Second-order cellular automaton

A second order cellular automaton is a reversible cellular automaton (CA) where the state of a cell at time t depends not only on its neighborhood at time t-1, but also on its state at time t-2. Specifically, the neighborhood at t-1 is used to choose a function f_n that maps the state of the cell at time t-2 to its state at time t. As long as each function f_n is invertible, it follows that the resulting automaton is reversible, regardless of how the functions at chosen.

In particular, for two-state cellular automata, any ordinary CA rule can be turned into a second order rule by xor-ing the new state of each cell at time t with its past state at time t-2. In fact, all two-state second order rules may be produced in this way. The resulting second order automaton, however, will generally bear little resemblance to the ordinary CA it was constructed from. Second order rules constructed in this way are commonly denoted by appending an "R" to the number or code of the base rule.

References

• Wolfram, Stephen, 2002. A New Kind of Science, pp. 437-440, 452. Wolfram Media. ISBN 1-57955-008-8.
• Kwanghyun Paek, 2003. "Reversible Cellular Automata"