6 Acknowledgements

We study the predictability of geometric concepts, in particular those de ned by boolean combinations of simple geometric objects. First, we give a negative result, showing that the problem of predicting the class of convex polytopes encoded by listing their vertices is prediction complete for P . Thus, an e cient solution to this prediction problem implies the existence of e cient solutions to all prediction problems whose associated evaluation problem is in P . Assuming the existence of a one-way function that is hard on iterates, there are such prediction problems which do not admit e cient solutions. Thus we show under minimalist cryptographic assumptions that the class of convex polytopes encoded by listing their vertices is not predictable. As a side e ect, we show that determining membership in the convex hull of a given set of points is complete for P with respect to log space reductions. Next, we establish the predictability of the class consisting of unions of a xed number of ats by reducing its prediction problem to that of the class of ats, which has previously been shown to be predictable. Finally, we give an Occam algorithm for predicting xed nite unions of boxes. Both constructive results for ats and boxes hold if the dimension is variable. This work supported by ONR grant N00014-86-K-0454. The authors' email addresses are [email protected] and [email protected].