Quantitative Coding and Complexity Theory of Compact Metric Spaces

Abstract

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.