In this talk, we will present a new way to do symbolic dynamics by using diagrams. Our diagrams are made of boxes with wires representing a generalized Wang tileset as relation with finite arity. The diagram obtained by connecting two or more copies of those boxes represents the composition of the underlying relations. By putting boxes on the vertices of a Cayley graph of a group G and connecting them along the edges, we obtain a diagram corresponding to a subshift.

More precisely, the boxes have an additional wire on a color alphabet A. We say that a configuration on the Cayley graph is accepted if imposing gradually the colors of it on the free wires of the boxes doesn't make the relation empty. The sets of accepted configurations are exactly the subshifts, and when the constraint alphabets are finite, we describe exactly the sofic subshifts. We show that these definitions don't depend on the choice of the Cayley graph of G. The periodicity property of a generalized tileset is characterized by its capacity to tile the Cayley graph of a quotient group of G.

Replacing relations by tensors with integer coefficients allows us to count the possibilities to tile some patterns. By putting real coefficients between 0 and 1 (resp. complex coefficients), we get a notion of probabilistic tilesets (resp. quantum tilesets, for which we observe interference phenomena). This is joint work with Titouan Carette and Étienne Moutot.