Computer Science 101

1. Computer Science 101 Circuit Design Algorithm

2. Circuit Design - The Problem • The problem is to design a circuit that accomplishes a specified task. • What does this mean? • What kind of task? • We would like to specify • The number of inputs to the circuit. • The number of outputs. • What value each output should have for each setting of the inputs.

3. Circuit Design - Observations • We can deal with one output at a time. • The truth table is a good way to organize the specifications. • If we can come up with a Boolean expression for the output, we’re home free.

4. Circuit Design and Truth Table • The problem reduces to this: • Given a truth table with all values for inputs. • And given a column of values for the output. • Find a Boolean expression that yields the specified column. • If we can do this, we can get the circuit from the Boolean expression. • We now develop an algorithm, the Sum of Products algorithm, to find the expression.

5. Circuit Design - Typical Problem • Find Boolean expression for the output. • Note that this is a “Majority Rules” circuit. • The output reflects the majority input.

6. Circuit Design - First Case • Consider case where output column has only one 1. • Want expression which is true when • A is false _ (A true) and • B is true and • C is true • _ ABC does the trick.

7. Sum of Products Algorithm- Single 1 • Note row in which the 1 occurs. • Make a product of all the input variables. • Put a bar over each variable that has a value of 0 in this row.

8. Sum of Products - Single 1 • Note row where the 1 occurs. • Make product of all variables. ABC • Put bar over variables with 0 value. _ ABC

9. Sum of Products - General Case • Want BE that is true if • _ ABC is true OR • _ ABC is true OR • _ ABC is true • _ _ _ ABC+ABC+ABC

10. Sum of Products Algorithm • Identify each row of the output that has a 1. • For each such row • Make a product of all the input variables. • Put bar over each variable with a 0 in this row. • Make a sum of all of these product terms.

11. Sum of Products - Example • Problem: Design a logic circuit with • 3 input lines, 1 output line • Anytime we change the value of any one of the inputs, the output value changes. • Think of a single light bulb controlled by three light switches that work independently. • Note: Once we set the output for one set of inputs, the rest is determined. • We’ll start with all inputs 0 giving output 0.

12. Example continued • 1’s in output give _ _ _ _ABC + ABC _ _+ ABC + ABC

13. A B C _ _ _ _ _ _ ABC+ABC+ABC+ABC Boolean Expression Circuit for the example

14. And a one and a two and ...