Minimal distance-based generalisation operators for first-order objects

Distance-based methods have been a successful family of machine learning techniques since the inception of the discipline. Basically, the classification or clustering of a new individual is determined by the distance to one or more prototypes. From a comprehensibility point of view, this is not especially problematic in propositional learning where prototypes can be regarded as a good generalisation (pattern) of a group of elements. However, for scenarios with structured data, this is no longer the case. In recent work, we developed a framework to determine whether a pattern computed by a generalisation operator is consistent w.r.t. a distance. In this way, we can determine which patterns can provide a good representation of a group of individuals belonging to a metric space. In this work, we apply this framework to analyse and define minimal distance-based generalisation operators (mgoperators) for first-order data. We show that Plotkin's lggis a mgoperator for atoms under the distance introduced by J. Ramon, M. Bruynooghe and W. Van Laer. We also show that this is not the case for clauses with the distance introduced by J. Ramon and M. Bruynooghe. Consequently, we introduce a new mgoperator for clauses, which could be used as a base to adapt existing bottom-up methods in ILP.