Title : On the Learnability of Counting Functions

We examine the learnability of concepts based on counting functions. A counting function is a generalization of a parity function in which the weighted sum of n inputs is tested for equivalence to some value k modulo N . The concepts we study therefore generalize many commonly studied boolean functions. We first show that disjunctions of counting functions (DOCFs) with modulus N are learnable by equivalence queries and determine the number of queries sufficient and necessary. We show that α(N)n + 1 equivalence queries are sufficient where α(N) = O(logN) is the sum of the exponents in the prime decomposition of N . When counting functions in the disjunction have distinct counting moduli {Ni}, we show that α(lcm(Ni))n + 1 equivalence queries are sufficient. We also give lower bounds on the number of equivalence queries required to learn diagonal DOCFs (and therefore general DOCFs) and provide a matching upper bound in the case of diagonal DOCFs. We then demonstrate how the additional power of membership queries allows improved learning in two different ways: (1) more efficient learning for some classes learnable with equivalence queries only, and (2) learnability of other classes not known to be learnable with equivalence queries only.