On the complexity of two-dimensional signed majority cellular automata

•The introduction of the family of signed majority cellular automata (SMCA).•Dynamics of the family of SMCA and their computational complexity.•A general framework that relates circuit simulation with CA complexity.•Examples of SMCA that are Turing-Universal and Intrinsic-Universal.•Three equivalence classes of uniform SMCA: symmetric, antisymmetric and asymmetric.

We study the complexity of signed majority cellular automata on the planar grid. We show that, depending on their symmetry and uniformity, they can simulate different types of logical circuitry under different modes. We use this to establish new bounds on their overall complexity, concretely: the uniform asymmetric and the non-uniform symmetric rules are Turing universal and have a P-complete prediction problem; the non-uniform asymmetric rule is intrinsically universal; no symmetric rule can be intrinsically universal. We also show that the uniform asymmetric rules exhibit cycles of super-polynomial length, whereas symmetric ones are known to have bounded cycle length.