Path rewriting cellular automata are a type of simple program that is based on ordinary cellular automaton systems. When a path rewriting cellular automaton evolves, both the cells and the topology on which they operate change at each step according to definite rules. Here, this concept is explored in the simplest of cases, using only elementary cellular automata and simple neighborhood rewrite rules. The rules specify cellular automaton uses the sequence obtained in its neighborhood not only to change its own color, but also to change which part of its neighborhood it is connected to on the following step.
Path rewriting cellular automata evolve in the following fashion: every system operates on a network where each node has one state and one outgoing connection. At each step, each node looks at it's own state and the states of the nodes following successive connections ahead of it. The node then uses these states to decide two things: the new state of the node, and which 'downstream' node it will be connected to on the following step. The rule icon here shows what a node will do with a particular sequence. The leftmost cell is the node itself, and there are two 'downstream' connections. A cell in the top center means that the node stays connected to the one it was already connected to, while a top right cell means that the node jumps ahead to the one following the one it was connected to.