### 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 |
---|---|

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 |

Externally published | Yes |

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) |
---|---|

Volume | 9709 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

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

Country | France |

City | Paris |

Period | 6/27/16 → 7/1/16 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

## Fingerprint Dive into the research topics of 'Towards computational complexity theory on advanced function spaces in analysis'. Together they form a unique fingerprint.

## 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