Evolutionary Trajectories by Recursive Analysis of 1D Cellular Automata

In this paper, extensive analysis has been done using the recursive evaluation technique on One-Dimensional Cellular Automata (1D CA). This approach focuses on tracking the formation of cell values and identifying the contributing cells from the initial configuration at a given generation. Coining the new naming convention of fundamental rules in 1D CA, along with their groups and different Boolean operators, the closed-form formula has been derived for certain CA. Cell values at the [Formula: see text]th generation without even explicitly computing the intermediate iteration steps have been attempted. These are derived through nested symbolic expressions that trace back to the initial state, enabling both symbolic and visual analysis of rule evolution. Our approach accommodates both uniform rules, characterized by a common update function applied to all cells and nonuniform rule sets, in which individual cells evolve under distinct logical functions.