On the Kolmogorov complexity of functions of finite smoothness

The complexity of a computational problem can be rigorously defined only in a model of computation including the description of the input information and the set of permissible primitive operations of which the solution algorithm must be constructed. The complexity of an algorithm is then defined as the number of primitive operations (they may be counted with weights) used in its program. If the input is available approximately or is incomplete the answer can be found only within certain error bounds. One may now consider all permissible algorithms (if there are any) to evaluate the solution within these bounds. The minimal complexity of such an algorithm is called the complexity of the problem. Rigorous definitions and detailed discussion of this concept can be found in Traub and Woiniakowski (1980). This paper deals with the Kolmogorov complexity of evaluation of a continuous function. The concept of the Kolmogorov complexity (which in the rest of this paper is referred to as simply “the complexity”) fits naturally into the general framework outlined above. For a continuous function y = f(x) one may consider both x and y represented approximately, with prescribed finite numbers of binary digits; the algorithm of evaluation can be described as a sequence of operations on binary digits with primitive operations being the elementary binary functions. Rigorous definitions are given below. Let B, be the set of n-dimensional binary vectors, i.e., all vectors of the form5= (6, 52,. . . , &), 6 = 0, 1. The functions B, + B,, for various