Tile Rewriting Grammars

Theoretical Computer Science, 2005

Tile Rewriting Grammars (TRG) are a new model for defining picture languages. A rewriting rule changes a homogeneous rectangular subpicture into a isometric one tiled with specified tiles. Derivation and language generation with TRG rules are similar to contextfree grammars. A normal form and some closure properties are presented. We prove this model has greater generative capacity than the Tiling Systems of Giammarresi and Restivo and the grammars of Matz, another generalization of context free string grammars to 2D. Examples are shown for pictures made by nested frames and spirals.

Theoretical Computer Science, 2006

Two formal models of pictures, i.e., two dimensional (2D) languages are compared: tiling systems and tile rewriting grammars, which resp. extend to 2D the regular and context-free languages. Two results extending classical language properties into 2D are proved. First, non-recursive tile writing grammars (TRG) coincide with tiling systems (TS). Second, non-self-embedding TRG are suitably defined as corner grammars, showing that they generate TS languages. The proofs exploit newly introduced language substitutions, also nested and iterated.

Lecture Notes in Computer Science

Several classical models of picture grammars based on array rewriting rules can be unified and extended by a tiling based approach. The right part of a rewriting rule is formalized by 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 Průša. 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 the tile grammars and languages, and are incomparable with Giammarresi-Restivo tiling systems (or Wang's tilings).

Discrete Applied …, 2009

A new syntactic model, called pure two-dimensional (2D) context-free grammar (P2DCFG), is introduced based on the notion of pure context-free string grammar. The rectangular picture generative power of this 2D grammar model is investigated. Certain closure properties are obtained. An analogue of this 2D grammar model called pure 2D hexagonal context-free grammar (P2DHCFG) is also considered to generate hexagonal picture arrays on triangular grids.

Information and Computation, 2011

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 a tiling based 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 Průša, 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).

Lecture Notes in Computer Science, 2012

Regional hexagonal Tile rewriting grammars(RHTRG) are the recently introduced hexagonal picture generating devices which used a simple type of tiling called regional hexagonal tiling. This model is having isometric rules of derivation and more general than context-free Hexagonal array grammars (HAG) but incomparable with hexagonal tiling systems (HTS). In this paper we compare RHTRG with some parallel generating formalisms like Extended pure 2D hexagonal context-free grammars with regular control and hexagonal array token Petri net Structure with respect to generating capacity..

Lecture Notes in Computer Science, 2009

We propose a new model for defining picture languages inspired by the idea of formalizing an assembly mechanism of tiles based on rules. More precisely, a picture language will be generated from a finite set of initial pictures by iteratively applying rewriting rules from a given finite set of rules, called a tiling rule system ( t-RS system). We prove this model has greater generative capacity than the tiling systems of Giammarresi and Restivo, even in the case they generate one-letter alphabet picture languages. Using tiling rules systems we are able to show a different way of assembling recognizable pictures languages w.r.t. the one of non recognizable languages.

2016

Here we introduce a variant of extended two-dimensional context-free picture grammar (E2DCF P G), called (l/u)E2DCF P G which allows rewriting only the leftmost column, or the uppermost row of variables in a picture array. Several theoretical properties of (l/u)E2DCF P G are obtained and an application in generating digitized picture arrays is discussed.

International Journal of Recent Technology and Engineering

In formal languages, picture language is generalization of string language theory to two dimensions. Pictures which may be regarded as two-dimensional objects occur in studies concerning recognition of patterns, images and various computational fields. Several studies have been done for generating and/or recognizing higher dimensional objects using formal models. Tile rewriting grammar (TRG) is yet another model introduced for generating picture languages. TRG combines isometric rewriting rules with the Giammaresi and Restivo’s Tiling system. This rewriting grammar generates spirals, square and rectangular grids. The power of generating pictures by tile rewriting grammar is more than REC .Sweety et al have generated hexagonal pictures, introducing hexagonal Tile Rewriting Grammar. Kuberalet al have introduced Triangular Tile Rewriting Grammar to generate Triangular Pictures. A special class of objects namely Oxide pictures have been of interest recently. Oxide network is a special c...

Journal of Computer and System Sciences, 1996

Language theoretic aspects and algorithmic properties of particular classes of context-free collage languages and of patterns generated by iterated function systems are studied. These classes are defined by restricting the allowed transformations to a sort of similarity transformations called grid transformations. It turns out that, thanks to this restriction, the language classes have nice closure properties, and non-trivial questions concerning the generated pictures can be decided.