Fast identification of geometric objects with membership queries

We investigate the number of membership queries that are needed to identify polygons (i.e., intersections of halfplanes) over a two dimensional grid {0, ..., n ? 1}2. We exhibit a learning algorithm that learns 100% correctly, while requiring no random examples and not more membership queries than previous algorithms needed in addition to their random examples for probably almost correctly learning (even for moderate values of ?, ?). Furthermore, the learning algorithm in this paper uses only grid points for its membership queries. This appears to be appropriate in situations where the probing device has only a limited resolution, and for applications to the set of pixels on a two-dimensional screen. The learning algorithm that is described in this paper overcomes a fundamental obstacle related to the zigzag borderline that is generated by a halfplane over a discrete grid. This obstacle had thwarted previous attempts to show that, in Angluin?s model for on-line learning with equivalence and membership queries, one can learn in an efficient manner even those classes of geometric objects which cannot be learnt fast by equivalence queries alone (such as rectangles in general position).