Intrinsic Complexity of Learning Geometrical Concepts from Positive Data

Intrinsic complexity is used to measure the complexity of learning areas limited by broken-straight lines (called open semi-hulls) and intersections of such areas. Any strategy learning such geometrical concepts can be viewed as a sequence of primitive basic strategies. Thus, the length of such a sequence together with the complexities of the primitive strategies used can be regarded as the complexity of learning the concepts in question. We obtained the best possible lower and upper bounds on learning open semi-hulls, as well as matching upper and lower bounds on the complexity of learning intersections of such areas. Surprisingly, upper bounds in both cases turn out to be much lower than those provided by natural learning strategies. Another surprising result is that learning intersections of open semihulls turns out to be easier than learning open semi-hulls themselves.