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Multiple Valued Logic

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

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Multiple Valued Logic

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  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 asX{j}Wherejis 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

  4. 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} is Full, So it is Don’t Care • implicant is X{1} Y{0,1} • minterm is X{1}Y{0} • prime implicants are X{1} and Y{2} X F Y

  5. Multi-output Binary Function x f0 y • Consider f1 z

  6. Characteristic Equation Multi-output Binary Function W x F y z • Consider x f0 y f1 z

  7. Characteristic Equation Sum of Minterms

  8. PCN for MVL Functions • Binary Variables, {0,1}, Represented by 2-bit Fields • 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()

  9. PCN for MVL Example • Positional Cube Corresponding to X{1} is C(X{1}) • Since Y{0,1,2} is Don’t Care

  10. PCN for MVI-BO Example • 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 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)

  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} {0,1}0 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)

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