Retreat Bounded Picture Languages

A well-known string-encoding of 2D pictures is to use the picture alphabet π = {u, d, r, l}, where u (d, r, and l) means "draw a unit-line by moving the pen up (down, right, and left) from the current point". A word over π is k-retreat-bounded (k?0) if it describes a picture in such a manner that the maximum distance of left-moves, ignoring up- and down-moves, from a rightmost point of any partially drawn picture is bounded by k. A set of such words forms a k-retreat-bounded language. A k-retreat-bounded picture language is a set of pictures described by a k-retreat-bounded language.There is a 1-retreat-bounded regular picture language (1-retreat-bounded vertical-stripe linear picture language) for which the membership problem is NP-complete. The membership problem can be solved in O(n4) time for each k-retreat-bounded nonvertical-stripe context-free picture language. The inclusion and intersection-emptiness problems are undecidable for a 1-retreat-bounded regular picture language and a 2-retreat-bounded regular picture language. The equivalence and ambiguity problems are undecidable for 2-retreat-bounded regular picture languages. The picture recognition algorithm presented is the first polynomial-time algorithm in the literature for a reasonably large subclass of context-free picture languages and the NP-completeness and undecidability results improve the previously known such results for regular picture languages with no structural restriction imposed.

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