Necessary and sufficient conditions for reversibility in one dimensional cellular automata

In this work it is proved that necessary and sufficient conditions for binary regular, non-bounded, one dimensional cellular automata do exist. These conditions are derived from the definition of sufficiency for non reversibility and using a well known property of Boolean algebra. These conditions are dependent on the relative neighbourhood of the considered automaton and a graph search algorithm is derived for the establishment of these conditions and the associated listing of solutions. It is shown, however, that this general solution has an exponential complexity nature, depending on the number of cell neighbours. A simple example is also presented