Symbolic systems set up simple idealizations of iterations of replacement rules in Mathematica. For example, , where can be any symbol, and and can be any expressions, says to replace the pattern on the left by what is on the right. For example, using the initial condition for this Demonstration, in the expression , is , is , and the result is then .
Any single rule may be used; another might be , and so forth (the eight rules used in this Demonstration are listed below in "Details").
Starting with the initial condition, each expression is the result of applying the rule to the previous expression, replacing all instances of the pattern. In the array, these expressions are reduced by dropping —for instance, the expression becomes —and mapping the opening and closing parentheses "" and "" to black and gray cells.
The Demonstration shows the evolution of eight different rules as an array in the lower part, the plot of steps versus step lengths at the top left, and a detailed version of the first five steps of the evolution at the top right.