Abstract
We study the dynamics of (synchronous) one-dimensional cellular automata with cyclical boundary conditions that evolve according to the majority rule with radius r. We introduce a notion that we term cell stability with which we express the structure of the possible configurations that could emerge in this setting. Our main finding is that apart from the configurations of the form \( (0^{r+1}0^* + 1^{r+1}1^*)^* \), which are always fixed-points, the other configurations that the automata could possibly converge to, which are known to be either fixed-points or 2-cycles, have a particular spatially periodic structure. Namely, each of these configurations is of the form \( s^* \) where s consists of \( O(r^2) \) consecutive sequences of cells with the same state, each such sequence is of length at most r, and the total length of s is \( O(r^2) \) as well. We show that an analogous result also holds for the minority rule.
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Notes
- 1.
In this work, a state is a value in \( \{0,1\}\).
- 2.
Which can be shown to be at most linear in n [10].
- 3.
Indeed, it is shown in [14] that the probability that a randomly selected configuration of length n being transient approaches 1 as \( n \longrightarrow \infty \). As such, the additional temporally periodic configurations that we address in this work are, in a sense, not “typical”. We, in contrast to [14], make no assumption about the distribution of the configuration space, and are therefore interested in understanding the structure of all configurations, not only the “typical” ones.
- 4.
e.g., for \( r=3 \), the transient configuration 001001001001001001 converges after one step to the fixed-point configuration \( (0)^* \).
- 5.
e.g., for \( r=4 \), the transient configuration 001011001011001011001011001011001011 converges after one step to the 2 cycle consisting of \( (111000)^6 \) and \( (000111)^6 \).
- 6.
Also \( (10)^* \), but since the configurations are cyclic, the patterns \( (01)^* \) and \( (10)^* \) correspond to equivalent sets of configurations.
- 7.
The string s could also be the mirror or the complement of any of the specified patterns, which we omit for the sake of conciseness. For example, since we explicitly specified that s could be 010011, it means that s could also be 110010 (which is the mirror of 010011) or 101100 (which is the complement of 010011), even though these two are not explicitly specified.
- 8.
The middle block is well defined, as it is shown in the full version [10] that the number of blocks between I and J must be odd.
- 9.
In the dominance problem, one asks how many cells must initially be at a certain state so that eventually all cells have the same state.
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Nakar, Y., Ron, D. (2022). The Structure of Configurations in One-Dimensional Majority Cellular Automata: From Cell Stability to Configuration Periodicity. In: Chopard, B., Bandini, S., Dennunzio, A., Arabi Haddad, M. (eds) Cellular Automata. ACRI 2022. Lecture Notes in Computer Science, vol 13402. Springer, Cham. https://doi.org/10.1007/978-3-031-14926-9_6
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