Fractal Dimension versus Process Complexity

Abstract

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 is basically 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, if and only if its dimension is 2, and its dimension is 1, if and only if it runs in superpolynomial 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 one side versus the time complexity of a computation on the other side.