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Conference on Computability in Europe

CiE 2016: Pursuit of the Universal pp 142–152Cite as

Towards Computational Complexity Theory on Advanced Function Spaces in Analysis

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,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.

Supported in part by JSPS Kakenhi projects 24106002 and 26700001, by EU FP7 IRSES project 294962, by DFG Zi 1009/4-1 and by IRTG 1529. We thank Daniel Graça and Elvira Mayordomo for inviting this extended abstract as opportunity to report on our progress since its CCA 2015 short version.

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Author information

Authors and Affiliations

  1. The University of Tokyo, Tokyo, Japan

    Akitoshi Kawamura & Florian Steinberg

  2. TU Darmstadt, Darmstadt, Germany

    Florian Steinberg & Martin Ziegler

  3. KAIST, Daejeon, The Republic of Korea

    Martin Ziegler

Authors
  1. Akitoshi Kawamura
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  2. Florian Steinberg
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  3. Martin Ziegler
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Corresponding author

Correspondence to Martin Ziegler .

Editor information

Editors and Affiliations

  1. Swansea University, Swansea, United Kingdom

    Arnold Beckmann

  2. Université Paris-Diderot - Paris 7, Paris, France

    Laurent Bienvenu

  3. University of South Florida, Tampa, Florida, USA

    Nataša Jonoska

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Kawamura, A., Steinberg, F., Ziegler, M. (2016). Towards Computational Complexity Theory on Advanced Function Spaces in Analysis. In: Beckmann, A., Bienvenu, L., Jonoska, N. (eds) Pursuit of the Universal. CiE 2016. Lecture Notes in Computer Science(), vol 9709. Springer, Cham. https://doi.org/10.1007/978-3-319-40189-8_15

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  • DOI: https://doi.org/10.1007/978-3-319-40189-8_15

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