Dynamical Rules Notation and Definitions

Although there is an incredibly rich variety of specific CA systems, each of which is carefully defined or selected to fit the requirements of a particular model, the definition of any of these specific, systems requires the specification of each of the following four generic characteristics  

Discrete Cellular State Space the discrete lattice of cells or sites upon which CA live , and their dynamics unfolds. C can be one-dimensional, two-dimensional 

Local Value Space E each cell of C, indexed by the elements of the s symmetry group, can assume only one of a finite number of different values  

Boundary Conditions although CA are a.ssumed to live on infinitely large lattices, computer simulations must necessarily be run on finite sets. For a one dimensional lattice with N cells, it is common to use periodic boundary conditions, in which ctn + i is identified with ai. Alternatively, all cells to the left and right of a finite block of N cells may be arbitrarily defined to possess value 0 for all time, so that their dynamics remains uncoupled with that taking place within the block. Similarly, in two dimensions, it is usual to have the dynamics take place on a torus, in which o m+i = 7, 2 and = cTi,j- As we will see later it turns one 

Dynamical Rule d E x E x x E — E, where n specifies the number of cells needed to define the neighborhood of a given cell. Defining M i] to be the neighborhood about cell i, the transition rule is most generally written as 

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