Analysis of Composite Rules of Classes II and II from Wolfram’s Elementary Cellular Automata

Abstract

Cellular automata are mathematical models used to simulate dynamic systems where time is discrete and state space is finite, evolving over time. They consist of a matrix, or grid, of cells, each of which has a state belonging to a finite set of alphabets, such as {0, 1}. Each cell evolves according to rules that depend solely on its individual state and the state of a finite number of neighboring cells. It is important to note that, in most cases, the evolution of a cellular automaton is irreversible, as multiple sets of states can lead to the same evolution. In his research, Wolfram classified Elementary Cellular Automata (ECA), the simplest type of cellular automaton, into four classes. In this work, the class II composite ECA are studied and analyzed in relation to the Wolfram classification and all the codes used were developed in the Python programming language. A cellular automaton is a quintuple C = (L, S, c, n, R) where L represents the size of the set of cells, also known as grid. Similarly, S is a finite set of states, such as {0, 1}. The parameter c defines the initial configuration, which is a specific mapping of states to the cells in the grid in the initial moment, and the parameter n specifies the size of the neighborhood, which consists of the number of adjacent cells to be considered. Finally, R is a local rule, defined as a function R : S²ⁿ⁺¹ → S. In this work, the composition of two rules was considered, showing that the compositions of elementary cellular automata from Wolfram’s Class II with Class II exhibit dynamic behaviors that can align with any of the four Wolfram classes. [...]