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DOI: 10.4230/OASIcs.CCA.2009.2277
URN: urn:nbn:de:0030-drops-22770
URL: http://drops.dagstuhl.de/opus/volltexte/2009/2277/
Ziegler, Martin
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:DSP:2009:2277,
  author =	{Martin Ziegler},
  title =	{Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability},
  booktitle =	{6th Int'l Conf. on Computability and Complexity in Analysis},
  year =	{2009},
  editor =	{Andrej Bauer and Peter Hertling and Ker-I Ko},
  publisher =	{Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2009/2277},
  annote =	{Keywords: Nonuniform computability, recursive analysis, topological complexity, linear algebra},
}

Keywords: Nonuniform computability, recursive analysis, topological complexity, linear algebra
Seminar: 6th International Conference on Computability and Complexity in Analysis (CCA'09)
Issue date: 2009
Date of publication: 25.11.2009

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