Linear cellular automata over Z

Reversibility of general 1D linear cellular automata over the binary fieldZ2under null boundary conditions

“…The reversibility problem of 1D CA with no boundary condition considered was already solved [3], while for two or higher dimensional CA, it was proved to be generally undecidable whether a CA rule is invertible or not [18]. When restricted to linear rules, however, there exist some solutions to the reversibility problem of a two or higher dimensional CA [17], and the inverse CA can also be computed [22]. For more information about the reversibility problem of these conventional CA please refer to [19,24].…”

Digital phagograms: predicting phage infectivity through a multilayer machine learning approach

“…The difficulties referred to above have been overcome in the case of a particular class of CAs, namely linear CAs (or additive CAs) with Z m as state set. It turned out that such CAs are particularly amenable to a theoretical analysis using tools from linear algebra, due to the linearity, or formal power series representations, due to the fact that the state set is a finite commutative ring (see, for example, [15][16][17][18] and [19][20][21][22][23]). More information about the reversibility problem of these types of CA can be found in [24][25][26].…”

A family of sand automata

“…In general we work exclusively with one dimensional sand automata (ie 1 those whose corresponding cellular automata act on two dimensional configuration space), whose local rules have radius 1. We frequently work with one fixed sand automaton, Γ : A Z → A Z , whose local rule is built using the rule γ : Z 3 5 → Z 5 defined by γ(x −1 x 0 x 1 ) = x −1 + x 1 , which is the local rule for the famous XOR cellular automaton (albeit on a different alphabet), and whose dynamical properties have been studied extensively, for example in [MM98], [CFMM97] and [IÔN83].…”