The Expected Value of Hierarchical Problem-Solving

In the best case using an abstraction hierarchy in problem solving can yield an exponential speed up in search e ciency Such a speed up is predicted by var ious analytical models developed in the literature and e ciency gains of this order have been con rmed empir ically However these models assume that the Down ward Re nement Property DRP holds When this property holds backtracking never need occur across abstraction levels When it fails search may have to consider many di erent abstract solutions before nd ing one that can be re ned to a concrete solution In this paper we provide an analysis of the expected search complexity without assuming the DRP We nd that our model predicts a phase boundary where abstraction provides no bene t if the probability that an abstract solution can be re ned is very low or very high search with abstraction yields signi cant speed up However in the phase boundary area where the probability takes on an intermediate value search e ciency is not nec essarily improved The phenomenon of a phase bound ary where search is hardest agrees with recent empirical studies of Cheeseman et al CKT