A Polynomial Time Algorithm for Finding Finite Unions of Tree Pattern Languages

A tree pattern is a structured pattern known as a term in formal logic, and a tree pattern language is the set of trees which are the ground instances of a tree pattern. In this paper, we deal with the class of tree languages whose language is de ned as a union of at most k tree pattern languages, where k is an arbitrary xed positive number. In particular, we present a polynomial time algorithm that, given a nite set of trees, to nd a set of tree patterns that de nes a minimal union of at most k tree pattern languages containing the given set. The algorithm can be considered as a natural extension of Plotkin's anti-uni cation algorithm, which nds a minimal single tree pattern language containing the given set. By using the algorithm, we can realize a consistent and conservative polynomial time inference machine that identi es the class of unions of k tree pattern languages in the limit from positive data for every k > 0.