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On the Learnability of Z~-DNF Formulas (Extended Abstract)

By Nader H. Bshouty, Zhixiang Chent, Scott E. Decatur and Steven Homer


Although many learning problems can be re-duced to learning Boolean functions, in many cases a more efficient learning algorithm can be derived when the problem is considered over a larger domain. In this paper we give a natu-ral generalization of DNF formulas, ZN-DNF formulas over the ring of integers modulo lV. We first show using elementary number theory that for almost all larger rings the learnability of ZjV-DNF formulas is easy. This shows that the difficulty of learning Boolean DNF formu-las lies in the fact that the domain is small. We then establish upper and lower bounds on the number of equivalence queries required for the exact learning of Z~-terms. We show that a(lV)n + 1 < (log AJ)n + 1 equivalence queries are sufficient and y(lV)n equivalence queries are necessary, where a(lV) is the sum of the exponents in the prime decomposition of N, and Y(N) is the sum of logarithms of the exponents in the prime decomposition of N. We also demonstrate how the additional power of membership queries allows improved learning in two different ways: (1) more efficient learning for some classes learnable with equivalenc

Year: 2009
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