Fast solutions to k-parity and k-synchronisation using parallel automata networks

We present a family of automata networks that solve the k-parity problem when run in parallel. These solutions are constructed by connecting cliques in a non-cyclical fashion. The size of the local neighbourhood is linear in the size of the alphabet, and the convergence time is proven to always be the diameter of the interaction graph. We show that this family of solutions can be slightly altered to obtain an equivalent family of solutions to the k-synchronisation problem, which means that these solutions converge from any initial configuration to the cycle which contains all the uniform configurations over the alphabet, in order.