Computer Logic and Digital Design Chapter 2 Henry Hexmoor AND logic

1
Computer Logic and Digital Design Chapter
2Henry HexmoorAND logic
input
output
AND Gate
AB
B
S1 . S2
S1 S2 path? off
off no on off
no off on no on
on yes
2
OR logic
input
S1 S2
S1 S2 path? off
off no on off
yes off on yes on
on yes
S1
S2
output
3
NOT logic
S
S 1 0 0 1
S
S'
4
Equivalent Symbols of NAND, NOR Gates
NAND Symbols
Normal Symbol
Alternate NAND Symbol
According to DeMorgans theorem T13 (X .
Y) X Y
NOR Symbols
Alternate NOR Symbol
Normal NOR Symbol
According to DeMorgans theorem T13 (X
Y) X . Y
5
Boolean Algebra

• Literals (n) and terms
• number of truth table rows 2n

X Y Z YZ XYZ 0 0 0
0 0 0 0 1 1
1 0 1 0 0 0 0 1 1
0 0 1 0 0 0
1 1 0 1 1 1 1 1
0 0 1 1 1 1 0
1

• See table 2-3
• Using demorgans laws OR and AND are
  interchanged the duality principle

6
Switching Algebra Axioms Theorems

• (A1) X 0 if X ? 1 (A1) X 1 if X ? 0
• (A2) If X 0, then X 1 (A2) if X 1,
  then, X 0
• (A3) 0 . 0 0 (A3) 1 1 1
• (A4) 1 . 1 1 (A4) 0 0 0
• (A5) 0 . 1 1 . 0 0 (A5) 1 0 0 1 1
• (T1) X 0 X (T1) X . 1 X
  (Identities)
• (T2) X 1 1 (T2) X . 0 0
  (Null elements)
• (T3) X X X (T3) X . X X
  (Idempotency)
• (T4) (X) X (Involution)
• (T5) X X 1 (T5) X . X 0
  (Complements)
• (T6) X Y Y X (T6) X . Y Y . X
  (Commutativity)
• (T7) (X Y) Z X (Y Z) (T7) (X . Y)
  . Z X . (Y . Z) (Associativity)
• (T8) X . Y X . Z X . (Y Z) (T8) (X
  Y) . (X Z) X Y . Z (Distributivity)
• (T9) X X . Y X (T9) X . (X Y) X
  (Covering)
• (T10) X . Y X . Y X (T10) (X Y) .
  (X Y) X (Combining)
• (T11) X . Y X. Z Y . Z X . Y X . Z
• (T11) (X Y) . ( X Z) . (Y Z) (X Y) .
  (X Z)
  (Consensus)
• (T12) X X . . . X X (T12) X . X
  . . . . . X X (Generalized idempotency)

7
Switching Algebra Axioms

• First two axioms state that a variable X can only
  take on only one of two values
• (A1) X 0 if X ¹ 1
• (A1) X 1 if X ¹ 0
• Not Axioms, formally define X (X prime or NOT
  X)
• (A2) If X 0, then X
  1
• (A2) if X 1, then, X 0

Note Above axioms are stated in pairs with only
difference being the interchange of the symbols
0 and 1.
8
Three More Switching Algebra Axioms

• The following three Boolean Algebra axioms state
  and formally define the AND, OR operations
• (A3) 0 . 0 0
• (A3) 1 1 1
• (A4) 1 . 1 1
• (A4) 0 0 0
• (A5) 0 . 1 1 .0 0
• (A5) 1 0 0 1 1

Axioms A1-A5, A1-A5 completely define switching
algebra.
9
Switching Algebra Single-Variables Theorems

• Switching-algebra theorems are statements known
  to be always true (proven using axioms) that
  allow us to manipulate algebraic logic
  expressions to allow for simpler analysis.
• (e.g . X 0 X allow us to replace every
  X 0 with X)
• The Theorems (T1-T5, T1-T5)
• (T1) X 0 X (T1) X . 1 X
  (Identities)
• (T2) X 1 1 (T2) X . 0 0
  (Null elements)
• (T3) X X X (T3) X . X X
  (Idempotency)
• (T4) (X) X
  (Involution)
• (T5) X X 1 (T5) X . X 0
  (Complements)

10
Perfect Induction

• Most theorems in switching algebra are simple to
  prove using perfect induction
• Since a switching variable can only take the
  values 0 and 1 we can prove a theorem involving a
  single variable X by proving it true for X 0
  and X 1
• Example To prove (T1) X 0 X
• X 0 0 0 0 true
  according to axiom A4
• X 1 1 0 1 true
  according to axiom A5

11
Switching Algebra Two- and Three-Variable
Theorems

• (Commutativity)
• (T6) X Y Y X
• (T6) X . Y Y . X
• (Associativity)
• (T7) (X Y) Z X (Y Z)
• (T7) (X . Y) . Z X . (Y . Z)

T6-T7, T6 -T7 are similar to commutative and
associative laws for addition and multiplication
of integers and reals.
12
Two- and Three-Variable Theorems (Continued)

• (Distributivity)
• (T8) X . Y X . Z X . (Y Z)
• (T8) (X Y) . (X Z) X Y . Z
• T8 allows to multiply-out an expression to get
  sum-of-products form (distribute logical
  multiplication over logical addition)
• For example
• V . (W X) . (Y Z) V .W . Y V.
  W. Z V. X . Y V. X . Z
• 
  sum-of-products form
• T8 allows to add-out an expression to get a
  product-of-sums form (distribute logical addition
  over logical multiplication)
• For example
• (V . W . X) (Y . Z ) (V Y) . (V Z) . (W
  Y) . (W Z) . (X Y) . (X Z)
• 
  product-of-sums form

13
Theorem Proof using Truth Table

• Can use truth table to prove T8 by perfect
  induction.
• i.e, Prove that X . Y X . Z X . (Y Z)
• (i) Construct truth table for both sides of
  above equality.

(ii) Check that from truth table check that
that X . Y X . Z X . (Y Z) This is
satisfied because output column values for X .
Y X . Z and output column values for X . (Y
Z) are equal for all cases.
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Two- and Three-Variable Theorems (Continued)

• (Covering)
• (T9) X X . Y X
• (T9) X . (X Y) X
• (Combining)
• (T10) X . Y X . Y X
• (T10) (X Y) . (X Y) X
• T9-T10 used in the minimization of logic
  functions.

15
Two- and Three-Variable Theorems (Continued)

• (Consensus)
• (T11) X . Y X. Z Y . Z X . Y X .
  Z
• (T11) (X Y) . ( X Z) . (Y Z) (X
  Y) . (X Z)
• In T11 the term Y. Z is called the consensus of
  the term X . Y and the term X . Z
• If Y . Z 1, then either X . Y or X . Z
  must also be 1.
• Thus the term Y . Z is redundant and may be
  dropped.

16
n-Variable Theorems

• (Generalized idempotency)
• (T12) X X . . . X X
• (T12) X . X . . . . . X X
• (DeMorgans theorems)
• (T13) (X1 . X2 . . . . . Xn) X1
  X2 . . . Xn
• (T13) (X1 X2 . . . Xn) X1 .
  X2 . . . . . Xn
• (T13), (T13) are probably the most
  commonly
• used theorems of switching algebra.

17
Examples Using DeMorgans theorems

• Example Equivalence of NAND Gate
• A two-input NAND Gate has the output expression
  Z (X . Y) using (T13) Z
  (X . Y) (X Y)
• The function of a NAND gate can be achieved with
  an OR gate with an inverter at each input.
• Example Equivalence of NOR Gate
• A two-input NOR Gate has the output expression
  Z(XY)
• using (T13) Z (X Y) X . Y
• The function of a NOR gate can be achieved with
  an AND gate with an inverter at each input.

18
n-Variable Theorems (Continued)

• (Generalized DeMorgrans theorem)
• (T14) F(X1, X2, . . ., Xn, , .) F(X1,
  X2, . . ., Xn, . , )
• States that given any n-variable logic expression
  its complement can be found by swapping and .
  and complementing all variables.
• Example
• F(W,X,Y,Z) (W.X) ( X.Y) (W.(X Z))
• ((W) . X) (X. Y)
  (W.((X) (Z)))
• F(W,X,Y,Z) ((W) X) .(X Y).(W
  ((X).(Z)))
• Using T4, (X) X simplifies it to
• F(W,X,Y,Z) (W X) . (X Y) . (W (X .
  Z))

19
Logic Function Representation Definitions

• A literal is a variable or a complement of a
  variable
• Examples X, Y, X, Y
• A product term is a single literal, or a
  product of two or more literals. Examples
  Z W.Y.Y X.Y.Z W.Y.Z
• A sum-of-products expression is a logical sum
  of product terms.
• Example Z W.X.Y X.Y.Z W.Y.Z
• A sum term is a single literal or logical sum
  of two or more literals
• Examples Z W X Y X
  Y Z W Y Z
• A product-of-sums expression is a logical
  product of sum terms.
• Example Z. (W X Y) . (X Y Z) .
  (W Y Z)
• A normal term is a product or sum term in
  which no variable appears more than once
• Examples of non-normal terms W.X.X.Y
  WWXY X.X.Y
• Examples of normal terms W . X . Y
  W X Y

20
Logic Expression Algebraic Manipulation Example

• Prove that the following identity is true using
  Algebraic expression Manipulation (one can
  also prove it using a truth table)
• X .Y X . Z ((X Y) . (X
  Z))
• Starting from the left hand side of the identity
• Let F X .Y X . Z
• A X . Y B X . Z
• Then F A B
• Using DeMorgans theorem T 13 on F
• F A B (A . B)
  (1)
• Using DeMorgans theorem T 13 on A, B
• A X . Y (X Y)
  (2)
• B X . Z (X Z)
  (3)
• Substituting A, B from (2), (3), back in F in (1)
  gives
• F (A . B) ((X Y)
  . (X Z))
• Which is equal to the right hand side of
  the identity.

21
Terminology Minterms

• A minterm is a special product of literals, in
  which each input variable appears exactly once.
• A function with n variables has 2n minterms
  (since each variable can appear complemented or
  not)
• A three-variable function, such as f(x,y,z), has
  23 8 minterms
• Each minterm is true for exactly one combination
  of inputs

xyz xyz xyz xyz xyz xyz xyz xyz
Minterm Is true when Shorthand xyz x0, y0,
z0 m0 xyz x0, y0, z1 m1 xyz x0, y1,
z0 m2 xyz x0, y1, z1 m3 xyz x1, y0,
z0 m4 xyz x1, y0, z1 m5 xyz x1, y1,
z0 m6 xyz x1, y1, z1 m7
22
Terminology Maxterms

• A maxterm is a special sum of literals, in which
  each input variable appears exactly once.
• A function with n variables has 2n maxterms
  (since each variable can appear complemented or
  not)
• A three-variable function, such as f(x,y,z), has
  23 8 maxterms

Maxterm Shorthand x y z M0 x y z
M1 x y z M2 x y z M3 x
y z M4 x y z M5 x y z
M6 x y z M7
23
Canonical Forms

• Minterms and Maxterms
• Index Representation of Minterms and Maxterms
• Sum-of-Minterm (SOM) Representations
• Product-of-Maxterm (POM) Representations
• Representation of Complements of Functions
• Conversions between Representations

24
Logic Function Representation Definitions

• Minterm
• An n-variable minterm is a normal product
  term with n literals.
• There are 2n such products terms.
• Example of 4-variable minterms
• W.X.Y.Z W.X.Y.Z
  W.X.Y.Z
• Maxterm
• An n-variable maxterm is a normal sum term
  with n literals.
• There are 2n such sum terms.
• Examples of 4-variable maxterms
• W X Y Z W X Y
  Z W X Y Z
• A minterm can be defined as as product term that
  is 1 in exactly one row of the truth table.
• A maxterm can similarly be defined as a sum term
  that is 0 in exactly one row in the truth table.

25
Minterms/Maxterms for A 3-variable function
F(X,Y,Z)

• Row X Y Z F Minterm
  Maxterm
• 0 0 0 0 F(0,0,0) X.Y.Z
  X Y Z
• 1 0 0 1 F(0,0,1) X.Y.Z
  X Y Z
• 2 0 1 0 F(0,1,0) X.Y.Z
  X Y Z
• 3 0 1 1 F(0,1,1) X.Y.Z
  X Y Z
• 4 1 0 0 F(1,0,0) X.Y.Z
  X Y Z
• 5 1 0 1 F(1,0,1) X.Y.Z
  X Y Z
• 6 1 1 0 F(1,1,0) X.Y.Z
  X Y Z
• 7 1 1 1 F(1,1,1) X.Y.Z
  X Y Z

26
Canonical Sum Example

• The function represented by the truth table
• has the canonical sum representation
• F S X,Y,Z m(0, 3, 4, 6, 7)
• X.Y.Z X.Y.Z X.Y.Z
  X.Y.Z X.Y.Z

Row X Y Z F 0 0 0 0
1 1 0 0 1 0 2
0 1 0 0 3 0 1 1
1 4 1 0 0 1 5 1
0 1 0 6 1 1 0
1 7 1 1 1 1
Minterm list using S notation
Algebraic canonical sum of minterms
27
Canonical Product Example

• The function represented by the truth table
• has the canonical product representation
• F P X,Y,Z M(1,2,5)
• (X Y Z) . (X Y Z) . (X Y
  Z)

Row X Y Z F 0 0 0 0
1 1 0 0 1 0 2
0 1 0 0 3 0 1 1
1 4 1 0 0 1 5 1
0 1 0 6 1 1 0
1 7 1 1 1 1
28
Minterm and Maxterm Relationship

• Review DeMorgan's Theorem
• and
• Two-variable example
• and
• M2 is the complement of m2 and vice-versa.
• Since DeMorgan's Theorem holds for n variables,
  the above holds for terms of n variables
• giving
• and
• Thus Mi is the complement of mi.

y
x

y


x
y
x
y
x



y
x


m

y

x


M
2
2
29
Conversion Between Minterm/Maxterm Lists

• To convert between a minterm list and a maxterm
  list
• take the set complement.
• Example (page 44, 3rd ed.)
• F(X,Y, Z) S m(1,3, 4, 6) m1 m3
  m4 m6
• F(X,Y, Z) P M(1,3,4,6) M1M3M4M6

30
Verbal Synthesis Example An Alarm Circuit

• A verbal logic description
• The ALARM output is 1 if the panic input is 1,
  or if the ENABLE input is 1, the EXISTING input
  is 0, and the house is not secure.
• The house is secure if the WINDOW, DOOR, GARAGE
  inputs are all 1
• This can be put in logic expressions as follows
• ALARM PANIC ENABLE . EXISTING . SECURE
• SECURE WINDOW. DOOR. GARAGE
• ALARM PANIC ENABLE . EXISTING. (WINDOW .
  DOOR . GARAGE)
• In sum of products form as (by using DeMorgan T13
  and multiplying out)
• ALARM PANIC ENABLE. EXISTING . WINDOW
• ENABLE . EXISTING. DOOR
  ENABLE. EXISTING. GARAGE

31
cost criteria2-4

1. Literal cost
2. Gate input cost literals terms

OR Gate
32
Combinational Circuit Minimization

• Canonical sum and product logic expressions do
  not provide a circuit realization with the
  minimum number of gates.
• Minimization methods reduce the cost of two level
  AND-OR, NAND-NAND, OR-AND, NOR-NOR circuits in
  three ways
• By minimizing the number of first level gates
• By minimizing the number of inputs of each
  first-level gate.
• Minimizing the inputs of the second level gate
• Most minimization methods are based on the
  combining theorems T10, T10

33
Karnaugh Maps

• A Karnaugh Map or (K-map for short) is a
  graphical representation of the truth table of a
  logic function.
• The K-map for an n-input logic function is an
  array with 2n cells or squares, one for each
  input combination or minterm.
• The rows and columns are labeled so that the
  input combination for any cell is determined from
  the row and column headings.
• The row and columns of the map are ordered in
  such a way that each cell differs from an
  adjacent cell in only one input variable
• Thus for an n-variable K-map, each cell has n
  adjacent cells.
• The K-map for a function is filled by putting
• a 1 in the square corresponding to a minterm
• a 0 otherwise (maybe omitted)

34
2-Variable K-map
For a 2-variable logic function F(X,Y)
Truth Table

• Row X Y F Minterm
• 0 0 0 F(0,0) X.Y
• 1 0 1 F(0,1) X.Y
• 2 1 0 F(1,0) X.Y
• 3 1 1 F(1,1) X .Y

K-map
Example For the function F(X,Y) S X,Y m(1,2,3)
Truth Table
K-map
Row X Y F 0 0 0 0
1 0 1 1 2 1 0
1 3 1 1 1
1
1
1
35
3 variable Karnaugh mapTextbook convention
YZ
00 01 11 10
0 1
X Y Z
X Y Z
X Y Z
X Y Z
X
X Y Z
XY Z
X YZ
X Y Z
36
3-Variable K-map
For a 3-variable logic function F(X,Y,Z)
Truth Table
K-map

• Row X Y Z F Minterm
• 0 0 0 0 F(0,0,0) X.Y.Z
• 1 0 0 1 F(0,0,1) X.Y.Z
• 2 0 1 0 F(0,1,0) X.Y.Z
• 3 0 1 1 F(0,1,1) X.Y.Z
• 4 1 0 0 F(1,0,0) X.Y.Z
• 5 1 0 1 F(1,0,1) X.Y.Z
• 6 1 1 0 F(1,1,0) X.Y.Z
• 7 1 1 1 F(1,1,1) X.Y.Z

Example For the function F(X,Y,Z) S X,Y,Z
(1,4,6,7)
K-map
Row X Y Z F 0 0 0 0
0 1 0 0 1 1 2
0 1 0 0 3 0 1 1
0 4 1 0 0 1 5 1
0 1 0 6 1 1 0
1 7 1 1 1 1
1
Truth Table
1
1
1
37
4-Variable K-map
For a 4-variable logic function F(W,X,Y,Z)
Truth Table
K-map

• Row W X Y Z F Minterm
• 0 0 0 0 0 F(0,0,0,0)
  W.X.Y.Z
• 1 0 0 0 1 F(0,0,0,1) W.
  X.Y.Z
• 2 0 0 1 0 F(0,0,1,0) W.
  X.Y.Z
• 3 0 0 1 1 F(0,0,1,1) W.
  X.Y.Z
• 4 0 1 0 0 F(0,1,0,0) W.
  X.Y.Z
• 5 0 1 0 1 F(0,1,0,1)
  W.X.Y.Z
• 6 0 1 1 0 F(0,1,1,0)
  W.X.Y.Z
• 7 0 1 1 1 F(0,1,1,1)
  W.X.Y.Z
• 8 1 0 0 0 F(1,0,0,0)
  W.X.Y.Z
• 9 1 0 0 1 F(1,0,0,1)
  W.X.Y.Z
• 10 1 0 1 0 F(1,0,1,0)
  W.X.Y.Z
• 11 1 0 1 1 F(1,0,1,1)
  W.X.Y.Z
• 12 1 1 0 0 F(1,1,0,0)
  W.X.Y.Z
• 13 1 1 0 1 F(1,1,0,1)
  W.X.Y.Z
• 14 1 1 1 0 F(1,1,1,0)
  W.X.Y.Z
• 15 1 1 1 1 F(1,1,1,1)
  W.X.Y.Z

38
Minimization Using K-maps

• Group or combine as many adjacent 1-cells as
  possible
• The larger the group is, the fewer the number of
  literals in the resulting product term.
• Grouping 2 adjacent 1-cells eliminates 1
  variable, grouping 4 1-cells eliminates 2
  variables, grouping 8 1-cells eliminates 3
  variables, and so on. In general, grouping 2n
  squares eliminates n variables.
• Select as few groups as possible to cover all the
  1-cells (minterms) of the function
• The fewer the groups, the fewer the number of
  product terms in the minimized function.

39
Karnaugh mapsExample 2-4 (Figure 2-14b, page 54,
3rd ed.)
000 100 010 110

• 1cell Square overlap is OK.

Z
YZ
00 01 11 10
1
0 1
1
X
1
1
1
XY
100 101
F2(X,Y,Z) m(0,2,4,5,6) Z XY
40
4 variable Karnaugh mapexample 2-5 (Figure 2-19,
page 57, 3rd ed.)
YZ
00 01 11 10
0000 0001 0100 0101 0010 0110
0000 0001 1100 1101 1000 1001
00 01 11 10
1
0100 1100 0110 1110
1
1
1
1
WX
1
1
1
1
1
1
F(W,X,Y,Z) S m(0,1,2,4,5,5,8,9,12,13,14) Y
WZ XZ
41
prime/essential implicants
WZ and XZ are prime implicants (each rectangle
needs all its cells) Y is an essential prime
implicant (the rectangle contains exclusive
(i.e., nonshared) cells
YZ
00 01 11 10
00 01 11 10
1
1
1
1
1
WX
1
1
1
1
1
1
F(X,Y,Z) m(0,1,2,4,5,5,8,9,12,13,14) Y
WZ XZ
Selection rule minimize overlap among prime
implicants
42
product of sums Maxterms
YZ
00 01 11 10
00 01 11 10
1
1
1
0
1
1
0
WX
1
1
0
1
1
1
1
0
0
F(X,Y,Z) m(0,1,2,4,5,5,8,9,12,13,14) YZ
WXY ? (Y Z) (W X Y)
Selection rule minimize overlap among prime
implicants
43
More Examples

• xz yz xyz.

xz xyz
44
multilevel optimization2-6
Manipulate equations algebraically G ACE
ACF ADE ADF BCDEF A (CE CF
DE DF) BCDEF A (C D)(E F)
BCDEF
OR Gate
45
other gate types
See figures 2-22 and 2-23 in 4th edition
OR Gate
46
HW 2

• Demonstrate by means of truth tables the validity
  of the following identity
• (XYZ) X Y Z
• (Q 2-1)
• 2. Optimize the following Boolean function by
  means a three-variable map
• F(X,Y,Z) Sm(1,3,6,7)
• (Q 2-14)

OR Gate