Learning Read-Once Functions Using Subcube Identity Queries

TL;DR

This paper introduces subcube identity queries for learning read-once Boolean functions, establishing conditions under which the problem can be solved efficiently and highlighting cases with exponential complexity.

Contribution

It presents a new query type for learning read-once functions, analyzes its relation to existing queries, and provides complexity bounds for different bases.

Findings

Polynomial upper bound on learning complexity with subcube identity and membership queries.

Existence of bases requiring exponential queries for exact identification.

Polynomial complexity for finite subsets of an infinite basis.

Abstract

We consider the problem of exact identification for read-once functions over arbitrary Boolean bases. We introduce a new type of queries (subcube identity ones), discuss its connection to previously known ones, and study the complexity of the problem in question. Besides these new queries, learning algorithms are allowed to use classic membership ones. We present a technique of modeling an equivalence query with a polynomial number of membership and subcube identity ones, thus establishing (under certain conditions) a polynomial upper bound on the complexity of the problem. We show that in some circumstances, though, equivalence queries cannot be modeled with a polynomial number of subcube identity and membership ones. We construct an example of an infinite Boolean basis with an exponential lower bound on the number of membership and subcube identity queries required for exact identification. We prove that for any finite subset of this basis, the problem remains polynomial.