Finite Automata Encoding Geometric Figures

Finite automata are used for the encoding and compression of images. For black-and-white images, for instance, using the quad-tree representation, the black points correspond to ω-words defining the corresponding paths in the tree that lead to them. If the ω-language consisting of the set of all these words is accepted by a deterministic finite automaton then the image is said to be encodable as a finite automaton. For grey-level images and colour images similar representations by automata are in use. In this paper we address the question of which images can be encoded as finite automata with full infinite precision. In applications, of course, the image would be given and rendered at some finite resolution – this amounts to considering a set of finite prefixes of the ω-language – and the features in the image would be approximations of the features in the infinite precision rendering. We focus on the case of black-and-white images – geometrical figures, to be precise – but treat this case in a d-dimensional setting, where d is any positive integer. We show that among all polygons and convex polyhedra in d-dimensional space exactly those with rational corner points are encodable as finite automata. In the course of proving this we show that the set of images encodable as finite automata is closed under rational affine transformations. Several properties of images encodable as finite automata are consequences of this result. Finally we show that many simple geometric figures such as circles and parabolas are not encodable as finite automata. Finite Automata Encoding Geometric Figures 3