Comparison of Upward and Downward Generalizations in CF-Induction

CF-induction is a sound and complete procedure for finding hypotheses in full clausal theories. It is based on the principle of Inverse Entailment (IE), and consists of two procedures: construction of a bridge theory and generalization of it. There are two possible ways to realize the generalization task in CF-induction. The one uses a single deductive operator, called γ-operator, and the other uses a recently proposed form of inverse subsumption. Whereas both are known to retain the completeness of CF-induction, their favorite hypotheses, which tend to be found, are different from each other. In this paper, we introduce those two generalization approaches in CF-induction, and then investigate their logical relationship and features especially on each favorite hypotheses.