Quantitatively Admissible Representations and the "Main Theorem" of Continuous COMPLEXITY Theory
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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 crucially affects the computability of problems involving continuous data (like real numbers), and even more so their computational complexity. We introduce 'quantitative admissibility' as condition for complexity-theoretically sensible encodings of arbitrary compact metric spaces, a refinement of qualitative 'admissibility' due to Kreitz and Weihrauch (1985): An admissible representation of a T_0 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→ Y is continuous iff it admits continuous a code-translating mapping on Cantor space, a so-called REALIZER. We define a LINEARLY/POLYNOMIALLY admissible representation of a compact metric space (X,d) 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,d) 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→ Y on the one hand and on the other hand that of its realizer: the MAIN THEOREM of computational complexity. It suggests (how) to take into account the entropies of the spaces under consideration when measuring algorithmic cost over continuous data.
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