Conserved Energy Functions for Cellular Automata: Finding Nontrivials Faster Through a Complete Theory of the Trivials

Conserved quantities indicate fundamental physical properties of systems, and are therefore eagerly sought after in science. Conservation of energy and conservation of angular momentum, for example, are two fundamental principles that offer profound insights into the nature of the world we live in. Cellular automata, as models of physical systems, can exhibit conserved functions of relevance to the system under study. Number conservation is a simple example, but far more sophisticated ones can be discovered with appropriate algorithmic techniques. Cellular automata with identical conserved functions are, in some sense, closely related to one another. Thus conserved functions can be used to classify cellular automata, and to identify connections between seemingly unrelated systems.