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On the Learnability of Counting Functions
BibTeX
@MISC{Bshouty_onthe,
author = {Nader Bshouty and Zhixiang Chen and Scott E. Decatur and Steven Homer},
title = { On the Learnability of Counting Functions},
year = {}
}
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Abstract
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(log N) 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.
