Fractal dimension versus process complexity (1309.1779v6)

Abstract: Complexity measures are designed to capture complex behavior and quantify how complex, according to that measure, that particular behavior is. It can be expected that different complexity measures from possibly entirely different fields are related to each other in a non-trivial fashion. Here we study small Turing machines (TMs) with two symbols, and two and three states. For any particular such machine $\tau$ and any particular input $x$ we consider what we call the 'space-time' diagram which is the collection of consecutive tape configurations of the computation $\tau(x)$. In our setting, we define fractal dimension of a Turing machine as the limiting fractal dimension of the corresponding space-time diagram. 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 $O(x^n)$ 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.