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URN: urn:nbn:de:0030-drops-22770
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Contributed Papers

### Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability

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### Abstract

It is folklore particularly in numerical and computer sciences that, instead of solving some general problem $f:A\to B$, additional structural information about the input $x\in A$ (that is any kind of promise that $x$ belongs to a certain subset $A'\subseteq A$) should be taken advantage of. Some examples from real number computation show that such discrete advice can even make the difference between computability and uncomputability. We turn this into a both topological and combinatorial complexity theory of information, investigating for several practical problem show much advice is necessary and sufficient to render them computable. Specifically, finding a nontrivial solution to a homogeneous linear equation $A\cdot\vec x=0$ for a given singular real $n\times n$-matrix $A$ is possible when knowing $\rank(A)\in\{0,1,\ldots,n-1\}$; and we show this to be best possible. Similarly, diagonalizing (i.e. finding a basis of eigenvectors of) a given real symmetric $n\times n$-matrix $A$ is possible when knowing the number of distinct eigenvalues: an integer between $1$ and $n$ (the latter corresponding to the nondegenerate case). And again we show that $n$--fold (i.e. roughly $\log n$ bits of) additional information is indeed necessary in order to render this problem (continuous and) computable; whereas finding \emph{some single} eigenvector of $A$ requires and suffices with $\Theta(\log n)$--fold advice.

### BibTeX - Entry

@InProceedings{ziegler:OASIcs:2009:2277,
author =	{Martin Ziegler},
title =	{{Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability}},
booktitle =	{6th International Conference on Computability and Complexity in Analysis (CCA'09)},
series =	{OpenAccess Series in Informatics (OASIcs)},
ISBN =	{978-3-939897-12-5},
ISSN =	{2190-6807},
year =	{2009},
volume =	{11},
editor =	{Andrej Bauer and Peter Hertling and Ker-I Ko},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},