Characterization of $({k^{}},1/2)$ reversible one dimensional cellular automata

A characterization of reversible one dimensional cellular automata is possible if we use block permutations and shifts, this process will be useful for analyzing dynamical behavior of such systems.

The main result in the work of Kari [Kar96] is that the action of every reversible one dimensional cellular automaton can be represented by the process of applying $2$ block permutations and a shift among them. To explain the previous affirmation, first we will explain which are the properties of these reversible systems.