Learning read once functions using subcube parity queries

The article considers the problem of constructing a conditional diagnostic test (the exact identification problem) on the set of all read-once functions essentially dependent on the specified variables in an arbitrary hereditary basis. We use standard queries on function values at points and also parity queries on the number of ones in subfunctions obtained by substituting constants for variables (subcube parity queries return the modulo-2 sum of the function values on a subcube of the Boolean cube). For the elementary basis consisting of conjunction, disjunction, and negation, we prove the existence of a conditional diagnostic test with a quadratic (in the number of variables) number of queries. For bases that allow read-once expression of all elementary-basis functions (containing at least one nonmonotone and at least one nonlinear function) and are essentially different from the elementary basis, we prove the necessity of using an exponential (in the number of variables) number of queries in the worst case. Our analysis thus establishes the boundary between polynomial and exponential complexity problems.