Learning Boolean Read-Once Formulas over Generalized Bases

A read-once formula is one in which each variable appears on at most a single input. Angluin, Hellerstein, and Karpinski give a polynomial time algorithm that uses membership and equivalence queries to identify exactly read-once boolean formulas over the basis fAND;OR;NOTg AHK93]. The goal of this work is to consider natural generalizations of these gates, in order to develop exact identiication algorithms for more powerful classes of formulas. We show that read-once formulas over a basis of arbitrary boolean functions of constant fan-in k or less (i.e. any f : f0;1g 1ck ! f0;1g) are exactly identiiable in polynomial time using membership and equivalence queries. We show that read-once formulas over the basis of arbitrary symmetric boolean functions are also exactly identiiable in polynomial time in this model. Given standard cryptographic assumptions, there is no polynomial time identiication algorithm for read-twice formulas over either of these bases using membership and equivalence queries. We further show that for any basis class B meeting certain technical conditions, any polynomial time identiication algorithm for read-once formulas over B can be extended to a polynomial time identiication algorithm for read-once formulas over the union of B and the arbitrary functions of fan-in k or less. As a result, read-once formulas over the union of arbitrary symmetric and arbitrary constant fan-in gates are also exactly identiiable in polynomial time using membership and equivalence queries.