A unifying approach to picture grammars

TL;DR

This paper introduces regional tile grammars, a unifying framework for picture grammars that extends and generalizes existing models using a tiling approach, enabling polynomial-time recognition.

Contribution

It proposes regional tile grammars as a simple, unifying extension for various picture grammars, including recognition algorithms and complexity analysis.

Findings

Regional tile grammars include several existing picture grammar classes.

Recognition of regional pictures is polynomial-time.

Regional tile grammars are strictly included in previous tile grammars.

Abstract

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 Prusa, as well as Drewes's 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).