Learning arithmetic read-once formulas

摘要

A formula is read-once if each variable appears at most once in it. An arithmetic read-once formula is one in which the operators are addition, subtraction, multiplication, and division. We present polynomial time algorithm for exactly learning (or interpolating) arithmetic read-once formulas computing functions over a field. We present an algorithm that uses randomized membership queries (or substitutions) to identify such formulas over large finite fields and infinite fields. We also present a deterministic algorithm that uses equivalence queries as well as membership queries to identify arithmetic read-once formulas over small finite fields. We then non-constructively show the existence of deterministic membership query (interpolation) algorithms for arbitrary formulas over fields of characteristic 0 and for division-free formulas over large or infinite fields. Our algorithms assume we are able to efficiently perform arithmetic operations on field elements and compute square roots in the field. It is shown that the ability to compute square roots is necessary, in the sense that the problem of computing n -- 1 square roots in a field can be reduced to the problem of identifying an arithmetic formula over n variables in that field. Our equivalence queries are of a slightly non-standard form, in which counterexamples are required to not be inputs on which the formula evaluates to 0/0. This assumption is shown to be necessary for fields of size o(n/log n), for which it is shown that there is no polynomial time identification algorithm that uses just membership and standard equivalence queries.