Abstraction and Phase Transitions

region πa2 (26,50) πa1 (13,98) Figure 7 – Phase Transition curves in (L,m) and (N,L) plane with the useful area for abstraction Ideally, if a ground problem lies in the mushy region and has solution, then a useful abstraction operator shall map it into an abstract problem lying in the YES region. On the contrary, if a ground problem lies in the mushy region and has no solutions, then a useful abstraction operator shall map it into an abstract problem lying in the NO region. 5.1. Uninformed abstraction operator A possible generic abstraction consists in reducing the cardinality N of the tables in such a way that problems will fall into the NO-region (see problem πa2 in Figure 7). This transformation satisfies the so-called downard solution property, i.e. a solution in the abstract space can always be mapped back to a solution in the ground space. Giunchiglia & Walsh (1992) calls such abstraction a Theorem-Decreasing (TD-)abstraction because solutions may be lost in the case all ground solutions require some of the deleted tuples. Finally, the drawback of such an operator is that it is blind.