### Abstract

We promote the theory of computational complexity on metric spaces: as natural common generalization of (i) the classical discrete setting of integers, binary strings, graphs etc. as well as of (ii) the bit-complexity theory on real numbers and functions according to Friedman, Ko (1982ff), Cook, Braverman et al.; as (iii) resource-bounded refinement of the theories of computability on, and representations of, continuous universes by Pour-El&Richards (1989) and Weihrauch (1993ff); and as (iv) computational perspective on quantitative concepts from classical Analysis: Our main results relate (i.e. upper and lower bound) Kolmogorov's entropy of a compact metric space X polynomially to the uniform relativized complexity of approximating various families of continuous functions on X. The upper bounds are attained by carefully crafted oracles and bit-cost analyses of algorithms perusing them. They all employ the same representation (i.e. encoding, as infinite binary sequences, of the elements) of such spaces, which thus may be of own interest. The lower bounds adapt adversary arguments from unit-cost Information-Based Complexity to the bit model. They extend to, and indicate perhaps surprising limitations even of, encodings via binary string functions (rather than sequences) as introduced by Kawamura&Cook (SToC'2010, 3.4). These insights offer some guidance towards suitable notions of complexity for higher types.

Original language | English |
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Title of host publication | Proceedings of the 31st Annual ACM-IEEE Symposium on Logic in Computer Science, LICS 2016 |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 837-846 |

Number of pages | 10 |

ISBN (Electronic) | 9781450343916 |

DOIs | |

Publication status | Published - Jul 5 2016 |

Externally published | Yes |

Event | 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2016 - New York, United States Duration: Jul 5 2016 → Jul 8 2016 |

### Publication series

Name | Proceedings - Symposium on Logic in Computer Science |
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Volume | Proceedings - Symposium on Logic in Computer Science |

ISSN (Print) | 1043-6871 |

### Other

Other | 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2016 |
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Country | United States |

City | New York |

Period | 7/5/16 → 7/8/16 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)

### Cite this

*Proceedings of the 31st Annual ACM-IEEE Symposium on Logic in Computer Science, LICS 2016*(pp. 837-846). (Proceedings - Symposium on Logic in Computer Science; Vol. Proceedings - Symposium on Logic in Computer Science). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1145/2933575.2935311

**Complexity Theory of (Functions on) Compact Metric Spaces.** / Kawamura, Akitoshi; Steinberg, Florian; Ziegler, Martin.

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

*Proceedings of the 31st Annual ACM-IEEE Symposium on Logic in Computer Science, LICS 2016.*Proceedings - Symposium on Logic in Computer Science, vol. Proceedings - Symposium on Logic in Computer Science, Institute of Electrical and Electronics Engineers Inc., pp. 837-846, 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2016, New York, United States, 7/5/16. https://doi.org/10.1145/2933575.2935311

}

TY - GEN

T1 - Complexity Theory of (Functions on) Compact Metric Spaces

AU - Kawamura, Akitoshi

AU - Steinberg, Florian

AU - Ziegler, Martin

PY - 2016/7/5

Y1 - 2016/7/5

N2 - We promote the theory of computational complexity on metric spaces: as natural common generalization of (i) the classical discrete setting of integers, binary strings, graphs etc. as well as of (ii) the bit-complexity theory on real numbers and functions according to Friedman, Ko (1982ff), Cook, Braverman et al.; as (iii) resource-bounded refinement of the theories of computability on, and representations of, continuous universes by Pour-El&Richards (1989) and Weihrauch (1993ff); and as (iv) computational perspective on quantitative concepts from classical Analysis: Our main results relate (i.e. upper and lower bound) Kolmogorov's entropy of a compact metric space X polynomially to the uniform relativized complexity of approximating various families of continuous functions on X. The upper bounds are attained by carefully crafted oracles and bit-cost analyses of algorithms perusing them. They all employ the same representation (i.e. encoding, as infinite binary sequences, of the elements) of such spaces, which thus may be of own interest. The lower bounds adapt adversary arguments from unit-cost Information-Based Complexity to the bit model. They extend to, and indicate perhaps surprising limitations even of, encodings via binary string functions (rather than sequences) as introduced by Kawamura&Cook (SToC'2010, 3.4). These insights offer some guidance towards suitable notions of complexity for higher types.

AB - We promote the theory of computational complexity on metric spaces: as natural common generalization of (i) the classical discrete setting of integers, binary strings, graphs etc. as well as of (ii) the bit-complexity theory on real numbers and functions according to Friedman, Ko (1982ff), Cook, Braverman et al.; as (iii) resource-bounded refinement of the theories of computability on, and representations of, continuous universes by Pour-El&Richards (1989) and Weihrauch (1993ff); and as (iv) computational perspective on quantitative concepts from classical Analysis: Our main results relate (i.e. upper and lower bound) Kolmogorov's entropy of a compact metric space X polynomially to the uniform relativized complexity of approximating various families of continuous functions on X. The upper bounds are attained by carefully crafted oracles and bit-cost analyses of algorithms perusing them. They all employ the same representation (i.e. encoding, as infinite binary sequences, of the elements) of such spaces, which thus may be of own interest. The lower bounds adapt adversary arguments from unit-cost Information-Based Complexity to the bit model. They extend to, and indicate perhaps surprising limitations even of, encodings via binary string functions (rather than sequences) as introduced by Kawamura&Cook (SToC'2010, 3.4). These insights offer some guidance towards suitable notions of complexity for higher types.

UR - http://www.scopus.com/inward/record.url?scp=84994620469&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84994620469&partnerID=8YFLogxK

U2 - 10.1145/2933575.2935311

DO - 10.1145/2933575.2935311

M3 - Conference contribution

AN - SCOPUS:84994620469

T3 - Proceedings - Symposium on Logic in Computer Science

SP - 837

EP - 846

BT - Proceedings of the 31st Annual ACM-IEEE Symposium on Logic in Computer Science, LICS 2016

PB - Institute of Electrical and Electronics Engineers Inc.

ER -