Computing Modulo-n by Composing Cellular Automata Rules

The understanding of how predefined computations can be attained by means of individual cellular automata rules, their spatial arrangements or their temporal sequences, is a key conceptual underpinning in the general notion of emergent computation. In this context, here we construct a solution to the MODn problem, which is the determination of whether the number of 1-bits in a cyclic binary string is perfectly divisible by the integer n > 1. Our solution is given for any lattice size N that is co-prime to n, and relies upon a set of one-dimensional rules, with maximum radius of n - 1, organised in a temporal sequence. Although the simpler cases of the problem for n = 2 and n = 3 have been addressed in the literature, this is the first account on the general case, for arbitrary n.