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Fractal dimension versus process complexity

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 $τ$ 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 $τ(x)$. 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.