Faster Methods for Identifying Nontrivial Energy Conservation Functions for Cellular Automata

The biggest obstacle to the efficient discovery of conserved energy functions for cellular auotmata is the elimination of the trivial functions from the solution space. Once this is accomplished, the identification of nontrivial conserved functions can be accomplished computationally through appropriate linear algebra. As a means to this end, we introduce a general theory of trivial conserved functions. We consider the existence of nontrivial additive conserved energy functions (”nontrivials”) for cellular automata in any number of dimensions, with any size of neighborhood, and with any number of cell states. We give the first known basis set for all trivial conserved functions in the general case, and use this to derive a number of optimizations for reducing time and memory for the discovery of nontrivials. We report that the Game of Life has no nontrivials with energy windows of size 13 or smaller. Other 2D automata, however, do have nontrivials. We give the complete list of those functions for binary outer-totalistic automata with energy windows of size 9 or smaller, and discuss patterns we have observed. 1. Preliminaries: basic definitions We consider cellular automata with k states in n dimensions. The neighborhood of a cellular automaton is the region of surrounding cells used to determine the next state of a given cell. The window of an energy function for a cellular automaton is the region of adjacent cells that contribute to the function. Both neighborhoods and windows are n-dimensional tensors, with the size of each dimension specified as a positive integer. Given the size of such a tensor, it is useful to define the following 3 sets of tensors. Definition 1.1. Cellular automata are composed of cells, each of which is in one of k states (or colors) at any given time. The set C is the set of such colors, and the set C∗ is that set augmented with another color named *. (* denotes a special state with certain properties that simplify our proofs. It is explained in more detail in the pages that follow.) C = {0, 1, 2, . . . , k − 1} (1.1) C∗ = C ∪ {∗} (1.2) It is sometimes useful to choose one color to be treated specially. In all such cases, the color 0 will be chosen. Definition 1.2. An n-dimensional cellular automaton rule is a function R that gives the color of a given cell on the next time step as a function of a neighborhood of cells centered on that cell on the current time step. The neighborhood is an ndimensional tensor of size w1 × · · · ×wn, where each wi is an odd, positive integer.