Abstract
We study one-dimensional cellular automata with local update rule defined by an extension of Wolfram’s Rule 90 over non-abelian group alphabets. In particular we develop necessary and sufficient conditions for a state in such an automaton to have a predecessor. We apply our results to compute the fraction of states that are reachable through evolution of an automaton over a finite dihedral group.
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Notes
- 1.
It is important to mention that \(S^{\prime }_0\) evolves to S 0 under the update rule which maps
Hence, the study of automata with even length, W, under the original update rule corresponds to study of automata of length \(\frac {W}{2}\) under the update rule written above. Corresponding results are inherent within the paper. While we note this here, we do not discuss it again explicitly.
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© 2019 The Author(s) and the Association for Women in Mathematics
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Craig, E., Poimenidou, E. (2019). An Extension of Wolfram’s Rule 90 for One-Dimensional Cellular Automata over Non-Abelian Group Alphabets. In: D'Agostino, S., Bryant, S., Buchmann, A., Guinn, M., Harris, L. (eds) A Celebration of the EDGE Program’s Impact on the Mathematics Community and Beyond . Association for Women in Mathematics Series, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-030-19486-4_20
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DOI: https://doi.org/10.1007/978-3-030-19486-4_20
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