Three-state one-dimensional cellular automata with memory

Standard cellular automata (CA) are ahistoric (memoryless): i.e., the new state of a cell depends on the neighborhood configuration only at the preceding time step. This article considers an extension to the standard framework of CA by considering automata implementing memory capabilities. While the update rules of the CA remain the same, each site remembers a weighted mean of all its past states, with a decreasing weight of states farther in the past. The historic weighting is defined by a geometric series of coefficients based on a memory factor (α). This paper considers the time evolution of one-dimensional, totalistic three-state CA with memory. The consideration of historic memory of past states has an inertial (or conservation) effect. According to this principle, on increasing the value of the memory factor from α 0.25, at which the evolution corresponds to the standard (ahistoric) model, to α=1.0 (fully historic), historic memory tends to introduce order even in the ahistoric chaotic CA rules. There is usually a gradual effect of memory, without the spurious extinctions found in the two-state scenario. Nevertheless the composition of the spatio-temporal patterns can vary notably at close α values for a considerable number of rules, some of which generate complex patterns with memory discounted (even if the ahistoric pattern is not complex at all). At variance with the two-state scenario, studied in earlier work, memory fires the pattern of some rules that die out in the ahistoric model.