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Towards Computational Complexity Theory on Advanced Function Spaces in Analysis

  • Akitoshi Kawamura
  • Florian Steinberg
  • Martin Ziegler
Conference paper
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9709)

Abstract

Pour-El and Richards [PER89], Weihrauch [Weih00], and others have extended Recursive Analysis from real numbers and continuous functions to rather general topological spaces. This has enabled and spurred a series of rigorous investigations on the computability of partial differential equations in appropriate advanced spaces of functions. In order to quantitatively refine such qualitative results with respect to computational efficiency we devise, explore, and compare natural encodings (representations) of compact metric spaces: both as infinite binary sequences (TTE) and more generally as families of Boolean functions via oracle access as introduced by Kawamura and Cook ([KaCo10], Sect. 3.4). Our guide is relativization: Permitting arbitrary oracles on continuous universes reduces computability to topology and computational complexity to metric entropy in the sense of Kolmogorov. This yields a criterion and generic construction of optimal representations in particular of (subsets of) \(L^p\) and Sobolev spaces that solutions of partial differential equations naturally live in.

Keywords

Dense Sequence Real Unit General Topological Space Oracle Access Cantor Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Akitoshi Kawamura
    • 1
  • Florian Steinberg
    • 1
    • 2
  • Martin Ziegler
    • 2
    • 3
  1. 1.The University of TokyoTokyoJapan
  2. 2.TU DarmstadtDarmstadtGermany
  3. 3.KAISTDaejeonThe Republic of Korea

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