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

TL;DR

The paper addresses distributed computation of parity-type properties and synchronization in automata networks. It introduces Clique Tree ($CT$) constructions that grow by attaching cliques of size $k+1$ to realize the sum modulo $k$ as the updating rule, ensuring convergence in $r$ steps to $s^n$ where $s$ is the initial sum modulo $k$, and a negation-based variant yields $k$-synchronisation. A central result is that, for any finite alphabet $\\Sigma$ of size $k$, the networks in $CT^k$ converge in time equal to the interaction graph diameter, with acceleration possible by projecting from larger alphabets via the projection $\\pi_{mk,k}$. The approach generalizes to any finite alphabet and provides a topologically driven method for reliable distributed convergence, with potential extensions to other tasks such as density classification.

Abstract

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.