Polynomial Time Inductive Inference of Ordered Tree Languages with Height-Constrained Variables from Positive Data

Abstract

Due to the rapid growth of tree structured data or semistructured data such as Web documents, efficient learning of structural features from tree structured data becomes more and more important. In order to represent tree structured patterns with rich structural features, we introduce a new type of structural variables, called height-constrained variables. An (i,j)-height-constrained variable can be replaced with any tree such that the trunk length of the tree is at least i and the height of the tree is at most j. Then, we define a term tree as a rooted tree pattern with ordered children and height-constrained variables. The minimal language (MINL) problem for term trees is to find a term tree t such that the language generated by t is minimal among languages, generated by term trees, which contains all given tree structured data. Let $\mathcal{OTT}^h$ be the set of all term trees with (i,j)-height-constrained variables for any i and j (1 ≤ i ≤ j) and no variable-chain. We assume that there are at least two edge labels. In this paper, we give a polynomial time algorithm for the MINL problem for $\mathcal{OTT}^h$. Thus we show that the class $\mathcal{OTT}^h$ is polynomial time inductively inferable from positive data.