# 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.

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