Universal functions using powerful oracles

The paper discusses a method for uniquely defining boolean functions from a certain set using the parity of the number of ones in subcubes of a function that does not belong to this set (the oracle is a parity counter). Functions suitable for such definitions were previously called «universal.» A criterion for the universality of functions for the class of linear functions is proved. The universality of the negation of the sum modulo two for the classes of monotonic and polarizable functions is also proven.