Picture Languages: From Wang Tiles to 2D Grammars
CAI '09 Proceedings of the 3rd International Conference on Algebraic Informatics
An Eilenberg Theorem for Pictures
CAI '09 Proceedings of the 3rd International Conference on Algebraic Informatics
Selected decision problems for square-refinement collage grammars
Algebraic Foundations in Computer Science
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Algebraic picture generation based on a pixel deformation theory is presented. The main tool used is the deformation monoid which simulates the algebraic structure of pictures viewed as rectangular arrays with operations the horizontal and vertical concatenation. Picture languages generated by grammatical systems are considered and a Chomsky-like normal form as well as an iteration lemma are established. Infinite pictures are obtained as the ω-completion of the set of finite pictures ordered by picture refinement. Regular fractal pictures (such as the Sierpinski Carpet, the Cantor dust, etc.) are defined as the components of the least solution of systems whose right hand side members are finite pictures. They constitute the least class of pictures containing the finite pictures and closed under substitution and the self similarity operation. Solving non deterministic picture program schemes we get the so called ∞-refinement languages which consist of finite and infinite pictures. For such languages the emptiness and finiteness problems are decidable.