Decision Tables with Many-Valued Decisions

Abstract

Decision tables with many-valued decisions arise often in various applications. In contrast to decision tables with one-valued decisions, in decision tables with many-valued decisions each row is labeled with a nonempty finite set of natural numbers (decisions). If we want to find all decisions corresponding to a row, we deal with the same mathematical object as decision table with one-valued decisions: it is enough to code different sets of decisions by different numbers. However, if we want to find one (arbitrary) decision from the set attached to a row, we have essentially different situation.

In particular, in rough set theory [70, 80] decision tables are considered often that have equal rows labeled with different decisions. The set of decisions attached to equal rows is called the generalized decision for each of these equal rows. The usual way is to find for a given row its generalized decision. However, the problems of finding an arbitrary decision or one of the most frequent decisions from the generalized decision look also reasonable.

This chapter consists of ten sections. Section 5.1 contains examples of decision tables with many-valued decisions. In Sect. 5.2, main notions are discussed. In Sect. 5.3, relationships among decision trees, rules and tests are considered. In Sects. 5.4 and 5.5, lower and upper bounds on complexity of trees, rules and tests are studied. Approximate algorithms for optimization of tests, rules and trees are considered in Sects. 5.6 and 5.7. Section 5.8 is devoted to the discussion of exact algorithms for optimization of trees, rules and tests. Section 5.9 contains an example which illustrates the constructions considered in this chapter. Section 5.10 contains conclusions.