ON LEARNING RING-SUM-EXPANSIONS

The problem of learning ring-sum-expansions from examples is studied. Ring-sum-expansions (RSE) are representations of Boolean functions over the base {AND, +, 1}, which reflect arithmetic operations in GF(2). k-RSE is the class of ring-sum-expansions containing only monomials of length at most k. k-term-RSE is the class of ring-sum-expansions having at most k monomials. It is shown that k-RSE, k greater-than-or-equal-to 1, is learnable while k-term-RSE, k greater-than-or-equal-to 2, is not learnable if RP not-equal NP. Without using a complexity-theoretical hypothesis, it is proven that k-RSE, k greater-than-or-equal-to 1, and k-term-RSE, k greater-than-or-equal-to 2 cannot be learned from positive (negative) examples alone. However, if the restriction that the hypothesis which is output by the learning algorithm is also a k-RSE is suspended, then k-RSE is learnable from positive (negative) examples only. Moreover, it is proved that 2-term-RSE is learnable by a conjunction of a 2-CNF and a 1-DNF. Finally the paper presents learning (on-line prediction) algorithms for k-RSE that are optimal with respect to the sample size (worst case mistake bound).