Complexity Theory of (Functions on) Compact Metric Spaces

Akitoshi Kawamura, Florian Steinberg, Martin Ziegler

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

5 Citations (Scopus)

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 languageEnglish
Title of host publicationProceedings of the 31st Annual ACM-IEEE Symposium on Logic in Computer Science, LICS 2016
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages837-846
Number of pages10
ISBN (Electronic)9781450343916
DOIs
Publication statusPublished - Jul 5 2016
Externally publishedYes
Event31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2016 - New York, United States
Duration: Jul 5 2016Jul 8 2016

Publication series

NameProceedings - Symposium on Logic in Computer Science
VolumeProceedings - Symposium on Logic in Computer Science
ISSN (Print)1043-6871

Other

Other31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2016
CountryUnited States
CityNew York
Period7/5/167/8/16

All Science Journal Classification (ASJC) codes

  • Software
  • Mathematics(all)

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