Elementary cellular automata deterministically map a binary sequence to another using simple local rules. Visualizing the structure of this mapping is difficult because the number of nodes (i.e. possible binary sequences) grows exponentially. If periodic boundary conditions are used, rotation of a sequence and rule application to that sequence commute. This allows us to recover the rotational invariance property of loops and to reduce the number of nodes by only considering binary necklaces , the equivalence class of n-character strings taking all rotations as equivalent. Combining together many equivalent histories reveals the general structure of the rule, both visually and computationally. In this work, we investigate the structure of necklace-networks induced by the 256 Elementary Cellular Automata rules and show how their network structure change as the length of necklaces grow.

Catastrophic forgetting continues to severely restrict the learnability of controllers suitable for multiple task environments. Efforts to combat catastrophic forgetting reported in the literature to date have focused on how control systems can be updated more rapidly, hastening their adjustment from good initial settings to new environments, or more circumspectly, suppressing their ability to overfit to any one environment. When using robots, the environment includes the robot's own body, its shape and material properties, and how its actuators and sensors are distributed along its mechanical structure. Here we demonstrate for the first time how one such design decision (sensor placement) can alter the landscape of the loss function itself, either expanding or shrinking the weight manifolds containing suitable controllers for each individual task, thus increasing or decreasing their probability of overlap across tasks, and thus reducing or inducing the potential for catastrophic forgetting.