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 machineτand any particular inputx, we consider what we call thespace-timediagram which is basically the collection of consecutive tape configurations of the computationτ(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 timeO(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.
Highlights
IntroductionWe look at small Turing machines (TMs) that work with just two colors (alphabet symbols) and either two or three states
We look at small Turing machines (TMs) that work with just two colors and either two or three states
It makes sense to define a fractal dimension for a Turing machine as the limiting fractal dimension for the corresponding space-time diagrams
Summary
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). We will fix on a computational model: small Turing machines with a one-way infinite tape For these machines, we will define the so-called spacetime diagrams which are a representation of the memory state throughout time. There is a geometrical framework in which the complexity of spatiotemporal objects is measured by their fractal dimension.
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