The Specialization Problem and the Completeness of Unfolding

We discuss the problem of specializing a deenite program with respect to sets of positive and negative examples, following BI94]. This problem is very relevant in the eld of inductive learning. First we show that there exist sets of examples that have no correct program, i.e., no program which implies all positive and no negative examples. Hence it only makes sense to talk about specialization problems for which a solution (a correct program) exists. To solve such problems, we rst introduce UD 1-specialization, based upon the transformation rule unfolding. We show UD 1-specialization is incomplete|some solvable specialization problems do not have a UD 1-specialization as solution|and generalize it to the stronger UD 2-specialization. UD 2 also turns out to be incomplete. An analysis of program specialization, using the subsumption theorem for SLD-resolution, shows the reason for this incompleteness. Based on that analysis, we then de-ne UDS-specialization (a generalization of UD 2-specialization), and prove that any specialization problem has a UDS-specialization as a solution. We also discuss the relationship between this specialization technique, and the generalization technique based on inverse resolution. Finally, we go into several more implementational matters, which outline an interesting topic for future research.