Infinite Binary Information Systems. Decision Trees of Types 4 and 5

In this chapter, based on results of exact learning, test theory, and rough set 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: (4) using only proper hypotheses (an analog of proper equivalence queries from exact learning), and (5) 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 in Chap. 7 for attributes and arbitrary hypotheses, we divide the set of all infinite binary information systems into seven complexity classes.