Learning Arithmetic Read-Once Formulas

• Authors: Nader H. Bshouty;Thomas R. Hancock;Lisa Hellerstein
• Venue: SIAM Journal on Computing
• Year: 1995

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 algorithms for exact learning of arithmetic read-once formulas over a field. We present a membership and equivalence query algorithm that identifies arithmetic read-once formulas over an arbitrary field. We present a randomized membership query algorithm (i. e. a randomized black box interpolation algorithm) that identifies such formulas over finite fields with at least $2n+5$ elements (where $n$ is the number of variables), and over infinite fields. We also show the existence of non-uniform deterministic membership query algorithms for arbitrary read-once formulas over fields of characteristic 0, and for division-free read-once formulas over fields that have at least $2n^3+1$ elements. For our algorithms, we assume we are able to efficiently perform arithmetic operations on field elements and to 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 not to be inputs on which the formula evaluates to $0/0$. This assumption is shown to be necessary for fields of size $o(n/\log {n})$, in the sense that we prove there exists no polynomial time identification algorithm that uses just membership and standard equivalence queries.