Multiple Valued Logic

1
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
  Multi-output Functions
• Characteristic Equation
• Positional Cube Notation (PCN) Extensions

2
MVI Functions

• Each Input can have Value in Set 0, 1, 2, ...,
  pi-1

• MVI Functions
• X is p-valued variable
• literal over X corresponds to subset of values of
  S ? 0, 1, ... , p-1 denoted by XS

3
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 Xj Where j is the
  Logic Value
• Empty Literal X?
• Full Literal has Values S0, 1, 2, , p-1
• X0,1,,p-1 Equivalent to Dont Care

4
MVL Example

• MVI Function with 2 Inputs X, Y
• X is binary valued 0, 1
• Y is ternary valued 0, 1, 2
• n2 pX2 pY3
• Function is TRUE if
• X1 and Y 0 or 1
• Y2
• SOP form is
• F X1Y0,1 X0,1Y2
• Literal X0,1 is Full, So it is Dont Care
• implicant is X1 Y0,1
• minterm is X 1Y0
• prime implicants are X1 and Y2

X
F
0 1
0 1
1 1
2 1 1
Y
5
Multi-output Binary Function

• Consider

x
f0
y
f1
z
6
Multi-output Binary Function
Characteristic Equation

• Consider

W
x
F
y
z
x
f0
y
f1
z
7
Characteristic Equation
Sum of Minterms
8
PCN for MVL Functions
? 00
0 10
1 01
11

• Binary Variables, 0,1,
• Represented by 2-bit Fields
• MV Variables, 0,1,,p-1, Represented by p-bit
  Fields
• BV Dont Care is 11
• MV Dont Care is 1111
• MV Literal or Cube is Denoted by C(?)

9
PCN for MVL Example

• Positional Cube Corresponding to X1 is C(X1)

 

• Since Y0,1,2 is Dont Care

10
PCN for MVI-BO Example
z
a b f1 f2 f3
a? b? 10 10 100
a? b 10 01 001
a b? 01 10 001
a b 01 01 110

• View This as a SOP of MVI Function

• F is the Characteristic Equation

11
List Oriented Manipulation

• Size of Literal Cardinality of Logic Value Set
• x0,2 ? 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)

12
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

13
MVL Logical Operations

• AND Operation MIN - Set Intersection
• OR Operation MAX - Set Union
• NOT Operation Set Complement

EXAMPLE
14
MVL Number of Functions of 1 Variable
15
MVL Circuits
MAX-gate
MIN-gate
16
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 0,10 1

17
Multi-Output Minimization Example
18
Minimization Example (cont)
Sum of Minterms (Fig. 10.7 PLA Implementation)
Merging

• Merge 1st and 2nd
• Merge 3rd and 4th
• Merge 5th and 6th
• Merge 7th and 8th

19
Minimization Example (cont)
Multi-Output Function Using of Multi-Output
Prime Implicants (Fig. 10.8 PLA Implementation)