On Learning Counting Functions With Queries

We investigate the problem of learning disjunctions of counting functions, which are general cases of parity and modulo functions, with equivalence and membership queries. We prove that, for any prime number p, the class of disjunctions of integer-weighted counting functions with modulus p over the domain Znq (or Zn) for any given integer q ≥ 2 is polynomial time learnable using at most n + 1 equivalence queries, where the hypotheses issued by the learner are disjunctions of at most n counting functions with weights from Zp. The result is obtained through learning linear systems over an arbitrary field. In general a counting function may have a composite modulus. We prove that, for any given integer q ≥ 2, over the domain Zn2, the class of read-once disjunctions of Boolean-weighted counting functions with modulus q is polynomial time learnable with only one equivalence query, and the class of disjunctions of log log n Boolean-weighted counting functions with modulus q is polynomial time learnable. Finally, we present an algorithm for learning graph-based counting functions.