Abstract
We study the asymptotic behavior of symbolic computing systems, notably one-dimensional cellular automata (CA), in order to ascertain whether and at what rate the number of complex versus simple rules dominate the rule space for increasing neighborhood range and number of symbols (or colors), and how different behavior is distributed in the spaces of different cellular automata formalisms. Using two different measures, Shannon's block entropy and Kolmogorov complexity, the latter approximated by two different methods (lossless compressibility and block decomposition), we arrive at the same trend of larger complex behavioral fractions. We also advance a notion of asymptotic and limit behavior for individual rules, both over initial conditions and runtimes, and we provide a formalization of Wolfram's classification as a limit function in terms of Kolmogorov complexity.
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