# Real computation

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Real computation

In computability theory, the theory of real computation deals with hypothetical computing machines using infinite-precision real numbers. They are given this name because they operate on the set of real numbers. Within this theory, it is possible to prove interesting statements such as "the complement of the Mandelbrot set is only partially decidable". For other such powerful machines, see the article Hypercomputation.

These hypothetical computing machines can be viewed as idealised analog computers which operate on real numbers and are differential, whereas digital computers are limited to computable numbers and are algebraic. Depending on the model chosen, this may enable real computers to solve problems that are inextricable on digital computers (for example, Hava Siegelmann's neural nets can have noncomputable real weights, making them able to compute nonrecursive languages), or vice versa (Claude Shannon's idealized analog computer can only solve algebraic differential equations, while a digital computer can solve some transcendental equations as well).

A canonical model of computation over the reals is Blum Shub Smale machine (BSS).

* Lenore Blum, Felipe Cucker, Michael Shub and Stephen Smale, "Complexity and Real Computation", ISBN 0387982817.

* "Neural Networks and Analog Computation: Beyond the Turing Limit" by Hava Siegelmann ISBN 0-8176-3949-7
* [http://www.lsm.tugraz.at/papers/lsm-telematik.pdf the Liquid Computer A Novel Strategy for Real-Time Computing on Time Series, by Thomas Natschläger et al]
* [ftp://ftp.cs.cuhk.hk/pub/neuro/papers/jcss1.ps.Z On the computational power of neural nets]
* [http://math.isa.utl.pt/~mlc/phdthesis.ps Computational complexity of real valued recursive functions and analog circuits. by Campagnolo]

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