Reducing the local alphabet size in tiling systems by means of 2D comma-free codes

The tiling systems define the family of recognizable picture languages as the projection of local picture languages, i.e., strictly-locally-testable (SLT) languages of order 2. A basic measure of the descriptive complexity of a picture language is given by the so-called alphabetic ratio, i.e., the size of the local alphabet divided by the size of the picture alphabet. The family of recognizable picture languages does not change when defined by the projection of SLT languages of order k>2, which use k by k tiles. In such a case, the natural question is how the alphabetic ratio changes We obtain the following main result: any recognizable picture language over an alphabet of size n is the projection of an SLT language over an alphabet of size 2n. Moreover, two is the minimal alphabetic ratio possible in general. The proof relies on a new family of comma-free picture, i.e., 2D, codes, for which a lower bound on numerosity is established; and on the relation between languages of encoded pictures and SLT languages. Our result reproduces in two dimensions a similar property (known as Extended Medvedev's theorem) of the regular word languages, concerning the minimal alphabetic ratio needed to define a language by means of a projection of an SLT word language.