The ability to represent Boolean expressions as digital logic circuits means we can built computers to evaluate Boolean expressions. This concept is at the very core of modern digital computer technology. CSC 314 (Computer Architecture) addresses the concepts of computer construction using the building blocks we have learned here. In this section, we will see two example circuits. These two circuits are all that are necessary to build a modern computer.
The first circuit is the complement circuit. This circuit accepts a binary number as input and outputs the complement of it. For example, if the number '0110' is input, the complement circuit outputs '1001'. This circuit is really quite trivial.
The second circuit performs addition. First, we'll study a circuit that performs the addition of just two bits, possibly producing a carry.
This circuit has two inputs and two outputs. The output labelled 'S' is the SUM of adding the two bits A and B. The output labelled 'C' is the CARRY which results from the addition. The truth table below illustrates the function of the circuit:
A |
B |
S |
C |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
1 |
0 |
1 |
In other words, the circuit above can compute all of the following binary additions:
A natural extension for this circuit is to be able to add two 2-bit numbers with a carry out. The pattern for this computation is shown below. There are two columns: 0 and 1. A_{0} and B_{0} are added together to produce S_{0} and C_{0}. Next, A_{1} is added to B_{1} to produce an intermediate sum and carry. The intermediate sum is added to C_{0} to produce S_{1} and another possible carry. Finally, the two possible carries are added together to produce C_{1}. An important observation is that the two additions performed in the second column can produce at most 1 carry.
The corresponding circuit, known as a ripple adder is shown below. This is called a "ripple" adder because the carry ripples through the circuit from the least significant bit to the most significant.:
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Boolean Expressions
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