  In this paper, based on results of exact learning and test theory, we study
arbitrary infinite binary information systems each of which consists of an
infinite set of elements and an infinite set of two-valued functions
(attributes) defined on the set of elements. We consider the notion of a
problem over information system, which is described by a finite number of
attributes: for a given element, we should recognize values of these
attributes. As algorithms for problem solving, we consider decision trees of
two types: (i) using only proper hypotheses (an analog of proper equivalence
queries from exact learning), and (ii) using both attributes and proper
hypotheses. As time complexity, we study the depth of decision trees. In the
worst case, with the growth of the number of attributes in the problem
description, the minimum depth of decision trees of both types either is
bounded from above by a constant or grows as a logarithm, or linearly. Based on
these results and results obtained earlier for attributes and arbitrary
hypotheses, we divide the set of all infinite binary information systems into
seven complexity classes.
