Learning unions of tree patterns using queries

Abstract

This paper characterizes the polynomial time learnability of TP ^k, the class of collections of at most k first-order terms. A collection in TPA ^k defines the union of the languages defined by each first-order terms in the set. Unfortunately, the class TP ^k not polynomial time learnable in most of learning frameworks under standard assumptions in computational complexity theory. To overcome this computational hardness, we relax the learning problem by allowing a learning algorithm to make membership queries. We present a polynomial time algorithm that exactly learns every concept in TP ^k using O(kn) equivalence and O(k ² n · max{k, n}) membership queries, where n is the size of longest counterexample given so far. In the proof, we use a technique of replacing each restricted subset query by several membership queries under some condition on a set of function symbols. As corollaries, we obtain the polynomial time PAC-learnability and the polynomial time predictability of TP ^k when membership queries are available. We also show a lower bound Ω(kn) of the number of queries necessary to learn TP ^k using both types of queries. Further, we show that neither types of queries can be eliminated to achieve efficient learning of TP ^k. Finally, we apply our results in learning of a class of restricted logic programs, called unit clause programs.