Handbook of formal languages, vol. 3
Model checking
A machine program for theorem-proving
Communications of the ACM
Wang Tiles for image and texture generation
ACM SIGGRAPH 2003 Papers
Picture languages: tiling systems versus tile rewriting grammars
Theoretical Computer Science - In honour of Professor Christian Choffrut on the occasion of his 60th birthday
Automata on a 2-dimensional tape
FOCS '67 Proceedings of the 8th Annual Symposium on Switching and Automata Theory (SWAT 1967)
Picture Languages: From Wang Tiles to 2D Grammars
CAI '09 Proceedings of the 3rd International Conference on Algebraic Informatics
A computational model for tiling recognizable two-dimensional languages
Theoretical Computer Science
Deterministic and Unambiguous Families within Recognizable Two-dimensional Languages
Fundamenta Informaticae
Hi-index | 0.01 |
Pictures or patterns have been formally specified by different methods such as grammars. An alternative approach is based on tiling systems (TS) (Wang tiles are an analogous and equivalent formalism), whereby the picture is obtained by first covering it with a specified set of 2x2 tiles, then by performing a pixel by pixel mapping. TS are a powerful technique: the corresponding pictures can be recognized by non-deterministic cellular automata, which are more powerful than the four-ways automata. The difficulty to write such specifications for non-elementary pictures, and the NP-complete computational complexity of TS picture recognition have so far blocked any attempt to application. We have implemented a recognizer and generator for TS pictures in a very attractive, unconventional way, by transforming the tiling problem into a SAT (Boolean satisfiability) one, then using an efficient off-the-shelf SAT-solver. The prototype is fast enough to experiment on reasonably sized samples, and has the bonus of being able to complete or extrapolate a partial or noisy picture. The tool is invaluable to assist in writing picture specification. A series of examples shows how to specify patterns using TS.