Multiple Valued Logic
• Currently Studied for Logic Circuits with More
Than 2 Logic States
– Intel Flash Memory – Multiple Floating Gate
Charge Levels – 2,3 bits per Transistor
http://www.ee.pdx.edu/~mperkows/ISMVL/flash.html
• Techniques for Manipulation Applied to Multioutput Functions
– Characteristic Equation
– Positional Cube Notation (PCN) Extensions
MVI Functions
• Each Input can have Value in Set {0, 1, 2, ..., pi-1}
F :{0,1,..., pi 1}  {0,1, X }
• MVI Functions
• X is p-valued variable
• literal over X corresponds to subset of values
of S  {0, 1, ... , p-1} denoted by XS
MVL Literals
• Each Variable can have Value in Set {0, 1, 2, ..., pi-1}
• X is a p-valued variable
• MVL Literal is Denoted as X{j} Where j is the Logic
Value
• Empty Literal: X{}
• Full Literal has Values S={0, 1, 2, …, p-1}
X{0,1,…,p-1} Equivalent to Don’t Care
MVL Example
• MVI Function with 2 Inputs X, Y
– X is binary valued {0, 1}
– Y is ternary valued {0, 1, 2}
– n=2 pX=2 pY=3
• Function is TRUE if:
–
–
–
–
X=1 and Y= 0 or 1
Y=2
SOP form is:
F = X{1}Y{0,1} + X{0,1}Y{2}
• Literal
X{0,1}
F
is Full, So it is Don’t Care
– implicant is
– minterm is X {1}Y{0}
– prime implicants are X{1} and Y{2}
X{1}
Y{0,1}
Y
X
0
1
0
1
1
1
2
1
1
Multi-output Binary Function
• Consider
f 0 ( x, y , z )  x z  x y  x z
x
y
z
f1 ( x, y, z)  z  x y
x
0
0
0
0
1
1
1
1
y
0
0
1
1
0
0
1
1
z
0
1
0
1
0
1
0
1
f0
1
1
1
0
0
1
0
1
f1
0
1
1
1
0
1
0
1
f0
f1
Multi-output Binary Function
• Consider f 0 ( x, y, z )  x z  x y  x z
f1 ( x, y, z)  z  x y
x
y
z
f0
f1
x
0
0
0
0
1
1
1
1
y
0
0
1
1
0
0
1
1
z
0
1
0
1
0
1
0
1
f0
1
1
1
0
0
1
0
1
f1
0
1
1
1
0
1
0
1
Characteristic
Equation
W
x
y
z
F
x
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
y
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
z
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
W
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
F
1
0
1
1
1
1
0
1
0
0
1
1
0
0
1
1
Characteristic Equation
Sum of Minterms
F  x{0} y{0} z{0}W {0}  x{0} y{0} z{1}W {0}
 x{0} y{0} z{1}W {1}  x{0} y{1} z{0}W {0}
 x{0} y{1} z{0}W {1}  x{0} y{1} z{1}W {1}
 x{1} y{0} z{1}W {0}  x{1} y{0} z{1}W {1}
 x{1} y{1} z{1}W {0}  x{1} y{1} z{1}W {1}
x
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
y
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
z
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
W
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
F
1
0
1
1
1
1
0
1
0
0
1
1
0
0
1
1
PCN for MVL Functions
• Binary Variables, {0,1},
Represented by 2-bit Fields
 00
0 10
1 01
* 11
• MV Variables, {0,1,…,p-1}, Represented
by p-bit Fields
• BV Don’t Care is 11
• MV Don’t Care is 111…1
• MV Literal or Cube is Denoted by C()
PCN for MVL Example
F  X {1}Y {0,1}  X {0,1}Y {2}
X
01
11
Y
110
001
• Positional Cube Corresponding to X{1} is C(X{1})
C ( X )  01 111
{1}
• Since Y{0,1,2} is Don’t Care
PCN for MVI-BO Example
z
f1  ab  ab
f 2  ab
f 3  ab  ab
f (a, b); f  ( f1 , f 2 , f 3 )
a
b
f1 f2 f3
a b
10
10
100
a b
10
01
001
a b
01
10
001
ab
01
01
110
• View This as a SOP of MVI Function:
F  f  a{0}b{0} z{0}  a{0}b{1} z{2}  a{1}b{0} z{2}  a{1}b{1} z{0,1}
• F is the Characteristic Equation
List Oriented Manipulation
• Size of Literal = Cardinality of Logic Value Set
x{0,2}  size = 2
• Size of Implicant (Cube, Product Term) = Integer
Product of Sizes of Literals in Cube
• Size of Binary Minterm = 1  Implicant of Unit Size
EXAMPLE f (x1,x2,x3,x4,x5,x6)
implicant  x1 x3 x4  x1{1} x3{1} x4{0} x2{0,1} x5{0,1} x6{0,1}
size  111 2  2  2  01 01 10 11 11 11  8
# Don ' t Cares  3 ( x2 , x5 , x6 )
Logic Operations
• Consider Implicants as Sets
– Apply (, , , etc)
• Apply Bitwise Product, Sum, Complement to PCN
Representation
• Bitwise Operations on Positional Cubes May Have
Different Meaning than Corresponding Set Operations
EXAMPLE
Complement of Implicant  Complement of Positional Cube
MVL Logical Operations
• AND Operation – MIN - Set Intersection
• OR Operation – MAX - Set Union
• NOT Operation – Set Complement
EXAMPLE
X {0,1,3}Y {1,2}  X {1}Y {2}  X {1}Y {2}
X {0,1,3}Y {1,2}  X {1}Y {2,3}  X {0,1,3}Y {1,2,3}
p  5; X {0,1,3}  X {2,4}
MVL Number of Functions of 1 Variable
p
2
3
4
5
6
7
8
p
p
4
27
256
3125
46656
823543
16777216
MVL Circuits
MAX-gate
MIN-gate
Cube Merging
•
Basic Operation – OR of Two Cubes
•
MVL Operation – MAX is Union of Two Cubes
EXAMPLE
 = 1 {0,1} 0 1
 = 0 {0,1} 0 1
Merge  and  into 
 = {0,1} {0,1}0 1
a c d  ac d c d (a  a )c d
a{1}b{0,1}c{0}d {1}  a{0}b{0,1}c{0}d {1}
 (b
c d )(a  a )
{0,1} {0}
{1}
{1}
 a{0,1}b{0,1}c{0}d {1}
{0}
Multi-Output Minimization Example
X1
0
0
0
0
1
1
1
1
Input
X2
0
0
1
1
0
0
1
1
X3
0
1
0
1
0
1
0
1
Output
f0 f1 f2
0 1 1
0 0 1
0 0 0
0 0 1
0 0 0
1 1 0
1 0 0
1 0 0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
X1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
X2
0
0
0
0
0
0
1
1
1
1
1
1
0
0
0
0
0
0
1
1
1
1
1
1
X3
0
0
0
1
1
1
0
0
0
1
1
1
0
0
0
1
1
1
0
0
0
1
1
1
X4
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
F
0
1
1
0
0
1
0
0
0
0
0
1
0
0
0
1
1
0
1
0
0
1
0
0
Minimization Example (cont)
Sum of Minterms (Fig. 10.7 PLA Implementation)
F  X1 X 2 X 3 X 4  X1 X 2 X 3 X 4
{0}
{0}
{0}
{1}
{0}
{0}
{0}
{2}
 X1 X 2 X 3 X 4
{2}
 X1 X 2 X 3 X 4
 X1 X 2 X 3 X 4
{0}
 X1 X 2 X 3 X 4
 X1 X 2 X 3 X 4
 X1 X 2 X 3 X 4
{0}
{1}
{1}
{0}
{0}
{1}
{1}
{1}
{0}
{0}
{0}
{1}
{1}
{1}
{0}
{1}
{2}
{1}
{1}
{1}
{1}
{0}
Merging
• Merge 1st and 2nd
• Merge 3rd and 4th
• Merge 5th and 6th
• Merge 7th and 8th
F  X1 X 2 X 3 X 4
{0}
{0}
{0}
{1,2}
 X1 X 2 X 3 X 4
{1}
{0}
{1}
{0,1}
 X1 X 2
{0}
{0,1}
{1}
 X1 X 2 X 3
{1}
{1}
{2}
X3 X4
{0,1}
{0}
X4
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
X1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
X2
0
0
0
0
0
0
1
1
1
1
1
1
0
0
0
0
0
0
1
1
1
1
1
1
X3
0
0
0
1
1
1
0
0
0
1
1
1
0
0
0
1
1
1
0
0
0
1
1
1
X4
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
F
0
1
1
0
0
1
0
0
0
0
0
1
0
0
0
1
1
0
1
0
0
1
0
0
Minimization Example (cont)
F  X1 X 2 X 3 X 4
{0}
{0}
{0}
{1,2}
 X1 X 2 X 3 X 4
{1}
{0}
{1}
{0,1}
 X1 X 2
{0}
{0,1}
{1}
 X1 X 2 X 3
{1}
{1}
{2}
X3 X4
{0,1}
{0}
X4
Multi-Output Function Using of Multi-Output Prime Implicants
(Fig. 10.8 PLA Implementation)
f0  X1 X 2 X 3  X1 X 2
f1  X 1 X 2 X 3  X 1 X 2 X 3
f2  X1 X 2 X 3  X1 X 2