Cellular Automata Homework Help - K-12 Grade Level, College Level Mathematics

Introduction of Cellular Automaton

A Cellular Automaton (abbreviated as CA) is a discrete model studied in the complexity science, computability theory, mathematics, physics, microstructure modeling and theoretical biology. Cellular automata are also known as tessellation automata, iterative arrays cellular spaces, cellular structures, tessellation structures, and homogeneous structures.

The cellular automaton consists of a regular grid of cells, each in one of a fixed number of states like on and off (in contrary to a coupled map lattice). The grid can be of any number of fixed dimensions. For each cell, a set of cells called its neighborhood (generally involving the cell itself) is defined relative to the identified cell. The initial state (i.e., time t=0) is selected by assigning a state for each cell. A new generation is generated (advancing t by 1), according to some fixed rule (usually, a mathematical function) which determines the new state of each cell in terms of the present state of the cell and the states of the cells in its neighborhood. Generally, the rule for updating the state of cells is similar for each cell and does not change over time and is applied to the entire grid at similar time, however exceptions are known, like the probabilistic cellular automaton and asynchronous cellular automaton.

The notion was originally introduced in the year1940 by Stanislaw Ulam and John von Neumann while they were contemporaries at Los Alamos National Laboratory. Whereas studied some during the year 1950s to1960s, it was not until the year 1970s and Conway's Game of Life a 2-dimensional cellular automaton which interest in the subject that prolonged beyond academia. In the year 1980, Stephen Wolfram engaged in a systematic study of 1-dimensional cellular automata or what he calls elementary cellular automata, displaying that some rule sets were Turing-complete. He published a book in the year 2002 declaring that cellular automata had applications in numerous fields of science.

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The simplest kind of cellular automaton is a binary nearest-neighbor, 1-dimensional automaton. This kind of automata was known as "elementary cellular automata" by the scientist S. Wolfram who studied their outstanding properties. There are 256 automata of such kind each of which can be indexed by an exclusive binary number whose decimal representation is termed as the "rule" for the specific automaton. An illustration of rule 30 is shown above altogether with the evolution it generates after 15 steps beginning from a single black cell.

The Stephen Wolfram introduced a numbering system for all the cellular automata that base only on themselves, the left-hand neighbor, and the right-hand neighbor, frequently known as the elementary cellular automata.

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