The Activity of a Variable and Its Relation to Decision Trees

The construction of sequential testing procedures from functions of discrete arguments is a common problem in switching theory, software engineering, pattern recognition, and management. The concept of the activity of an argument is introduced, and a theorem is proved which relates it to the expected testing cost of the most general type of decision trees. This result is then extended to trees constructed from relations on finite sets and to decision procedures with cycles. These results are used, in turn, as the basis for a fast heuristic selection rule for constructing testing procedures. Finally, some bounds on the performance of the selection rule are developed.