Reversibility of general 1D linear cellular automata over the binary field [formula omitted] under null boundary conditions

Abstract

The reversibility of cellular automata(CA) has long been studied but there are still many problems untouched. This paper tackles the reversibility problem of the general case concerning 1D CA under null boundary conditions defined by linear rules over the binary field Z2. Although transition matrix has been widely used for some special linear CA(LCA) rules, it has severe limitations: its complexity depends on the number of cells, and determining its reversibility becomes another unsolved tough problem for general rules. By constructing deterministic finite automata(DFA) of the de Bruijn graph presentation of CA, we conclude the reversibility problem of all 1D linear rules over Z2 under null boundary conditions. It turns out that any 1D LCA, except for some unilateral rules, over Z2 under null boundary conditions can be reversible once having a proper number of cells, and DFA provides an efficient way to find those numbers. Moreover, the complexity of the DFA solution is independent of the number of cells. In addition, the relative low complexity of DFA can be further significantly reduced by using only a small part of it.