1. CS/COE0447 Computer Organization & Assembly Language
Logic Design
Appendix C

2. Outline Example to begin: let’s implement a MUX
Gates, Truth Tables, and Logic Equations
Combinatorial Logic
Constructing an ALU
Memory Elements: Flip-flops, Latches, and Registers

3. Logic Gates 2-input AND Y=A&B Y=A|B 2-input OR 2-input NAND Y=~(A&B)
2-input NOR
Y=~(A|B) Y B

4. Multiplexor If S then C=B else C = A A C B 1 How many bits is S? SC B 1
How many bits is S? S
1, since it is choosing between 2 values
Let’s see how to implement a 2-input MUX using gates.
Hint: the answer uses AND gates, an OR gate, and one INVERTER
Answer in lecture

5. Computers and Logic
Digital electronics operate with only two voltage levels of interest: high and low voltage.
All other voltage levels are temporary and occur while transitioning between values
We’ll talk about them as signals that are
Logically true; 1; asserted
Logically false; 0, deasserted
0 and 1 are complements and inverses of each other

6. Combinational vs. Sequential Logic
Combinational logic
A function whose outputs depend only on the current input
Sequential logic
Memory elements, i.e., state elements
Outputs are dependent on current input and current state
Next state is also dependent on current input and current state

7. Combinational Logic … …
inputs
outputs

8. Sequential Logic … …
inputs
outputs
next
state
current
state
clock

9. The next set of topics [until the sequential logic picture we just saw pops up again] will only be about combinatorial logic

10. Functions Implemented Using Gates… ? …
inputs
outputs
Combinatorial logic blocks implement logical functions, mapping inputs
to outputs

11. Describing a Function OutputA = F(Input0, Input1, …, InputN-1)
OutputB = F’(Input0, Input1, …, InputN-1)
OutputC = F’’(Input0, Input1, …, InputN-1)
Methods
Truth table (since combinatorial logic has no memory, it can be completely specified by a truth table)
…[in a moment]

12. Truth Table
Input
Output A B
Cin S
Cout 1

13. Truth Tables
In a truth table, there is one row for every possible combination of values of the inputs
Specifically, if there are N inputs, the possible combinations are the binary numbers 0 through 2EN For example:
3 bits (0-7): through 111
4 bits (0-15): through 1111
5 bits (0-31): through 11111
While we could always use a truth table, they quickly grow in size and become hard to understand and work with
Boolean logic equations are more succinct

14. Describing a Function OutputA = F(Input0, Input1, …, InputN-1)
OutputB = F’(Input0, Input1, …, InputN-1)
OutputC = F’’(Input0, Input1, …, InputN-1)
Methods
Truth table
Boolean logic equations
Sum of products
Products of sums

15. Truth Table and Equations
Input
Output A B
Cin S
Cout 1
S = A’B’Cin+A’BCin’+AB’Cin’+ABCin
Cout = A’BCin+AB’Cin+ABCin’+ABCin
Each output has its own…?
Column in the truth table
And its own Boolean equation

16. Truth Tables and Equations
All functions specified by truth tables can also be specified by Boolean formulas [and vice versa]
So, let’s look more closely at Boolean algebra

17. Boolean Algebra
Boole, George (1815~1864): mathematician and philosopher; inventor of Boolean Algebra, the basis of all computer arithmetic
Binary values: 0, 1
Two binary operations: AND (/), OR ()
AND is also called the logical product since its result is 1 only if both operands are 1
OR is also called the logical sum since its result is 1 if either operand is 1
One unary operation: NOT (~)

18. Laws of Boolean Algebra
Identity, Zero, and One laws
aa = a+a = a
a1 =a; a+0 = a [“copy” operations]
a0 =0; a+1 = 1 [deassert by ANDing with 0; assert by ORing with 1]
Inverse
aa = 0; a+a = 1
Commutative
ab = ba
a+b = b+a
Associative
a(bc) = (ab)c
a+(b+c) = (a+b)+c
Distributive
a(b+c) = ab + ac
a+(bc) = (a+b)(a+c)

19. Laws of Boolean Algebra
De Morgan’s laws
~(a+b) = ~a~b
~(ab) = ~a+~b
More…
a+(ab) = a
a(a+b) = a
~~a=a
You’ll see this again in CS441 and CS1502

20. Examples To get used to Boolean equations
To see the relationships among Truth Tables, Boolean Equations, and hardware implementations in gates
To see that a “sum of products” formula can always be derived from a truth table
To see that equations can often be simplified

21. Example equation E = (A’ B C) + (A B’ C) + (A B C’)
What is the value of the equation if A = 1, B = 0 and C = 0?
E = (1’ 0 0) + (1 0’ 0) + (1 0 0’)
E = (0 0 0) + (1 1 0) + (1 0 1) = 0
What is the value of the equation if A = 0, B = 1, and C = 1?
E = (0’ 1 1) + (0 1’ 1) + (0 1 1’)
E = (1 1 1) + (0 0 1) + (0 1 0) = 1

22. Truth Table for E A B C D E F 1
You can read our equation for E right from the truth table:
E = (A’ B C) + (A B’ C) + (A B C’)
These are the three cases when E is 1.
Now, give a Boolean equation for F:
F = A B C

23. Give a Boolean Equation for DC D E F 1
D = (A’ B’ C) + (A’ B C’) + (A’ B C) + (A B’ C’) + (A B’ C) + (A B C’) + (A B C)
There are many logically equivalent equations (in general)
D = (A’ B’ C’)’ [D is true in all cases except A=0 B=0 C=0.]
Apply DeMorgan’s law:
D = A’’ + B’’ + C’’ = A + B + C

24. Example: boolean equation of a circuit First add the boolean equations at the output for each AND gate A B Y C
A•B
B•C

25. Example: Next add the Boolean equations at the output for the OR gate The Boolean equation for the complete logic circuit is: Y = (A•B)+(B•C) A B Y C
A•B
(A•B) + (B•C)
B•C

26. Example: Truth Table Y = (A•B)+(B•C)1
Reading an equation from the Table:
Y = (A’ B C) + (A B C’) + (A B C)
The equations are logically equivalent: one way to see this is to consider
each row in the truth table. If the two equations have the same outputs for
each input, then they are logically equivalent.

27. Example: MUX (A S’) A (A S’) + (B S) C B (B S) S
If the equation below were implemented directly:
four (3-input) AND gates and one (4-input) OR
gate would be needed
(B S) S A B S C 1
Again, the two formulas
are equivalent
[next slide]
C = (A’ B S) + (A B’ S’) + (A B S’) + (A B S)
Equation read from the Table:

28. Example: MUX BS C = (A S’) + (B S)
C = (A’ B S) + (A B’ S’) + (A B S’) + (A B S)
AS’
If B ==0: (AS’) + 0
If B == 1: 0 + (AS’)
So, this is the same as AS’
Methods
perform such simplifications
automatically
You can see they
are equivalent by
comparing values
for each row A B S C 1

29. Expressive Power
Any Boolean algebra function can be constructed using AND gates, OR gates, and Inverters
[For your interest: NAND and NOR are both universal: any logic function can be built with just that one gate type]
There are “canonical forms” for Boolean functions: all equations can be expressed in these forms
This made it possible to create translation programs that, given a logic equation or truth table as input, can automatically design a circuit that implements it

30. Outline Example to begin: let’s implement a MUX!
Gates, Truth Tables, and Logic Equations
Combinatorial Logic
Constructing an ALU
Memory Elements: Flip-flops, Latches, and Registers

31. Since we were talking about MUXs…
How are larger MUXs implemented
Wider inputs than 1 bit
More choices

32. A 32-bit wide 2-to-1 Multiplexor
1-bit input to
to all 32 MUXs
Choosing between 2
32-bit wide buses
Bus: collection of data lines
treated as a single value.
E.g., MUX controlled by
MemtoReg.
Each MUX is the
same; just like the
one we saw earlier

33. Use a Decoder to build a MUX with more choices     Decoder n bit input value and 2^n outputs  1
This is a 2-to-4 decoder
Appendix C shows the truth table for a 3-to-8 decoder

34. Decoder: implementation with gates    Decoder n bit input value and 2^n outputs A = X • Y B = X • Y C = X • Y D = X • Y X Y A B C D 1 X A Y B C C D

35. N input MUX using a decoder
Example in lecture

36. Implementing Combinatorial Logic
PLA (Programming Logic Array)
A direct implementation of sum of products form pla.html (thanks to:
ROM (Read Only Memory)
Interpret the truth table as fixed values stored in memory
Using logic gate chips (74LS…)

37. 74LS Series Chips contain several logic gates 2-input OR gate
SN 74LS04 Hex
inverter gate
SN 74LS08 Quad
2-input AND gate
SN 74LS32 Quad
2-input OR gate

38. ALU Symbol
Note that it’s combinational logic

39. Building a 1-bit ALU
ALU = Arithmetic Logic Unit

40. Building a 1-bit Adder S = A’B’Cin+A’BCin’+AB’Cin’+ABCin
Input
Output A B
Cin S
Cout 1
S = A’B’Cin+A’BCin’+AB’Cin’+ABCin
Cout = AB+BCin+ACin (after simplification)
E.g., build a pla

41. Building a 32-bit ALU

42. Implementing “SUB”

43. Implementing “NAND”/”NOR”

44. Implementing “SLT”

45. Implementing “SLT”, cont’d

46. 4-bit datapath
“Operation” same for all
“Binvert” same for all
“Ainvert” same for all
Bit 0
Bit 1
Bit 3
Bit 2

47. Supporting “BEQ”/”BNE”
Need a “zero-detector”

48. ALU Symbol
Note that it’s a combinational logic

49. Sequential Logic … …
inputs
outputs
next
state
current
state
clock

50. RS Latch
Note that there are feedbacks!

51. RS Latch, cont’d 1 1 1
When R=0, S=1

52. RS Latch, cont’d 1 1 1
When R=1, S=0

53. RS Latch, cont’d 1 1
When R=0, S=0, and Q was 0

54. RS Latch, cont’d 1 1
When R=0, S=0, and Q was 1

55. RS Latch, cont’d 1 1
What happens if R=S=1?

56. D Latch
Note that we have an R-S latch as a back-end

57. D Latch, cont’d R S
Note that S, R inputs always get D and inverted input of D when C=1
When C=0, S=R=0, remembering the previous value

58. D Latch, cont’d R C D
Q(t)
Q(t-1) 1 S

59. D Latch, cont’d D Q
D Latch C Q’

60. D Flip-Flop (D-FF)
Two D latches are cascaded, with opposite clock

61. D Flip-Flop, cont’d D Q
D-FF C Q’