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

Abstract

A recognizable picture language is defined as the projection of a local picture language defined by a set of two-by-two tiles, i.e. by a strictly-locally-testable (SLT) language of order 2. The family of recognizable picture languages is also defined, using larger $k$ by $k$ tiles, $k>2$, by the projection of the corresponding SLT language. A basic measure of the descriptive complexity of a picture language is given by the size of the SLT alphabet using two-by-two tiles, more precisely by the so-called alphabetic ratio of sizes: SLT-alphabet / picture-alphabet. We study how the alphabetic ratio changes moving from tiles of size two to tiles of larger size, and we obtain the following 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 codes, for which a lower bound on numerosity is established; and on the relation of languages of encoded pictures with 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.