### 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.

Original language | English |
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Title of host publication | Pursuit of the Universal - 12th Conference on Computability in Europe, CiE 2016, Proceedings |

Editors | Nataša Jonoska, Laurent Bienvenu, Arnold Beckmann |

Publisher | Springer Verlag |

Pages | 142-152 |

Number of pages | 11 |

ISBN (Print) | 9783319401881 |

DOIs | |

Publication status | Published - Jan 1 2016 |

Event | 12th Conference on Computability in Europe, CiE 2016 - Paris, France Duration: Jun 27 2016 → Jul 1 2016 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 9709 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 12th Conference on Computability in Europe, CiE 2016 |
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Country | France |

City | Paris |

Period | 6/27/16 → 7/1/16 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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

}

TY - GEN

T1 - Towards computational complexity theory on advanced function spaces in analysis

AU - Kawamura, Akitoshi

AU - Steinberg, Florian

AU - Ziegler, Martin

PY - 2016/1/1

Y1 - 2016/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84976609574&partnerID=8YFLogxK

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U2 - 10.1007/978-3-319-40189-8_15

DO - 10.1007/978-3-319-40189-8_15

M3 - Conference contribution

AN - SCOPUS:84976609574

SN - 9783319401881

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 142

EP - 152

BT - Pursuit of the Universal - 12th Conference on Computability in Europe, CiE 2016, Proceedings

A2 - Jonoska, Nataša

A2 - Bienvenu, Laurent

A2 - Beckmann, Arnold

PB - Springer Verlag

ER -