Generalisation Operators for Lists Embedded in a Metric Space

In some application areas, similarities and distances are used to calculate how similar two objects are in order to use these measurements to find related objects, to cluster a set of objects, to make classifications or to perform an approximate search guided by the distance. In many other application areas, we require patterns to describe similarities in the data. These patterns are usually constructed through generalisation (or specialisation) operators. For every data structure, we can define distances. In fact, we may find different distances for sets, lists, atoms, numbers, ontologies, web pages, etc. We can also define pattern languages and use generalisation operators over them. However, for many data structures, distances and generalisation operators are not consistent. For instance, for lists (or sequences), edit distances are not consistent with regular languages, since, for a regular pattern such as *a, the covered set of lists might be far away in terms of the edit distance (e.g. bbbbbba and aa). In this paper we investigate the way in which, given a pattern language, we can define a pair of generalisation operator and distance which are consistent. We define the notion of (minimal) distance-based generalisation operators for lists. We illustrate positive results with two different pattern languages.