Logic Circuits

  Many important functions are naturally computed with straight-line programs, programs without loops or branches. Such computations are conveniently described with circuits, directed acyclic graphs of straight-line programs. Circuit vertices are associated with program steps, whereas edges identify dependencies between steps. Circuits are characterized by their size, the number of vertices, and their depth, the length (in edges) of their longest path. Circuits in which the operations are Boolean are called logic circuits, those using algebraic operations are called algebraic circuits, and those using comparison operators are called comparator circuits. In this chapter we examine logic circuits. Algebraic and comparator circuits are examined in Chapter 6.

Logic circuits are the basic building blocks of real-world computers. As shown in Chapter 3, all machines with bounded memory can be constructed of logic circuits and binary memory units. Furthermore, machines whose computations terminate can be completely simulated by circuits.

In this chapter circuits are designed for a large number of important functions. We begin with a discussion of circuits, straight-line programs, and the functions computed by them. Normal forms, a structured type of circuit, are examined next. They are a starting point for the design of circuits that compute functions. We then develop simple circuits that combine and select data. They include logical circuits, encoders, decoders, multiplexers, and demultiplexers. This is followed by an introduction to prefix circuits that efficiently perform running sums. Circuits are then designed for the arithmetic operations of addition (in which prefix computations are used), subtraction, multiplication, and division. We also construct efficient circuits for symmetric functions. We close with proofs that every Boolean function can be realized with size and depth exponential and linear, respectively, in its number of inputs, and that most Boolean functions require such circuits.

The concept of a reduction from one problem to a previously solved one is introduced in this chapter and applied to many simple functions. This important idea is used later to show that two problems, such as different NP-complete problems, have the same computational complexity. (See Chapters 3 and 8.)