A unifying approach to picture grammars

Several old and recent classes of picture grammars, that variously extend context-free string grammars in two dimensions, are based on rules that rewrite arrays of pixels. Such grammars can be unified and extended using an approach, whereby the right part of a rule is formalized by means of a finite set of permitted tiles. We focus on a simple type of tiling, named regional, and define the corresponding regional tile grammars. They include both Siromoney’s (or Matz’s) Kolam grammars and their generalization by Pru˚ ša, as well as Drewes’s grid grammars. Regionally defined pictures can be recognized with polynomialtime complexity by an algorithm extending the CKY one for strings. Regional tile grammars and languages are strictly included into our previous tile grammars and languages, and are incomparable with Giammarresi–Restivo tiling systems (or Wang systems).