Construction of One-Dimensional Nonuniform Number Conserving Elementary Cellular Automata Rules

Abstract

An effort to study one-dimensional nonuniform elementary number conserving cellular automata (NCCA) rules from an exponential order rule space of cellular automata is an excellent computational task. To perform this task effectively, a mathematical heritage under the number of conserving functions over binary strings of length [Formula: see text] has been highlighted along with their number conserving cellular automata rules (either uniform or nonuniform). A basic approach for the construction of some feasible nonuniform NCCA rules of any finite configuration with the assistance of nine uniform elementary CA rules has been investigated. From our construction procedure, recurrence equations are formulated as suitably solved to ascertain the actual range of NCCA rules. The state transition diagrams (STDs) of NCCA rules are analyzed. While classifying the binary strings through STDs, we found a fascinating optical insight that equal weight strings from a class whose cardinality is the same as the binomial coefficient [Formula: see text] where [Formula: see text] is the length and [Formula: see text] is the weight of the binary string.

Non-uniform number-conserving elementary cellular automata

“…We start with the following known fact, which is independent of the type of boundary condition used (see, for instance, [29]).…”

“…We take this point of view for two reasons. First of all, if one is not interested in cyclic independence, then for n ≥ 5 the answer is already given in [29] (and for n < 5 in the previous sections). The second reason is that in the case of periodic boundary conditions, we believe that for a given H, it is quite natural to consider each of its cyclic shifts as being equivalent to H, rather than as different from H.…”

“…Recently, a very interesting paper about the construction of ν-ECAs was published [29]. On the basis of a computational programming approach, Pal et al formulate several theorems concerning ν-ECAs with periodic boundary conditions.…”