Complexity of the Freezing Majority Rule with L-shaped Neighborhoods

Abstract: In this article we investigate the computational complexity of predicting two dimensional freezing majority cellular automata with states ${-1,+1}$, where the local interactions are based on an L-shaped neighborhood structure. In these automata, once a cell reaches state $+1$, it remains fixed in that state forever, while cells in state $-1$ update to the most represented state among their neighborhoods. We consider L-shaped neighborhoods, which mean that the vicinity of a given cell $c$ consists in a subset of cells in the north and east of $c$. We focus on the prediction problem, a decision problem that involves determining the state of a given cell after a given number of time-steps. We prove that when restricted to the simplest L-shaped neighborhood, consisting of the central cell and its nearest north and east neighbors, the prediction problem belongs to $\mathsf{NC}$, meaning it can be solved efficiently in parallel. We generalize this result for any L-shaped neighborhood of size two. On the other hand, for other L-shaped neighborhoods, the problem becomes $\mathsf{P}$-complete, indicating that the problem might be inherently sequential.