Operads of cellular automata and little n - cubes October 25

We demonstrate how to organize 1-dimensional cellular automata into an operad of spaces. The nth term C(k) is the space of radius r = k − 1 automata. The operad composition operation involves both automata composition and shifting of domain. Pointwise operations such as addition of automata become important when we look at the structure of the individual terms in the operad, the spaces of automata with a given radius. Having adopted the discrete topology on such a space, we demonstrate an action of the little n-cubes operad on (n−1)-dimensional radius r automata. There are clear applications of this action to parallel programming issues. Finally we discuss ways of generalizing both the idea of an operad of automata and the n-cubes action to higher dimensional automata, using higher dimensional operads.