Characteristic Sets for Inferring the Unions of the Tree Pattern Languages by the Most Fitting Hypotheses

A tree pattern p is a first-order term in formal logic, and the language of p is the set of all the tree patterns obtainable by replacing each variable in p with a tree pattern containing no variables. We consider the inductive inference of the unions of these languages from positive examples using strategies that guarantee some forms of minimality during the learning process. By a result in our earlier work, the existence of a characteristic set for each language in a class ${\mathcal L }$ (within ${\mathcal L }$) implies that ${\mathcal L }$ can be identified in the limit by a learner that simply conjectures a hypothesis containing the examples, that is minimal in the number of elements of up to an appropriate size. Furthermore, if there is a size l such that each candidate hypothesis has a characteristic set (within the languages in ${\mathcal L }$ that intersects non-emptily with the examples) that consists only of elements of up to size l, then the hypotheses containing the least number of elements of up to size l are at the same time minimal with respect to inclusion. In this paper we show how to determine such a size l for the unions of the tree pattern languages, and hence allowing us to learn the class using hypotheses that fulfill the two mentioned notions of minimality.