Representation Theory of Compact Metric Spaces and Computational Complexity of Continuous Data

TL;DR

This paper develops a refined theory of representations for compact metric spaces, linking the complexity of continuous functions to the entropy of the spaces, and establishing a framework for measuring algorithmic cost over continuous data.

Contribution

It introduces QUANTITATIVE admissibility for representations, connecting entropy and modulus of continuity, and proves a main theorem relating function complexity to space entropy.

Findings

Category of such representations is Cartesian closed.

Every compact space admits a linearly-admissible representation.

Quantitative correspondence between function and realizer moduli of continuity.

Abstract

Choosing an encoding over binary strings for input/output to/by a Turing Machine is usually straightforward and/or inessential for discrete data (like graphs), but delicate -- heavily affecting computability and even more computational complexity -- already regarding real numbers, not to mention more advanced (e.g. Sobolev) spaces. For a general theory of computational complexity over continuous data we introduce and justify QUANTITATIVE admissibility as requirement for sensible encodings of arbitrary compact metric spaces, a refinement of qualitative 'admissibility' due to [Kreitz&Weihrauch'85]: An admissible representation of a T0 space $X$ is a (i) continuous partial surjective mapping from the Cantor space of infinite binary sequences which is (ii) maximal w.r.t. continuous reduction. By the Kreitz-Weihrauch (aka "Main") Theorem of computability over continuous data, for fixed spaces $X,Y$ equipped with admissible representations, a function $f:X\to Y$ is continuous iff it admits continuous a code-translating mapping on Cantor space, a so-called REALIZER. We define a QUANTITATIVELY admissible representation of a compact metric space $X$ to have (i) asymptotically optimal modulus of continuity, namely close to the entropy of $X$, and (ii) be maximal w.r.t. reduction having optimal modulus of continuity in a similar sense. Careful constructions show the category of such representations to be Cartesian closed, and non-empty: every compact $X$ admits a linearly-admissible representation. Moreover such representations give rise to a tight quantitative correspondence between the modulus of continuity of a function $f:X\to Y$ on the one hand and on the other hand that of its realizer: the MAIN THEOREM of computational complexity. This suggests (how) to take into account the entropies of the spaces under consideration when measuring algorithmic cost over continuous data.