Quine-McCluskey Method

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Quine-McCluskey Method

• ECE-331, Digital Design
• Dr. Ron Hayne
• Electrical and Computer Engineering

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Quine-McCluskey Minimization

• wrt to minimization, K-maps Only good for
• Small functions (lt5variables)
• Single output function at a time
• Not Implementable on Computer
• Subjective Interpretation, Different Coverings
• Q-M (Tabular Minimization) Solves These Problems

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Basic Definitions

• Prime Implicant
• A group of adjacent 1s that is not contained
  within any larger implicant
• Essential Prime Implicant
• A prime implicant which includes a minterm that
  is covered by only one Prime Implicant

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Basic Definitions

• Complete Sum
• A representation of a function by the sum of ALL
  possible prime implicants
• Not necessarily irredundant
• Irredundant Sum
• The representation of a function by A sum of PIs
  such that the removal of any one of the PIs
  changes the value of the function for some
  combination of input values

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Basic Definitions

• Cost Function
• A measure of the effort or hardware associated
  with the implementation of something or the
  accomplishment of a task
• Digital Design Cost Functions
• Number of terms
• Number of occurrences of a literal (not the
  number of distinct literals).

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Quine-McCluskey Method

• Construct the Implicant Table
• Group minterms by number of 1s
• Apply Adjacency Theorem to pairs of implicants
• Mark all implicants that are used in larger groups

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Q-M Example

Karim-Johnson, p. 59
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Step 1a Implicant Table
Size Minterms
0 0
1 2 8
2 3 5 10
3 7 13
4 15
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Step 1b,c Adjacency Theorem
Size Minterms One-Cube
0 0 0,2 (2) 0,8 (8)
1 2 8
2 3 5 10
3 7 13
4 15
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Step 1b,c Adjacency Theorem
Size Minterms One-Cube
0 0 0,2 (2) 0,8 (8)
1 2 8 2,3 (1) 2,10 (8) 8,10 (2)
2 3 5 10 3,7 (4) 5,7 (2) 5,13 (8)
3 7 13 7,15 (8) 13,15 (2)
4 15
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Step 1b,c Adjacency Theorem
Size Minterms One-Cube Two-Cube
0 0 0,2 (2) 0,8 (8) 0,2,8,10 (2,8)
1 2 8 2,3 (1) 2,10 (8) 8,10 (2)
2 3 5 10 3,7 (4) 5,7 (2) 5,13 (8) 5,7,13,15 (2,8)
3 7 13 7,15 (8) 13,15 (2)
4 15
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Quine-McCluskey Method

• Construct the Covering Table
• Prime Implicants and Minterms
• Mark rows and columns
• Identify Essential Prime Implicants
• Construct the Reduced Covering Table
• Secondary Prime Implicants
• Mark rows and columns
• Minimum cover for remaining minterms
• Simplified Logic Expression

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Step 2a Covering Table
Prime Implicants Minterms 0 2 3 5 7 8 10 13 15
0,2,8,10 (2,8) 5,7,13,15 (2,8)
2,3 (1) 3,7 (4)
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Step 2b Covering Table
Prime Implicants Minterms 0 2 3 5 7 8 10 13 15
0,2,8,10 (2,8) 5,7,13,15 (2,8) X X X X X X X X
2,3 (1) 3,7 (4) X X X X
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2c Essential Prime Implicants
Prime Implicants Minterms 0 2 3 5 7 8 10 13 15
0,2,8,10 (2,8) 5,7,13,15 (2,8) X X X X X X X X
2,3 (1) 3,7 (4) X X X X
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Step 2c Covering Table
Prime Implicants v v v v v v v v 0 2 3 5 7 8 10 13 15
0,2,8,10 (2,8) 5,7,13,15 (2,8) X X X X X X X X
2,3 (1) 3,7 (4) X X X X
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3a,b Reduced Covering Table
Prime Implicants Minterms 3

2,3 (1) 3,7 (4) X X
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3c Reduced Covering Table
Prime Implicants Minterms 3

2,3 (1) 3,7 (4) X X
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4 Simplified Logic Expression
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Summary

• Q-M
• Implicant Table
• Covering Table
• Essential Prime Implicants
• Simplified Logic Expression