Generalisation of a synchronous solution of the parity problem on cyclic configurations over a non-circulant graph

Abstract

The parity problem is a classical binary benchmark for addressing the computational ability and limitations of automata networks. It refers to conceiving a local rule to allow deciding whether the number of 1-states in the nodes of an arbitrary network is an odd or even number, without global access to the nodes. In its standard formulation, the automata network has an odd number of nodes whose states, arranged as a cyclic configuration, should converge to a fixed point of all 0s, if the initial configuration has an even number of 1s, or to a fixed point of all 1s, otherwise. It has been shown that a local rule alone is able to solve the problem in this formulation, including a synchronous solution we have recently shown, totally based on the local parity rule of the cellular automata elementary space (number 150), with a certain pattern of connection between the nodes. Here, we generalise this solution, showing how to construct many others, combining rule 150 with rules 170 and 240, which are the local shifts of the same space, in such a way that the original solution is just one among countless possibilities. Such solutions may have different convergence times for specific configurations, but are equivalent in the context of all configurations of a given size; also, various solutions form symmetric pairs, in terms of the actual rules they used. The solutions were evaluated computationally and presented here without their formal proofs, but empirical evidences strongly suggest that they can be obtained by the same kind of technique we used in the solution exclusively with rule 150.