Quantitative Coding and Complexity Theory of Compact Metric Spaces

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Specifying a computational problem requires fixing encodings for input and output: encoding graphs as adjacency matrices, characters as integers, integers as bit strings, and vice versa. For such discrete data, the actual encoding is usually straightforward and/or complexity-theoretically inessential (up to polynomial time, say); but concerning continuous data, already real numbers naturally suggest various encodings with very different computational properties. With respect to qualitative computability, Kreitz and Weihrauch (1985) had identified admissibility as crucial property for “reasonable” encodings over the Cantor space of infinite binary sequences, so-called representations. For (precisely) these does the Kreitz-Weihrauch representation (aka Main) Theorem apply, characterizing continuity of functions in terms of continuous realizers. We similarly identify refined criteria for representations suitable for quantitative complexity investigations. Higher type complexity is captured by replacing Cantor’s as ground space with more general compact metric spaces, similar to equilogical spaces in computability.
Publisher
Springer International Publishing
Issue Date
2020-06-29
Language
English
Citation

16th Beyond the Horizon of Computability, pp.205 - 214

DOI
10.1007/978-3-030-51466-2_18
URI
http://hdl.handle.net/10203/275001
Appears in Collection
CS-Conference Papers(학술회의논문)
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