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For example, in a 1-dimensional cellular automaton like the examples below, the neighborhood of a cell x i t is {x i−1 t−1, x i t−1, x i+1 t−1}, where t is the time step (vertical), and i is the index (horizontal) in one generation.
Class 1: Cellular automata which rapidly converge to a uniform state. Examples are rules 0, 32, 160 and 232. Class 2: Cellular automata which rapidly converge to a repetitive or stable state. Examples are rules 4, 108, 218 and 250. Class 3: Cellular automata which appear to remain in a random state. Examples are rules 22, 30, 126, 150, 182.
Among the 88 possible unique elementary cellular automata, Rule 110 is the only one for which Turing completeness has been directly proven, although proofs for several similar rules follow as simple corollaries (e.g. Rule 124, which is the horizontal reflection of Rule 110). Rule 110 is arguably the simplest known Turing complete system.
Technically, they are not cellular automata at all, because the underlying "space" is the continuous Euclidean plane R 2, not the discrete lattice Z 2. They have been studied by Marcus Pivato. [24] Lenia is a family of continuous cellular automata created by Bert Wang-Chak Chan. The space, time and states of the Game of Life are generalized to ...
Time-space diagram of Rule 90 with random initial conditions. Each row of pixels is a configuration of the automaton; time progresses vertically from top to bottom. In the mathematical study of cellular automata, Rule 90 is an elementary cellular automaton based on the exclusive or function. It consists of a one-dimensional array of cells, each ...
A cellular automaton is defined by its cells (often a one- or two-dimensional array), a finite set of values or states that can go into each cell, a neighborhood associating each cell with a finite set of nearby cells, and an update rule according to which the values of all cells are updated, simultaneously, as a function of the values of their neighboring cells.
In von Neumann's cellular automaton, the finite state machines (or cells) are arranged in a two-dimensional Cartesian grid, and interface with the surrounding four cells. As von Neumann's cellular automaton was the first example to use this arrangement, it is known as the von Neumann neighbourhood. The set of FSAs define a cell space of ...
An automaton that is both injective and surjective is called a reversible cellular automaton. [3] The Garden of Eden theorem, due to Edward F. Moore and John Myhill , asserts that a cellular automaton in a Euclidean space is locally injective if and only if it is surjective. In other words, it asserts that a cellular automaton has a Garden of ...