Proceedings ArticleDOI
Complexity Theory of (Functions on) Compact Metric Spaces
Akitoshi Kawamura,Florian Steinberg,Martin Ziegler +2 more
- pp 837-846
TLDR
The main results relate Kolmogorov’s entropy of a compact metric space X polynomially to the uniform relativized complexity of approximating various families of continuous functions on X, and offer some guidance towards suitable notions of complexity for higher types.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.read more
Citations
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Journal Article
Computational Complexity on Computable Metric Spaces
TL;DR: In this paper, a new Turing machine based concept of time complexity for functions on computable metric spaces was introduced, which generalizes the ordinary complexity of word functions and the complexity of real functions studied by Ko [19] et al.
Polynomial Running Times for Polynomial-Time Oracle Machines
TL;DR: The notion of strongly polynomial-time computability of functionals on Baire space has been introduced in this article, which is a stronger requirement than polynomially time computability.
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Polynomial running times for polynomial-time oracle machines
TL;DR: A more restrictive notion of feasibility of functionals on Baire space than the established one from second-order complexity theory is introduced, making it possible to consider functions on the natural numbers as running times of oracle Turing machines and avoiding second- order polynomials, which are notoriously difficult to handle.
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Computable Operations on Compact Subsets of Metric Spaces with Applications to Fréchet Distance and Shape Optimization.
TL;DR: The thus obtained Cartesian closure is shown to exhibit the same structural properties as in the Euclidean case, particularly regarding function pre/image.
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Computing Haar Measures
TL;DR: It is established that in fact every computably compact computable metric group renders the Haar integral computable: once asserting computability using an elegant synthetic argument, exploiting uniqueness in a computable compact space of probability measures; and once presenting and analyzing an explicit, imperative algorithm based on 'maximum packings' with rigorous error bounds and guaranteed convergence.
References
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TL;DR: This paper discusses Fixed-Parameter Algorithms, Parameterized Complexity Theory, and Selected Case Studies, and some of the techniques used in this work.
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Jörg Flum,Martin Grohe +1 more
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