Computational Complexity of Fractal Sets

Abstract

In studies on fractal geometry, it is important to determine whether the classification by means of computational complexity is independent of the classification by means of fractal dimension. In this paper, we show that each self-similar set defined by polynomial time computable functions is polynomial time computable, if the self-similar set satisfies a polynomial time open set condition. This fact provides us examples of sets whose computational complexity are polynomial time computable, and which have non integer Hausdorff dimension. We also construct a set with computational complexity NP-complete and with an integer Hausdorff dimension. These two examples establish the independence of computational complexity and Hausdorff dimension.