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example, consider the skeletal Mux.hdl program supplied in this project. Suppose that for one reason or another you did not complete this program’s implementation, but you still want to use Mux gates as internal parts in other chip designs. This is not a problem, thanks to the following convention. If our simulator fails to find a Mux.hdl file in the current directory, it automatically invokes a built-in Mux implementation, pre-supplied with the simulator’s software. This built-in implementation—a Java class stored in the built In directory—has the same interface and functionality as those of the Mux gate described in the book. Thus, if you want the simulator to ignore one or more of your chip implementations, simply move the corresponding .hdl files out of the current directory.
     
    Steps We recommend proceeding in the following order:
     
    0. The hardware simulator needed for this project is available in the tools directory of the book’s software suite.
    1. Read appendix A, sections A1-A6 only.
    2. Go through the hardware simulator tutorial, parts I, II, and III only.
    3. Build and simulate all the chips specified in the projects/01 directory.

2

    Boolean Arithmetic

    Counting is the religion of this generation, its hope and salvation.
    —Gertrude Stein (1874-1946)

     
    In this chapter we build gate logic designs that represent numbers and perform arithmetic operations on them. Our starting point is the set of logic gates built in chapter 1, and our ending point is a fully functional Arithmetic Logical Unit. The ALU is the centerpiece chip that executes all the arithmetic and logical operations performed by the computer. Hence, building the ALU functionality is an important step toward understanding how the Central Processing Unit (CPU) and the overall computer work.
    As usual, we approach this task gradually. The first section gives a brief Background on how binary codes and Boolean arithmetic can be used, respectively, to represent and add signed numbers. The Specification section describes a succession of adder chips, designed to add two bits, three bits, and pairs of n -bit binary numbers. This sets the stage for the ALU specification, which is based on a sophisticated yet simple logic design. The Implementation and Project sections provide tips and guidelines on how to build the adder chips and the ALU on a personal computer, using the hardware simulator supplied with the book.
    Binary addition is a simple operation that runs deep. Remarkably, most of the operations performed by digital computers can be reduced to elementary additions of binary numbers. Therefore, constructive understanding of binary addition holds the key to the implementation of numerous computer operations that depend on it, one way or another.

2.1 Background

    Binary Numbers Unlike the decimal system, which is founded on base 10, the binary system is founded on base 2. When we are given a certain binary pattern, say “10011,” and we are told that this pattern is supposed to represent an integer number, the decimal value of this number is computed by convention as follows:

    (1)
     
    In general, let x = x n x n -1 ... x 0 be a string of digits. The value of x in base b, denoted ( x ) b , is defined as follows:

    (2)
     

The reader can verify that in the case of (10011) two , rule (2) reduces to calculation (1).
    The result of calculation (1) happens to be 19. Thus, when we press the keyboard keys labeled ‘1’, ‘9’ and ENTER while running, say, a spreadsheet program, what ends up in some register in the computer’s memory is the binary code 10011. More precisely, if the computer happens to be a 32-bit machine, what gets stored in the register is the bit pattern 00000000000000000000000000010011.
     
    Binary Addition A pair of binary numbers can be added digit by digit from right to left, according to the same elementary school method used in decimal addition. First, we add the two right-most digits, also called the least