Characterizing PAC-Learnability of Semilinear Sets

The learnability of the class of letter-counts of regular languages (semilinear sets) and other related classes of subsets of N d with respect to the distribution-free learning model of Valiant (PAC-learning model) is characterized. Using the notion of reducibility among learning problems due to Pitt and Warmuth called \prediction preserving reducibility," and a special case thereof, a number of positive and partially negative results are obtained. On the positive side the class of semilinear sets of dimension 1 or 2 is shown to be learnable when the integers are encoded in unary. On the neutral to negative side it is shown that when the integers are encoded in binary the learning problem for semilinear sets as well as a class of subsets of Z d much simpler than semilinear sets is as hard as learning DNF, a central open problem in the eld. A number of hardness results for related learning problems are also given. Most of the research reported herein was conducted while the author was with the Department of Computer and Information Science of the University of Pennsylvania. The author was supported in part by an IBM graduate fellowship and by the O ce of Naval Research under contract number N00014-87-K-0401.