Ulam's cellular automaton and Rule 150

Kawaharada, Akane

doi:10.14492/hokmj/1416837570

Cited by 5 publications

(13 citation statements)

References 8 publications

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“…The relation between the two spatio-temporal patterns has been demonstrated numerically [3]. The number of "0" state cells in T 2n−1 x o is the same as in T 4n−2 x o .…”

Section: Definition 22 the Rule Of The Symmentioning

confidence: 87%

“…However, singular functions are continuous everywhere and differentiable almost everywhere [5][6][7]. The number of "0" states in the configuration of a two-dimensional nonlinear cellular automaton has been studied numerically, as shown in Figure 1 [3,8]. We can observe that the number of "0" states at each time step is described well by a singular function, which comprises a self-affine function on a unit interval.…”

Section: Introductionmentioning

confidence: 98%

“…However, it is difficult to apply this analysis to nonlinear cellular automata, and not even a case study has been presented on this topic for some time. Kawaharada studied a two-dimensional nonlinear cellular automaton that generates a spatio-temporal pattern with a fractal structure [3]. It was found that a family of unstable invariant sets exists for which the dynamics are conjugate to the one-dimensional elementary cellular automaton Rule 150 [3].…”

Section: Introductionmentioning

confidence: 99%

“…Kawaharada studied a two-dimensional nonlinear cellular automaton that generates a spatio-temporal pattern with a fractal structure [3]. It was found that a family of unstable invariant sets exists for which the dynamics are conjugate to the one-dimensional elementary cellular automaton Rule 150 [3]. A further study demonstrated that the automaton admits a limit set starting from a certain initial configuration, and showed the box-counting dimension of its boundary [4].…”

Section: Introductionmentioning

confidence: 99%

“…It is also shown that some of the patterns are related to the singular functions. For six of the patterns, the number of "0" states at each time step is described by a singular function, which represents a generalization of a result for a particular cellular automaton [3,8]. In addition, for two of the patterns the cumulative sum of the number of "0" states is also described by this function.…”

Section: Introductionmentioning

confidence: 99%

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Relation between spatio-temporal patterns generated by two-dimensional cellular automata and a singular function

Kawaharada

Namiki

2019

IJNC

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In this study, we examine the relation between the spatio-temporal patterns generated by two-dimensional symmetrical elementary cellular automata and a singular function. In a previous paper, we proved that a specific cellular automaton admits a "limit set" (a limit on the series of spatio-temporal patterns contracted with time), and we calculated the fractal dimension of the boundary of this limit set. In this paper, we provide an overview of the previous results and a more precise analysis. Numerical simulations demonstrate that the essential fractal-like patterns created by two-dimensional cellular automata are also related to a singular function.

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“…The relation between the two spatio-temporal patterns has been demonstrated numerically [3]. The number of "0" state cells in T 2n−1 x o is the same as in T 4n−2 x o .…”

Section: Definition 22 the Rule Of The Symmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Relation between spatio-temporal patterns generated by two-dimensional cellular automata and a singular function

Kawaharada

Namiki

2019

IJNC

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Number of nonzero states in prefractal sets generated by cellular automata

Kawaharada

Namiki

2020

Journal of Mathematical Physics

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We count the number of nonzero states in the spatio-temporal or spatial patterns of cellular automata. By observing self-similar structures, we determine the number of nonzero states in the pattern. For Rule 90 and Rule 150 of one-dimensional elementary cellular automata, we provide an overview of previous studies on the number of nonzero states in spatio-temporal patterns until the finite time step 2n − 1. In this study, we calculate the numbers in spatial patterns for each time step t∈Z≥0 that allow us to compute future activities more efficiently than that by simulating the entire spatio-temporal pattern. We obtain results regarding the numbers for Rule 90, Rule 150, a generalization of Rule 90 to a two-dimensional case, and a modified Ulam’s model, which is a two-dimensional crystal growth model. In addition, we examine the relationship between spatio-temporal patterns generated by cellular automata and a singular function. Representing the number of nonzero states by specific numeric values, we show that they are described by a difference form of the singular function.

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Constructing Cellular Automaton Models from Observation Data

Kawaharada

Iima

2013

2013 First International Symposium on Computing and Networking

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Ulam's cellular automaton and Rule 150

Cited by 5 publications

References 8 publications

Relation between spatio-temporal patterns generated by two-dimensional cellular automata and a singular function

Relation between spatio-temporal patterns generated by two-dimensional cellular automata and a singular function

Number of nonzero states in prefractal sets generated by cellular automata

Constructing Cellular Automaton Models from Observation Data

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