question discussed the now-classical dynamic programming algorithm for TSP, due independently to Bellman and Held-Karp. The algorithm is universally reported to run in $O(2^n n^2)$ time. ...
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31 votes
3 answers
2k views

Consequences of existence of a strongly polynomial algorithm for linear programming?

One of the holy grails of algorithm design is finding a strongly polynomial algorithm for linear programming, i.e., an algorithm whose runtime is bounded by a polynomial in the number of variables and ...
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  • 2,679
43 votes
3 answers
4k views

What are the reasons that researchers in computational geometry prefer the BSS/real-RAM model?

Background The computation over real numbers are more complicated than computation over natural numbers, since real numbers are infinite objects and there are uncountably many real numbers, therefore ...
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27 votes
6 answers
2k views

How are real numbers specified in computation?

This may be a basic question, but I've been reading and trying to understand papers on such subjects as Nash equilibrium computation and linear degeneracy testing and have been unsure of how real ...
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1 vote
3 answers
908 views

Complexity of a variant of the Mandelbrot set decision problem?

Mandelbrot set is defined using the complex equation $P_c (z)=z^2 +c$ where $c$ is a complex number Let Set $M=${$(c,k,m) |$ the sequence $P_c (0),P_c (P_c (0)), P_c (P_c (P_c (0)))...$ is unbounded ...
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