Towards computational complexity theory on advanced function spaces in analysis

Akitoshi Kawamura, Florian Steinberg, Martin Ziegler

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Citations (Scopus)

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) Lp and Sobolev spaces that solutions of partial differential equations naturally live in.

Original languageEnglish
Title of host publicationPursuit of the Universal - 12th Conference on Computability in Europe, CiE 2016, Proceedings
EditorsNataša Jonoska, Laurent Bienvenu, Arnold Beckmann
PublisherSpringer Verlag
Pages142-152
Number of pages11
ISBN (Print)9783319401881
DOIs
Publication statusPublished - Jan 1 2016
Event12th Conference on Computability in Europe, CiE 2016 - Paris, France
Duration: Jun 27 2016Jul 1 2016

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume9709
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other12th Conference on Computability in Europe, CiE 2016
CountryFrance
CityParis
Period6/27/167/1/16

Fingerprint

Complexity Theory
Computability
Function Space
Partial differential equations
Computational complexity
Computational Complexity
Partial differential equation
Relativization
Metric Entropy
Binary sequences
Sobolev spaces
Boolean functions
Binary Sequences
Compact Metric Space
Lp Spaces
Boolean Functions
Computational efficiency
Set theory
Topological space
Computational Efficiency

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Kawamura, A., Steinberg, F., & Ziegler, M. (2016). Towards computational complexity theory on advanced function spaces in analysis. In N. Jonoska, L. Bienvenu, & A. Beckmann (Eds.), Pursuit of the Universal - 12th Conference on Computability in Europe, CiE 2016, Proceedings (pp. 142-152). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9709). Springer Verlag. https://doi.org/10.1007/978-3-319-40189-8_15

Towards computational complexity theory on advanced function spaces in analysis. / Kawamura, Akitoshi; Steinberg, Florian; Ziegler, Martin.

Pursuit of the Universal - 12th Conference on Computability in Europe, CiE 2016, Proceedings. ed. / Nataša Jonoska; Laurent Bienvenu; Arnold Beckmann. Springer Verlag, 2016. p. 142-152 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9709).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Kawamura, A, Steinberg, F & Ziegler, M 2016, Towards computational complexity theory on advanced function spaces in analysis. in N Jonoska, L Bienvenu & A Beckmann (eds), Pursuit of the Universal - 12th Conference on Computability in Europe, CiE 2016, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 9709, Springer Verlag, pp. 142-152, 12th Conference on Computability in Europe, CiE 2016, Paris, France, 6/27/16. https://doi.org/10.1007/978-3-319-40189-8_15
Kawamura A, Steinberg F, Ziegler M. Towards computational complexity theory on advanced function spaces in analysis. In Jonoska N, Bienvenu L, Beckmann A, editors, Pursuit of the Universal - 12th Conference on Computability in Europe, CiE 2016, Proceedings. Springer Verlag. 2016. p. 142-152. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-319-40189-8_15
Kawamura, Akitoshi ; Steinberg, Florian ; Ziegler, Martin. / Towards computational complexity theory on advanced function spaces in analysis. Pursuit of the Universal - 12th Conference on Computability in Europe, CiE 2016, Proceedings. editor / Nataša Jonoska ; Laurent Bienvenu ; Arnold Beckmann. Springer Verlag, 2016. pp. 142-152 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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