ON EXPLICIT EXPRESSIONS IN ADDITIVE CELLULAR-AUTOMATA THEORY

In the present paper, the problem of additive cellular automata is posed as a second-order linear difference equation in two indices-time step t and position i subject to a periodicity condition for all t and an initial condition for t = 0. An explicit solution of this mathematical problem is obtained by using the method of generating function. This explicit formula enables us to compute the configuration at any time without any explicit simulation. The explicit formula has been derived for all important additive rules; with the help of these, many salient characteristics of the transition may be obtained with surprising ease. The explicit formula method appears to be very efficient and has possibilities to be applied to diverse situations in cellular automata theory.