Finding tree patterns consistent with positive and negative examples using queries

Abstract

This paper is concerned with the problem of finding a hypothesis in \mathcal{TP}^2 consistent with given positive and negative examples. The hypothesis class \mathcal{TP}^2 consists of all sets of at most two tree patterns and represents the class of unions of at most two tree pattern languages. Especially, we consider the problem from the point of view of the consistency problem for \mathcal{TP}^2. The consistency problem is a problem for deciding whether there exists a consistent hypothesis with given positive and negative examples within some fixed hypothesis space. Efficient solvability of that problem is closely related to the possibility of efficient machine learning or machine discovery. Unfortunately, however, the consistency problem is known to be NPdcomplete for many hypothesis spaces. In this paper, the problem for the class \mathcal{TP}^2 is also shown to be NPdcomplete. In order to overcome this computational hardness, we try to use additional information obtained by making queries. First, we give an algorithm that, using restricted subset queries, solves the consistency problem for \mathcal{TP}^2 in time polynomial in the total size of given positive and negative examples. Next, we show that each subset query made by the algorithm can be replaced by several membership queries under some condition on a set of function symbols. As a result, we have that the consistency problem for \mathcal{TP}^2 is solved in polynomial time using membership queries.