Regular chain code picture languages of nonlinear descriptional complexity

Abstract

A picture description is a word over an alphabet such as {u,d,r,l}, where u means "go one unit line up from the current point", and d,r,l are interpreted analogously with "down", "right", "left" instead of "up". A picture description describes a walk in the plane. The set of unit lines traversed by this walk is the corresponding picture. A regular (chain code) picture language is a set of pictures corresponding to a regular set of picture descriptions. It is shown in this paper that there exists a regular picture language P which is of non-linear descriptional complexity (in the sense that, for every regular picture description language corresponding to P, the growth of the length of picture descriptions is quadratic in the size of the corresponding pictures). This solves a problem raised by Maurer, Rozenberg and Welzl.