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Questions tagged [computable-analysis]

computability and complexity in real or complex analysis

Computable analysis studies questions like computability and complexity of real numbers, real functions and operators. A major difference with classical computability theory is that inputs and outputs can be infinite objects (like a real number), whereas in classical computability theory the input and the outputs are represented by finite strings from an alphabet.

  • Use this tag if you are interested in computability or complexity of functions over higher-type objects.
  • Use for questions where inputs are floating point numbers or any other fixed approximation representation (i.e. the algorithm cannot ask for arbitrary amount of precision approximation to the inputs).
  • Use or for questions about classical models of computation (inputs are represented as natural numbers or finite strings).
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Are there established complexity classes with real numbers?

A student recently asked me to check an NP-hardness proof for them. They performed a reduction along the lines of: I reduce this problem $P'$ that is known to be NP-complete to my problem $P$ (with a poly-time many-one reduction), so $P$ is…
Raphael
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Decidable properties of computable reals

Is "Rice's theorem for the computable reals" -- that is, no nontrivial property of the number represented by a given computable real is decidable -- true? Does this correspond in some direct way to the connectedness of the reals?
Shachaf
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A continuous optimization problem that reduces to TSP

Suppose I am given a finite set of points $p_1,p_2,..p_n$ in the plane, and asked to draw a twice-differentiable curve $C(P)$ through the $p_i$'s, such that its perimeter is as small as possible. Assuming $p_i=(x_i,y_i)$ and $x_i
PKG
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Is a PDA as powerful as a CPU?

This is a question I have stumbled upon in my exam revision and I find it intriguing: My computer is blue and it has a massive graphics card and a DVD and every- thing so which is more powerful: my computer or a Pushdown Automaton? My Thoughts When…
Mike Howard
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How does automatic differentiation work?

What is the intuitive idea behind automatic differentiation? If I have a program which computes $f(x, y)=x^2+yx$, which steps lead to the program which computes the derivative $df/dx$ of f? double f(double x, double y) { double res = x*x; …
user13187
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What is the "continuity" as a term in computable analysis?

Background I once implemented a datatype representing arbitrary real numbers in Haskell. It labels every real numbers by having a Cauchy sequence converging to it. That will let $\mathbb{R}$ be in the usual topology. I also implemented addition,…
Dannyu NDos
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Semidecidable properties of computable reals

By computable real I mean $x\in\mathbb{R}$ such that there is some computable total function $p_x$ that takes a natural number $n$ and returns a dyadic rational $r$ such that $|x-r|<2^{-n}$. I think this is probably equivalent to any other…
user96217
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What is the computational cost of automatic differentiation compared to symbolic and numerical differentiation?

Automatic differentiation is a set of techniques to numerically evaluate the derivative of a function. Quoting from Wikipedia (emphasis mine): These classical methods run into problems: symbolic differentiation leads to inefficient code (unless…
glS
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What does Sigma notation mean, in the context of computability of functions?

I was reading a paper on the computability of AIXI [1] and came across the notion of $\Sigma^0_n$-computability for real-valued functions in section 2.3. I'd like to read about this in more detail. Unfortunately I couldn't find this definition…
Manlio
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Is Newton's Method to compute the zeros of a function an algorithm?

Looking for Newton's method in Wikipedia, I read the following: In numerical analysis, Newton's method (also known as the Newton-Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better…
babou
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Poly-time computability of inversion of poly-time real functions

At pp. 7-8 of Ker-I Ko's Computational Complexity of Real Functions (1991), the following is stated for one dimensional cases: Let $INV_1$ be the operator that maps a one-to-one function $f:[0,1]\rightarrow [0,1]$ to its inverse function $f^{-1}$.…
user13675
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Decide whether a polynomial has a root

Let $A$ be a ring such that all elements of $A$ are complex computable numbers. I'm interested in knowing whether the decision problem that asks, given $P\in A[X]$, if $P$ has a root in $A$ is decidable. I know that this problem is decidable for the…
Rémi
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What is the relationship between two definitions of Turing-computability of a partial function?

When one wants to know that whether a partial function $f \colon \Sigma^{*} \supsetneq \mathrm{dom}(f) \rightarrow \Sigma^{*}$ is Turing-computable, there are two methods that I think they are both useful. We can define a total function $\bar{f}$…
Blanco
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