On Learning Ring-Sum-Expansions

Abstract.  The problem of learning ring-sum-expansions from examples is studied. Ring-sum-expansions (RSE) are representations of Boolean functions over the base {^,\oplus,1}, which reflect arithmetic operations in GF(2). k-RSE is the class fo 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>=1, is learnable while k-term-RSE, k>=2, is not learnable if RP \neq NP. Without using a complexity-theoretical hypothesis, it is proven that k-RSE, k>=1, and k-term-RSE, k>=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).