Picture languages: Tiling systems versus tile rewriting grammars

Theoretical Computer Science (Impact Factor: 0.52). 05/2006; 356(1-2):90-103. DOI: 10.1016/j.tcs.2006.01.038
Source: DBLP

ABSTRACT 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.

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    • "Recently the investigation of two dimensional (picture) languages has moved towards the definition of formal models capable of characterizing special classes of languages that are not included in the family of recognizable languages generated with tiling systems of Gianmmaresi and Restivo [6]. An example of such models is tile rewriting grammars (TRG) defined in [5] and further investigated in [4]. Indeed, while tiling systems represent an extension to the two dimensional case of regular string languages, TRG provide an analogue of context-free grammars in the two dimensions, thus showing the capability of this approach of generating interesting picture languages that generalize context-free string languages, such as Dyck languages. "
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    ABSTRACT: [9]. Such abstract machines,are more powerful then the four ways automata of [10]. However TS definitions are hard to write and error-prone for non elementary pictures. Moreover the NP-complete computational complexity of picture recognition has until now blocked any attempt to realistic experimentation and application of TS, in spite of a large amount of theoretical work. Our work is concerned with a practical experimentation of tiling systems/Wang tiles in conjunction with a new approach for performing pattern recognition and image gen- eration or completion, based on powerful logical tools, the SAT-solvers, whose task is to find Boolean values which make a propositional formula true. We have implemented a recognizer/generator for TS defined pictures in a very attractive, unconventional way, by transforming the tiling problem into a Boolean satisfiability one, then using an effi- cient off-the-shelf SAT-solver. The tool is invaluable to assist in writing picture speci-
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