Questions of decidability for context-free chain code picture languages

Abstract

A picture description is a word over the alphabet {u,d,r,l,↑↓} with the following interpretation: "move one unit line up (down, right, left, resp.) from the current point" for letter u (d,r,l, resp.) and "lift (sink) the pen" for ↑ (↓). The set of unit lines traversed with sunk pen is the picture described. A set of pictures given by a set of picture descriptions generated by a context-free grammar is called a context-free (chain code) picture language.

We show that the membership problem and the superpicture problem are undecidable for context-free and for linear context-free picture languages, while the subpicture problem is decidable. This result is in contrast to the situation for regular picture languages and for context-free picture languages generated without using a "lift the pen"-symbol. For these classes all three problems are known to be decidable, and the subpicture problem is harder than the superpicture problem (NP-complete instead of polynomial time).

The research was done during the author's stay at IIG, Institutes for Information Processing, Graz University of Technology, Schießstattgasse 4a, A-8010 Graz, Austria

To the glory of God, who gave wisdom to man