Quantitatively Admissible Representations and the "Main Theorem" of Type-2 COMPLEXITY Theory
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Choosing an encoding over binary strings is usually straightforward or inessential for computations over countable universes (like of graphs), but crucially affects already the computability of problems involving continuous data (like real numbers), and even more their complexity. We introduce a condition for complexity-theoretically reasonable encodings of arbitrary compact metric spaces, quantitatively strengthening qualitative ADMISSIBILITY due to [Kreitz/Weihrauch'85]: The MAIN THEOREM of Computable Analysis asserts that a real function f is continuous iff there exists a continuous mapping on the Cantor space of infinite binary sequences translating codes (such as the signed-digit expansion) of real arguments x to codes of real values f(x). Generalizing this characterization to functions f:X→ Y between arbitrary topological T_0 spaces X and Y, [KW85] had identified ADMISSIBILITY as essential for any computably 'reasonable' encoding (called REPRESENTATIONS in Type-2 Computability Theory) of X's elements as infinite binary strings. We generalize the signed-digit representation of the real unit interval, having a modulus of continuity linear in the entropy, to arbitrary compact metric spaces X. We establish this representation to satisfy a carefully crafted quantitative refinement of the aforementioned qualitative admissibility: (i) its modulus of continuity is optimal, i.e., linear in the metric entropy; and (ii) it is maximal with respect to reductions that are metrically optimal in the same sense. The category of such representations is shown closed under countable Cartesian products generalizing the Hilbert Cube. And we deduce a tight quantitative correspondence between the modulus of continuity of functions f among compact metric spaces on the one hand and on the other hand those of code-translating mappings (known as REALIZERS) on Cantor space.
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