We present in this paper the original notion of natural relation, a quasi order that extends the idea of generality order: it allows the sound and dynamic pruning of hypothese that do not satisfy a property, be it completeness or correctness with respect to the training examples, or hypothesis language restriction. Natural relations for conjunctions of such properties are characterized. Learning operators that satisfy these complex natural relations allow pruning with respect to this set of properties to take place before inappropriate hypotheses are generated. Once the natural relation is defined that optimally prunes the search space with respect to a set of properties, we discuss the existence of ideal operators for the search space ordered by this natural relation. We have adapted the results from [vdLNC94a] on the non-existence of ideal operators to those complex natural relations. We prove those nonexistence conditions do not apply to some of those natural relations, thus overcoming the previous negative results about ideal operators for space ordered by theta-subsumption only.
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