# Questions tagged [computing-over-reals]

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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 ...
1answer
45 views

### 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 ...
1answer
499 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 ...
1answer
67 views

### 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 ...
2answers
494 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 ...
0answers
107 views

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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 ...
1answer
334 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 ...
2answers
295 views

### 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 ...
3answers
813 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 ...
5answers
4k 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 <...
2answers
323 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 ...
1answer
287 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 ...
2answers
5k 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, ...
0answers
80 views