Irreducible Rules and Equivalence Classes of One-dimensional Cellular Automata
Martin Schaller, Karl Svozil
TL;DR
The paper develops a symmetry-driven framework for classifying one-dimensional cellular automata by introducing irreducible local rules as canonical representatives and embedding them into a comprehensive group action that includes state-permutation, lattice reflection, translation, and neighbourhood scaling. It proves a bijection between irreducible local rules and global maps, derives exact counts for binary irreducible rules and their stabilizer types, and counts equivalence classes of binary global maps for contiguous neighborhoods. Notably, including scaling reduces the classical 88 equivalence classes of elementary cellular automata to 81, and accounting for topological conjugacies further yields 76, with additional Omega-relations collapsing 19 classes into 7. Overall, the work provides a rigorous, geometric, and algebraic coarse-graining of CA dynamics that complements CHL and topological-conjugacy perspectives while highlighting irreducible rules as canonical representatives for symmetry-based classification.
Abstract
One-dimensional cellular automata are discrete dynamical systems that operate on an infinite lattice of sites and are characterized by the locality and uniformity of their update rule. Permutations of the state set and isometric transformations of the lattice induce symmetry transformations on the set of local rules and the set of global maps of cellular automata, resulting in a partitioning of the set of cellular automata into equivalence classes. The concept of an irreducible local rule that depends on all its coordinates is used to analyse the equivalence classes and results on the number of equivalence classes of irreducible binary local rules and binary global maps are presented. Finally, another symmetry operator based on the scaling of neighbourhoods is introduced and the change in the number of equivalence classes is analysed.