Uniform computational complexity of the derivatives of C-functions

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Abstract

We discuss the uniform computational complexity of the derivatives of C-functions in the model of Ko and Friedman (Ko, Complexity Theory of Real Functions, Birkhäuser, Basel, 1991; Ko, Friedman, Theor. Comput. Sci. 20 (1982) 323–352). We construct a polynomial time computable real function gC[−1,1] such that the sequence {|g(n)(0)|}n∈N is not bounded by any recursive function. On the other hand, we show that if fC[−1,1] is polynomial time computable and the sequence of the derivatives of f is uniformly polynomially bounded, i.e., |f(n)(x)| is bounded by 2p(n) for all x∈[−1,1] for some polynomial p, then the sequence {f(n)}n∈N is uniformly polynomial time computable.

Keywords

Computational complexity
C-function
Derivative
Recursive analysis

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Completely revised and expanded version of a talk given at the First Workshop on Computability and Complexity in Analysis (Hagen, Germany, 1995).