to try reﬁning. This is thus another case where backtracking between levels occurs. This problem was further analysed and 
quantiﬁed by Bacchus and Yang [3], who suggested the stronger downward reﬁnement property (DRP). The DRP guarantees 
that every abstract plan can be reﬁned into a ground plan, but there are obviously fewer abstractions satisfying the DRP 
than  the  ordered  monotonicity  criterion.  Both  these  criteria  also  suffer  from  the  problem  that  they  give  no  quantitative 
guarantees on the solutions. In the best case, hierarchical abstraction can give an exponential speed-up over searching in 
the  ground  graph  [3,64].  However,  hierarchical  abstraction  can  be  counter-productive  in  the  worst  case.  Bäckström  and 
Jonsson [5] demonstrated two different abstraction hierarchies that both satisfy the DRP, but where the abstract solutions in
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