Cycle chain code picture languages

Abstract

A picture word is a string over the alphabet {u, d, l, r}. These symbols mean drawing of a unit line in direction up, down, left, and right, respectively. A picture word describes a walk in the plane, its trace is the picture it describes. A set of picture words describes a (chain code) picture language. A cycle means a closed curve in the discrete Cartesian plane. It is elementary, if the curve is simple and has no crossings. Cycles are among the most important features for chain code pictures. They are used as fundamental objects to build more complex objects, and occur, e.g., in the pictures of the digits "0", "6", "8" and "9", in icons for snowflakes, houses or trees and in complex kolam patterns. In general, cycles are non-context-free constructs and cannot be captured by context-free picture grammars. Therefore, we extend the concept of context-free picture grammars attaching cycles by the requirement that certain subpictures must be cycles or elementary cycles. We investigate basic properties of context-free cycle and elementary cycle grammars with emphasis on the complexity of the recognition problem. In particular, it is shown that the description complexity is polynomial for cycle languages and for unambiguous elementary cycle languages, and is NP-complete for ambiguous elementary cycle languages.