1. Logic Design CS 270: Mathematical Foundations of Computer Science Jeremy Johnson

2. 2 Logic Design  Objective: To provide an important application of propositional logic to the design and simplification of logic circuits.

3. 3 Topics  Logic gates  and, or, inverter  nand  Logic circuits  Encoder, decoder, multiplexor  Simplification of logic circuits  Full adder

4. 4 Logic Circuits  A single line labeled x is a logic circuit. One end is the input and the other is the output. If A and B are logic circuits so are:  and gate  or gate  inverter (not) A B A A B

5. 5 Logic Circuits  Given a boolean expression it is easy to write down the corresponding logic circuit  Here is the circuit for the original multiplexor expression x0x0 x1x1 s

6. 6 Logic Circuits  Here is the circuit for the simplified multiplexor expression x0x0 x1x1 s

7. 7 Nand  Nand – negation of the conjunction operation:  A nand gate is an inverted and gate: x y x | y 0 0 1 0 1 1 1 0 1 1 1 0

8. Nand is functionally complete  All boolean functions can be implemented using nand gates (and, or and not can be implemented using nand)  not:  and:  or:

9. 9 Implementing Logic Gates with Transistors output gate +V ground A Transistor NOT Gate A NAND B A +V ground A Transistor NAND Gate B

10. 10 Decoder  A decoder is a logic circuit that has n inputs (think of this as a binary number) and 2 n outputs. The output corresponding to the binary input is set to 1 and all other outputs are set to 0. d0d0 d1d1 d2d2 d3d3 b0b0 b1b1

11. 11 Encoder  An encoder is the opposite of a decoder. It is a logic circuit that has 2 n inputs and n outputs. The output equal to the input line (in binary) that is set to 1 is set to 1. d0d0 d1d1 d2d2 d3d3 b0b0 b1b1

12. 12 Multiplexor  A multiplexor is a switch which routes n inputs to one output. The input is selected using a decoder. d0d0 d1d1 d2d2 d3d3 s0s0 s1s1

13. XOR  “One or the other, but not both”  Notation for circuits: x y x y x  y 0 0 0 0 1 1 1 0 1 1 1 0

14. 14 Full Adder  Used to add to binary numbers stored as an array of bits using carry ripple addition  Three binary inputs  a, b and CarryIn  Two binary outputs  Sum and CarryOut such that a + b + CarryIn = 2*CarryOut + Sum Carry 110 A 101 B 111 A+B = 1100

15. 15 Exercise  Derive a truth table for the output bits (Sum and CarryOut) of a full adder.  Using the truth table derive a sum of products expression for Sum and CarryOut. Draw a circuit for these expressions.  Using properties of Boolean algebra simplify your expressions. Draw the simplified circuits.

16. 16 Solution

17. 17 Solution

18. 18 Solution

19. 19 Full Adder  Sum = parity(a,b,CarryIn)  a  b  c + a  b  c  a  b  c  CarryOut = majority(a,b,CarryIn)  b  CarryIn + a  CarryIn + a  b + a  b  CarryIn   b  CarryIn + a  CarryIn + a  b b a CarryIn Sum