Fractal Dimension Versus Computational Complexity

In this paper we look at small Turing machines (TMs) that work with just two colors (alphabet symbols) and either two or three states. For any particular such machine τ and any particular input x we consider what we call the space-time diagram which basically is just the collection of consecutive tape configurations of the computation τ(x). In our setting it makes sense to define a fractal dimension for a Turing machine as the limiting fractal dimension for the corresponding space-time diagrams. It turns out that there is a very strong relation between the fractal dimension of a Turing machine of the above specified type and its runtime complexity. In particular, a TM with three states and two colors runs in at most linear time iff its dimension is 2, and its dimension is 1 iff it runs in super-polynomial time and it uses polynomial space. If a TM runs in time O(xn) we have empirically verified that the corresponding dimension is n+1 n, a result that we can only partially prove. We find the results presented here remarkable because they relate two completely different complexity measures: the geometrical fractal dimension on the one side versus the time complexity of a computation on the other side