1. CS 121 Digital Logic Design
Chapter 3
Gate-Level Minimization

2. Outline 3.1 Introduction 3.2 The Map Method 3.3 Four-Variable Map
3.5 Product of sums simplification
3.6 Don‘t Care Conditions
3.7 NAND and NOR Implementaion
3.8 Other Two-Level Implementaion
3.9 Exclusive-OR function

3. 3.1 Introduction (1-1)
Gate-Level Minimization refers to the design task of finding an optimal gate-level implementation of the Boolean functions describing a digital circuit.
Notes about simplification of Boolean expression:
Minimum number of terms and literals in each term (minimum number of gates and inputs in the digram).
Reduce the complexity of the digital gates.
The simplest expression is not unique.
Simplification Methods:
Algebraic minimization  lack on specific rules. (section 2.4).
Karnaugh map or K-map.

4. Outline 3.1 Introduction 3.2 The Map Method 3.3 Four-Variable Map
3.5 Product of sums simplification
3.6 Don‘t Care Conditions
3.7 NAND and NOR Implementaion
3.8 Other Two-Level Implementaion
3.9 Exclusive-OR function

5. 3.2 The Map Method (1-12)
A Karnaugh map is a graphical tool for assisting in the general simplification procedure.
Combination of 2, 4, … adjacent squares
The relation is:
Logic circuit ↔ Boolean function ↔ Truth table ↔ K-map ↔ conical form ↔ satndrad form.
Conical form: ( sum of minterms , product of maxterms.
Standrad form: ( simplifier : sum of product , product of sum

6. 3.2 The Map Method (2-12) Two-variable maps: Y’ Y X’ X
Number of sequares (minterms) is 2 𝑛 , where n is the number of variables.
So in tow-variable map there are 4 squares(minterms). Y’ Y X’ X

7. 3.2 The Map Method (3-12) Rules for K-map:
We can reduce functions by circling 1’s in the K- map
Each circle represents a minterm reduction
Following circling, we can deduce minimized and- or form.
Rules to consider
Every cell containing a 1 must be included at least once.
The largest possible “power of 2 rectangle” must be enclosed.
The 1’s must be enclosed in the smallest possible number of rectangles.

8. 3.2 The Map Method (4-12) Two-Variable maps (cont.) Example 1:
F(X,Y) = XY’ + XY
From the map, we see that
F (X,Y) = X.
Note: There are implied 0s in other boxes.
This can be justified using algebraic manipulations: F(X,Y) = XY’ + XY
= X(Y’ +Y)
= X.1
= X
1 1 X

9. 3.2 The Map Method (5-12) Two-Variable maps (cont.) Example 2: Y 1 X
G(x,y) = m1 + m2 + m3 Y
G(x,y) = m1 + m2 + m3
= X’Y + XY’ + XY
From the map, we can see that :
G = X + Y 1 X

10. 3.2 The Map Method (6-12) Two-Variable maps (cont.) Example 3: X’ 1
x y F
0 0 1
0 1 1
1 0 0
1 1 0
Example 3:
F = Σ(0, 1)
Using algebraic manipulations:
F = Σ(0,1)
= x’y + x’y’
= x’ (y+y’)
= x’ 1 X’

11. 3.2 The Map Method (7-12) Three-variable maps:
3 variables  8 squares ( minterms).
On a 3-variable K-Map:
One square represents a minterm with three variables
Two adjacent squares represent a product term with two variables
Four “adjacent” terms represent a product term with one variables
Eight “adjacent” terms is the function of all ones (logic 1).

12. Three-variable maps (cont.):
3.2 The Map Method (8-12)
Three-variable maps (cont.):
Example 1:
F(X,Y) = X’Y’Z’ + X’YZ’ + XY’Z’ + XYZ’
using algebraic manipulations: F = X’Y’Z’ + X’YZ’ + XY’Z’ + XYZ’
= Z’ (X’Y’ + X’Y + XY’ + XY)
= Z’ (X’ (Y’+Y) + X (Y’+Y))
= Z’ (X’+ X)
= Z’
Y Z x 1 1 1 1

13. 3.2 The Map Method (9-12) three-Variable maps (cont.) 00 01 11 10 1
Example 2:
F=AB’C’ +ABC +ABC +ABC + A’B’C + A’BC’
From the map, we see that
F=A+BC +BC
B C 00 01 11 10 A 1 1 1 1 1 1 1

14. 3.2 The Map Method (10-12) three-Variable maps (cont.) 00 01 11 10 1
Example 4 :
F (x, y, z)= Σ (2, 3, 6, 7) Y
using algebraic manipulations:
F(x , y, z) = x’yz + xyz + x’yz’ + xyz’
= yz (x’ + x) + yz’ (x’ + x)
= yz + yz’
= y (z + z’)
= y
y z 00 01 11 10 x 1 1 1 1 1

15. 3.2 The Map Method (11-12) three-Variable maps (cont.)
Example (3-1) , (3-2) :

16. 3.2 The Map Method (12-12) three-Variable maps (cont.)
Example (3-3) , (3-4) :

17. Outline 3.1 Introduction 3.2 The Map Method 3.3 Four-Variable Map
3.5 Product of sums simplification
3.6 Don‘t Care Conditions
3.7 NAND and NOR Implementaion
3.8 Other Two-Level Implementaion
3.9 Exclusive-OR function

18. 3.3 Four-Variables Map (1-9)
4 variables  16 squares ( minterms).
On a 4-variable K-Map:
Two adjacent squares represent a term of three literals.
Four adjacent squares represent a term of two literals.
Eight adjacent squares represent a term of one literal.
Note: The larger the number of squares combined, the smaller the number of literals in the term.

19. 3.3 Four-Variables Map (2-9)
Flat Map Vs. Torus

20. 3.3 Four-Variables Map (3-9)
Example 1 (3-5) :
F(w,x,y,z) = ∑ ( 0,1,2,4,5,6,8,9,12,13,14)
y z 00 01 11 10
w x 00 1 1 1
W’YZ’ 01 1 1 1 Y’
XYZ’ 11 1 1 1 10 1 1
F = y‘ + w‘yz‘ + xyz‘

21. 3.3 Four-Variables Map (4-9)
Example 2 (3-6) :
F = A’B’C’ + B’CD’ + A’BCD’ + AB’C’
C D
B’C’ 00 01 11 10
A B 00 1 1 1
A’CD’ 01 1 11 10 1 1 1
B’D’
F = B‘D‘ + B‘C‘ + A‘CD‘

22. 3.3 Four-Variables Map (5-9)
Simplification using Prime Implicants
A Prime Implicant is a product term obtained by combining the maximum possible number of adjacent squares in the map .
If a minterm in a square is covered by only one prime implicant, that implicant is called Essential .
Prime Implicants and Essential Prime Implicants can be determined by inspection of a K-Map.
Notes:
Two adjacent 1’s form prime implicant, if they are not within a group of four adjacent squares.
Four adjacent 1’s form prime implicant, if they are not within a group of eight adjacent squares and so on.

23. ESSENTIAL Prime Implicants
3.3 Four-Variables Map (6-9)
Simplification using Prime Implicants
Example 1:
F(A,B,C,D) = ∑ (0,2,3,5,7,8,9,10,11,13,15)
ESSENTIAL Prime Implicants CD 1 B D A C 1 B C D A D B D B C B
Minterms covered by single prime implicant BD BD AD B A

24. 3.3 Four-Variables Map (7-9)
Simplification using Prime Implicants
Example 1:
F(A,B,C,D) = ∑ (0,2,3,5,7,8,9,10,11,13,15)
Essential prim implicants:BD , B’D’
Prime implicant: CD , B’C, AD , AB’.
The minterms that not cover by essential implicants are: m3, m9, m11.
The simplified expression is optained from the sum of the essential implicants and other prime implicants that may be needed to cover any remaining minterms.
So this function can be written with these ways:
F = BD + B’D’ + CD + AD
F = BD + B’D’ + CD + AB’
F = BD + B’D’ + B’C + AD
F = BD + B’D’ + B’C + AB’

25. Note: that all of these prime implicants are essential.
3.3 Four-Variables Map (8-9)
Simplification using Prime Implicants
Example 2:
F(W,X,Y,Z) = ∑ (0,2,3,8,9,10,11,12,13,14,15) X Y Z W
X’Y
X’Z’ 1 1 1
Note: that all of these prime implicants are essential. 1 1 1 1 1 1 1 1 W

26. 3.3 Four-Variables Map (9-9)
Simplification using Prime Implicants
Example 3:
F(W,X,Y,Z) = ∑ (0,2,3,4,7,12,13,14,15)
W’YZ
W’X’Z’ X Y Z W 1 1 1
W’X’Y
Essential: WX
Prime: XYZ , XY’Z’ , W’Y’Z’, W’YZ, W’X’Y , W’X’Z’
W’Y’Z’ 1 1 1 1 1 1
XYZ
XY’Z’ WX

27. Outline 3.1 Introduction 3.2 The Map Method 3.3 Four-Variable Map
3.5 Product of sums simplification
3.6 Don‘t Care Conditions
3.7 NAND and NOR Implementaion
3.8 Other Two-Level Implementaion
3.9 Exclusive-OR function

28. 3.5 Producut-of-Sum simplification (1-3)
Mark with 1’s the minterms of F.
Mark with 0’s the minterms of F’.
Circle 0’s to express F’.
Complement the result in step 3 to obtain a simplified F in product-of-sums form.

29. 3.5 Producut-of-Sum simplification (2-3)
Example 1:
Simplify : F= ∑(0,1,2,5,8,9,10) in Product-of-Sums Form
F’ = AB + CD + BD’
F = (F’)’ = (A’+B’) . (C’+D’) . (B’+D) CD B C D A 1 1 1 1
BD’ AB 1 1 1

30. 3.5 Producut-of-Sum simplification (3-3)
Example 2:
Simplify :
F(x, y, z) =(0, 2, 5,7)in Product-of-Sums Form
X’Z’
y z 00 01 11 10
F’ = XZ + X’Z’
F = (F’)’ = (X’+Z’) + (X+Z) x 1 XZ

31. Outline 3.1 Introduction 3.2 The Map Method 3.3 Four-Variable Map
3.5 Product of sums simplification
3.6 Don‘t Care Conditions
3.7 NAND and NOR Implementaion
3.8 Other Two-Level Implementaion
3.9 Exclusive-OR function

32. 3.6 Don't Cares Condition (1-4)
Sometimes a function table or map contains entries for which it is known:
The input values for the minterm will never occur, or
The output value for the minterm is not used.
Functions that have unspecified outputs for some input combinations are called incompletely specified functions.
In these cases, the output value is defined as a “don't care” ( an “x” entry) assumed to be either 0 or 1.
The choice between 0 and 1 is depending on the way the incompletely specified function is simplied.
By placing “don't cares” in the function table or map, the cost of the logic circuit may be lowered.

33. 3.6 Don't Cares Condition (2-4)
Example :
A logic function having the binary codes for the BCD digits as its inputs. Only the codes for 0 through 9 are used.
The six codes, 1010 through 1111 never occur, so the output values for these codes are “x” to represent “don’t cares.”

34. 3.6 Don't Cares Condition (3-4)
Example (3.9) :
F(W,X,Y,Z) = ∑ (1,3,7,11,15)
d(W,X,Y,Z) = ∑ (0,2,5) X Y Z W X Y Z W x 1 x 1 x 1 1 x x x 1 1 1 1 1 1
F = YZ + W’Z
F = YZ + W’X’

35. 3.6 Don't Cares Condition (4-4)
Example (3.9) :
F(W,X,Y,Z) = ∑ (1,3,7,11,15)
d(W,X,Y,Z) = ∑ (0,2,5) X Y Z W x 1 1 x x 1 1 1
F’ = Z’ + WY’
F = Z ( W’ + Y)

36. Outline 3.1 Introduction 3.2 The Map Method 3.3 Four-Variable Map
3.5 Product of sums simplification
3.6 Don‘t Care Conditions
3.7 NAND and NOR Implementaion
3.8 Other Two-Level Implementaion
3.9 Exclusive-OR function

37. 3.7 NAND and NOR Implementation (1-15)
Digital circuits are frequently constructed with NAND or NOR gates rather than with AND and OR gates.

38. 3.7 NAND and NOR Implementation (2-15)
NAND Implementation
NAND gate: a universal gate.
Any digital system can be implemented with it.

39. 3.7 NAND and NOR Implementation (3-15)
NAND Implementation
To facilitate the conversion to NAND logic, there are alternative graphic symbol for it.

40. 3.7 NAND and NOR Implementation (4-15)
NAND Implementation
Two-Level Implementation
Procedures (steps) of Implementation with two levels of NAND gates:
Express simplified function in sum of products form.
Draw a NAND gate for each product term that has at least two literals to constitute a group of first-level gates
Draw a single gate using AND-invert or invert-OR in the second level
A term with a single literal requires an inverter in the first level.

41. 3.7 NAND and NOR Implementation (5-15)
NAND Implementation
Two-Level Implementation
F = AB + CD
= [(AB + CD)’]’
= [(AB)’*(CD)’]’

42. 3.7 NAND and NOR Implementation (6-15)
NAND Implementation
Two-Level Implementation
Example (3.10):
F(X,Y,Z) = ∑ (1,2,3,4,5,7)
X’Y
F = XY’ + X’Y + Z
y z 00 01 11 10 x 1 1 1 1 1 1 1 Z
XY’

43. 3.7 NAND and NOR Implementation (7-15)
NAND Implementation
Multilevel Implementation
Procedures (steps) of Implementation with multilevel of NAND gates:
Convert all AND gates to NAND gates with AND-invert graphic symbols
Convert all OR gates to NAND gates with invert-OR graphic symbols
Check all the bubbles in the diagrams. For a single bubble, invert an inverter (one-input NAND gate) or complement the input literal

44. 3.7 NAND and NOR Implementation (8-15)
NAND Implementation
Multilevel Implementation
EXAMPLE 1:
F = A(CD + B) + BC’ B’

45. 3.7 NAND and NOR Implementation (9-15)
NAND Implementation
Multilevel Implementation
EXAMPLE 2:
F = (AB’ + A’B).(C + D’) F C’ D

46. 3.7 NAND and NOR Implementation (10-15)
The NOR operation is the dual of the NAND operation.
The NOR gate is anothar universal gate to implement any Boolean function.

47. 3.7 NAND and NOR Implementation (11-15)
To facilitate the conversion to NOR logic, there are alternative graphic symbol for it.

48. 3.7 NAND and NOR Implementation (12-15)
Two-Level Implementation
Procedures of Implementation with two levels of NOR gates:
Express simplified function in product of sums form.
Draw a NOR gate for each product term that has at least two literals to constitute a group of first-level gates
Draw a single gate using OR-invert or invert-AND in the second level
A term with a single literal requires an inverter in the first level.

49. 3.7 NAND and NOR Implementation (13-15)
Two-Level Implementation
Example :
F = (A+B).(C+D).E E

50. 3.7 NAND and NOR Implementation (14-15)
Multilevel Implementation
Procedures of Implementation with multilevel of NOR gates:
Convert all OR gates to NOR gates with OR-invert graphic symbols
Convert all AND gates to NOR gates with invert-AND graphic symbols
Check all the bubbles in the diagrams. For a single bubble, invert aninverter (one-input NAND gate) or complement the input literal

51. 3.7 NAND and NOR Implementation (15-15)
Multi-Level Implementation
Example :
F = (A B’ + A’B).(C+D’) A B’ A’ B

52. Outline 3.1 Introduction 3.2 The Map Method 3.3 Four-Variable Map
3.5 Product of sums simplification
3.6 Don‘t Care Conditions
3.7 NAND and NOR Implementaion
3.8 Other Two-Level Implementaion
3.9 Exclusive-OR function

53. 3.8 Other Two-Level Implementations (1-7)
Non-degeneratd forms Implementation
16 possible combinations of two-level forms with 4 types of gates: AND, OR, NAND, and NOR
8 are degenerate forms: degenerate to a single operation.
(AND-AND , AND-NAND, OR-OR , OR-NOR , NAND-NAND , NAND-NOR , NOR-AND , NOR-NAND)
8 are non-degenerate (generate) forms:
NAND-AND = AND-NOR = AND-OR-INVERT
OR-NAND = NOR-OR = OR-AND-INVERT

54. 3.8 Other Two-Level Implementations (3-7)
Non-degeneraetd forms Implementation
NOR
NAND OR
AND
2nd level
1st level #
(3.4) $
(3.6)
Discussed before
Generated forms
Discuss now

55. 3.8 Other Two-Level Implementations (4-7)
Non-degeneraetd forms Implementation
AND-OR-INVERT
AND-NOR = NAND-AND = AND-OR-INVERT
Similar to AND-OR, AND-OR-INVERT requires an expression in sum of products
Example: F = (AB + CD + E) ‘

56. 3.8 Other Two-Level Implementations (5-7)
Non-degeneraetd forms Implementation
OR-AND-INVERT
AND-NOR = NAND-AND = AND-OR-INVERT
Similar to OR-AND, OR-AND-INVERT requires an expression in product of sums
Example: F = [(A+B) . (C+D) . E ] ‘

57. Implements the Function Equivalent Non-degenerate form
3.8 Other Two-Level Implementations (6-7)
Non-degenerated forms Implementation
To Get an Output of
Simplify F’ into
Implements the Function
Equivalent Non-degenerate form b a F
sum-of-products form by combining 0’s in the map
AND-OR-INVERT
NAND-AND
AND-NOR
product-of-sums form by combining 1’s in the map and then complementing
OR-AND-INVERT
NOR-OR
OR-NAND

58. 3.8 Other Two-Level Implementations (7-7)
Non-degenerated forms Implementation
Example (3.11) :
F(x,y,z) = ∑ (0,6)
AND-OR-INVERT:
F’ = x’y + xy’ + z
F = ( x’y + xy’ + z ) ‘
OR-AND-INVERT:
F = x’y’z’ + xyz’
F’ = [ (x’y’z’ + xyz’)’ ] ‘
F = [ (x+y+z) . (x’+y’+z) ] ‘

59. Outline 3.1 Introduction 3.2 The Map Method 3.3 Four-Variable Map
3.5 Product of sums simplification
3.6 Don‘t Care Conditions
3.7 NAND and NOR Implementaion
3.8 Other Two-Level Implementaion
3.9 Exclusive-OR function

60. 3.9 Exclusive-OR Function (1-7)
Exclusive-OR (XOR) denoted by the symbol :
x  y = xy‘ + x‘y
Exclusive-OR is equal to 1, when the values of x and y are diffrent.
Exclusive-NOR (XNOR):
(x  y )‘ = xy + x‘y‘
Exclusive-NOR is equal to 1, when the values of x and y are same.
Only a limited number of Boolean functions can be expressed in terms of XOR operations, but it is particularly useful in arithmetic operations and error-detection and correction circuits.

61. 3.9 Exclusive-OR Function (2-7)
Exclusive-OR principles:
x  0 = x
x  1 = x‘
x  x = 0
x  x‘ = 1
x  y‘ = x‘ y = (x  y)‘
x  y = y  x
(x  y)  z = x  (y  z)

62. 3.9 Exclusive-OR Function (3-7)
Implementaion Exclusive-OR with AND-OR-NOT:
x  y = xy‘ + x‘y
Implementaion Exclusive-OR with NAND:
= x (x‘+y‘) + y (x‘+y‘)
= x (xy)‘ + y (xy)‘
= [ (x(xy)‘ + y(xy)‘)‘]‘
= [ (x(xy)‘)‘ + (y(xy)‘)‘ ]‘

63. 3.9 Exclusive-OR Function (4-7)
Odd Function:
The 3-variable XOR function is equal to 1 if only one variable is equal to 1 or if all three variables are equal to 1.
Multiple-variable exclusive OR operation = odd function : odd number of variables be equal to 1.
(A  B  C) = (AB‘ + A‘B) C‘ + (A‘B‘ + AB) C
= AB‘C‘ + A‘BC‘ + A‘B‘C + ABC
= ∑ (1,2,4,7)

64. 3.9 Exclusive-OR Function (5-7)
Odd Function:

65. 3.9 Exclusive-OR Function (6-7)
Odd Function:
A  B  C  D= ∑ (1,2,4,7,8,11,13,14)

66. 3.9 Exclusive-OR Function (7-7)
Parity Generation and Checking:
Exclusive-OR function is useful in systems requiring error-detection and correction circuits.
A parity bit is used for purpose of detection errors during transmission.
Parity bit : an extra bit included with a binary message to make the number of 1’s either odd or even.
The circuit generates the parity bit in transmitter is called parity generator.
The circuit checks the parity bit in receiver is called parity checker.

67. 3.9 Exclusive-OR Function (8-7)
Parity Generation and Checking:
Example : Three-bit message with even parity
Three-bit Massage
Parity bit X Y Z P 1
From the truth table , P constitutes an odd function.
It is equal 1 when numerical value of 1’s in a minterm is odd
P = x  y  z

68. 3.9 Exclusive-OR Function (8-7)
Parity Generation and Checking:
Example : Three-bit message with even parity
From the truth table , C constitutes an odd function.
It is equal 1 when numerical value of 1’s in a minterm is odd
C = x  y  z  P