Downward Re nement and the E ciency of Hierarchical Problem Solving

Analysis and experiments have shown that hierarchical problem solving is most ef fective when the hierarchy satis es the downward re nement property DRP whereby every abstract solution can be re ned to a concrete level solution without backtracking across abstraction levels However the DRP is a strong requirement that is not often met in practice In this paper we examine the case when the DRP fails and provide an analytical model of search complexity parameterized by the probability of an abstract solution being re nable Our model provides a more accurate picture of the e ective ness of hierarchical problem solving We then formalize the DRP in Abstrips style hierarchies providing a syntactic test that can be applied to determine if a hierarchy satis es the DRP Finally we describe an algorithm called Highpoint that we have developed This algorithm builds on the Alpine algorithm of Knoblock in that it au tomatically generates abstraction hierarchies However it uses the theoretical tools we have developed to generate hierarchies superior to those generated by Alpine This superiority is demonstrated empirically Abbreviated Title The Downward Re nement Property This work is supported by grants from the Natural Science and Engineering Council of Canada and ITRC Information Technology Research Centre of Ontario The authors e mail addresses are fbacchus logos waterloo edu and qyang logos waterloo edu Parts of Section have previously appeared at AAAI and parts of Section have previously appeared at IJCAI