Constructing refinement operators by decomposing logical implication
Inductive learning models [Plotkin 1971; Shapiro 1981] often use a search space of clauses, ordered by a generalization hierarchy. To find solutions in the model, search algorithms use different generalization and specialization operators. In this article we will decompose the quasi-ordering induced by logical implication into six increasingly weak orderings. The difference between two successive orderings will be small, and can therefore be understood easily. Using this decomposition, we will describe upward and downward refinement operators for all orderings, including $theta$-subsumption and logical implication.