Chapter 3: The World of Simple Programs

Section 10: Symbolic Systems

Other [symbolic systems] rules

If only a single variable appears in the rule, then typically only nested behavior can be generated—though in an example like [x_][_] [x[[][]][]] it can be quite complex. The left-hand side of each rule can consist of any expression; [[x_]][y_] and [][x_[y_]] are two possibilities. However, at least with small initial conditions it seems easier to achieve complex behavior with rules based on [x_][y_]. Note that rules with no explicit e's on the left-hand side always give trees with regular nested structures; x_[y_] x[y][x[y]] (or x_ x[x] in Mathematica), for example, yields balanced binary trees.

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From Stephen Wolfram: A New Kind of Science [citation]