New Conservation Functions and a Partial Taxonomy for 1-D Cellular Automata

We present algorithms that permit increased efficiency in the calculation of conservation functions for cellular automata, and report results obtained from implementations of these algorithms to report conservation laws for 1-D cellular automata of higher order than any previously known. We introduce the notion of trivial and core conservation functions to distinguish truly new conservation functions from simple extensions of lower-order ones. We give new theorems related to these concepts, and show our use of them to derive more efficient algorithms for finding conservation functions. We then present the complete list of conservation functions up to order 16 for the 256 elementary 1-D binary cellular automata. These include CAs that were not previously known to have nontrivial conservation functions.