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 Siromoneyʼs (or Matzʼs) Kolam grammars and their generalization by Průsa, as well as Drewesʼs grid grammars. Regionally defined pictures can be recognized with polynomial-time 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).

A unifying approach to picture grammars

The aim of this paper is to collect definitions and results on the main classes of 2D languages introduced with the attempt of generalizing regular and context-free string languages and in same time preserving some of their nice properties. Almost all the models here described are based on tiles. So we also summarize some results on Wang tiles and its applications.

Picture Languages: From Wang Tiles to 2D Grammars

A CKY parser for picture grammars

A Siromoney matrix grammar is a simple device for the generation of rectangular pictures. In this paper we discuss Siromoney matrix grammars whose picture language are slender, i.e., they contain only a bounded number of pictures for each size. In particular, it is shown that the slenderness decision problem is decidable for Siromoney matrix grammars with context-free rules. Moreover, some closure and decidability questions for slender Siromoney matrix languages are discussed.

Slender Siromoney matrix languages

In formal language theory, the Siromoney matrix grammars generate matrix languages. They are two dimensional languages which are \(m \times n\) arrays of terminals. In string languages, the ability of a regular language to dissect an infinite language into two partitions of infinite size has already been studied under the dissecting power of regular languages. In this paper we extend this special dissecting capacity of certain classes of string languages to matrix languages. The results demonstrate the matrix dissectibility of certain classes of matrix languages like infinite recursive matrix languages, constantly growing matrix languages (CGML), languages that are not CGML immune and CF:CF Siromoney matrix languages. In this paper the objectives of the study, extension methodology and results are discussed in detail.

Viswanathan Radhakrishnan

Papers

Pattern matching in matrix grammars