Logical Design of Optimal Digital Networks by Integer Programming

Abstract

Logical design of an “optimal” digital network has been one of the most important objectives of switching theory. Optimality of a network is the minimization of the number of gates, interconnections, levels, or whatever the designer wants to minimize, under restrictions on a network such as fan-in and fan-out restrictions. In practice, only a limited number of inputs can be connected to any switching gate, and the output of the gate can be connected, as inputs, to only a limited number of switching gates. So, if we do not impose fan-in and fan-out restrictions,* logical design is practically meaningless. However, there are no general design methods known in switching theory which can take into consideration the arbitrary restrictions which are important from an engineering viewpoint. These methods are useful only under very special types of restrictions. For example, the minimization of a prime implicant expression discussed in textbooks on switching theory yields an optimal network only when a network is to be realized with AND and OR gates exactly in two levels, without fan-in and fan-out restrictions. If fan-in and fan-out restrictions are imposed, if more than two levels are allowed, or if gates other than AND and OR (for example, NOR gates) are to be used, the minimization of a prime implicant expression does not always yield a network of the minimum number of gates. There are other design methods usable only under specific restrictions. But no efficient design method has been known for designing optimal networks with arbitrary types of gates under arbitrary network restrictions, except the exhaustive method, i.e., a method which exhausts all conceivable networks and chooses a best one. The exhaustive method is very time-consuming, as Hellerman found in his pioneering work (16).