1. Technical Seminar II Implementation of
Quine-McCluskey Minimization method
Using ‘C’ code

2. Contents Introduction Brief explain of Quine-McCluskey Method.
Algorithm
Execution of algorithm
Application
Advantage
Conclusion
References

3. Introduction
The Karnaugh map is a method to simplify Boolean algebra expressions.
McCluskey method is functionally identical to Karnaugh mapping, but the tabular form makes it more efficient for use in computer algorithms
This method involves two steps:
Finding all prime implicants of the function.
Use those prime implicants in a prime implicant chart to find the essential prime implicants of the function, as well as other prime implicants that are necessary to cover the function.

4. (Example) Bcd checker
Actual given input bits are 4.

5. Features Two step process utilizing tabular listings to:
Identify prime implicants (implicant tables)
Identify minimal PI set (cover tables)
All work is done in tabular form
Number of variables is not a limitation
Basis for many computer implementations
Don’t cares are easily handled
Proper organization and term identification are key factors for correct results

6. Quine-McCluskey Minimization
Terms are initially listed as
Each group contains terms with the same number of true and complemented variables
Terms are listed in numerical order within group
Terms and implicants are identified using one of three common notations
full variable form
cellular form
1,0,- form

7. Notation Forms
Full variable form - variables and complements in algebraic form
hard to identify when adjacency applies
very easy to make mistakes
Cellular form - terms are identified by their decimal index value
Easy to tell when adjacency applies; indexes must differ by power of two (one bit)
Implicants identified by term nos. separated by comma; differing bit pos. in () following terms

8. Notation Forms (cont.)
1,0,- form - terms are identified by their binary index value
Easier to translate to/from full variable form
Easy to identify when adjacency applies, one bit is different
‘-’ shows variable(s) dropped when adjacency is used
Different forms may be mixed during the minimization

9. Example of Different Notations
F(A, B, C, D) =  m(4,5,6,8,10,13)
Full variable Cellular ,0,-
1 ABCD
ABCD
2 ABCD
ABCD
ABCD
3 ABCD

10. NOTE: Don’t Cares are included
Implication Table (1,0,-)
Quine-McCluskey Method
Tabular method to systematically find all prime implicants
ƒ(A,B,C,D) = Σ m(4,5,6,8,9, 10,13) + Σ d(0,7,15)
Part 1: Find all prime implicants
Step 1: Fill Column 1 with active-set and DC-set minterm indices. Group by number of true variables (# of 1’s).
Implication Table
Column I
0000
0100
1000
0101
0110
1001
1010
0111
1101
1111
NOTE: Don’t Cares are included
in this step!

11. Minimization - First Pass (1,0,-)
Quine-McCluskey Method
Tabular method to systematically find all prime implicants
ƒ(A,B,C,D) = Σ m(4,5,6,8,9,10,13) + Σ d(0,7,15)
Part 1: Find all prime implicants
Step 2: Apply Adjacency - Compare elements of group with N 1's against those with N+1 1's. One bit difference implies adjacent. Eliminate variable and place in next column.
E.g., 0000 vs yields 0-00
0000 vs yields -000
When used in a combination, mark with a check. If cannot be combined, mark with a star. These are the prime implicants.
Repeat until nothing left.
Implication Table
Column I
0000 
0100 
1000 
0101 
0110 
1001 
1010 
0111 
1101 
1111 
Column II
0-00
-000
010-
01-0
100-
10-0
01-1
-101
011-
1-01
-111
11-1

12. Minimization - Second Pass (1,0,-)
Quine-McCluskey Method
Step 2 cont.: Apply Adjacency - Compare elements of group with N 1's against those with N+1 1's. One bit difference implies adjacent. Eliminate variable and place in next column.
E.g., 0000 vs yields 0-00
0000 vs yields -000
When used in a combination, mark with a check. If cannot be combined, mark with a star. These are the prime implicants.
Repeat until nothing left.
Implication Table
Column I
0000 
0100 
1000 
0101 
0110 
1001 
1010 
0111 
1101 
1111 
Column II
0-00 *
-000 *
010- 
01-0 
100- *
10-0 *
01-1 
-101 
011- 
1-01 *
-111 
11-1 
Column III
01-- *
-1-1 *

13. Coverage Table
Coverage Chart
0,4(0-00)
0,8(-000)
8,9(100-)
8,10(10-0)
9,13(1-01)
4,5,6,7(01--)
5,7,13,15(-1-1) 4 X 5 6 8 9 10 13
Note: Don’t include DCs in coverage table; they don’t have covered by the final logic expression!
rows = prime implicants
columns = ON-set elements
place an "X" if ON-set element is
covered by the prime implicant

14. Coverage Table (cont.) Coverage Chart rows = prime implicants
0,4(0-00)
0,8(-000)
8,9(100-)
8,10(10-0)
9,13(1-01)
4,5,6,7(01--)
5,7,13,15(-1-1) 4 X 5 6 8 9 10 13
0,4(0-00)
0,8(-000)
8,9(100-)
8,10(10-0)
9,13(1-01)
4,5,6,7(01--)
5,7,13,15(-1-1) 4 X 5 6 8 9 10 13
rows = prime implicants
columns = ON-set elements
place an "X" if ON-set element is
covered by the prime implicant
If column has a single X, than the
implicant associated with the row
is essential. It must appear in
minimum cover

15. Coverage Table (cont.) Eliminate all columns covered by
0,4(0-00)
0,8(-000)
8,9(100-)
8,10(10-0)
9,13(1-01)
4,5,6,7(01--)
5,7,13,15(-1-1) 4 X 5 6 8 9 10 13
Eliminate all columns covered by
essential primes

16. Coverage Table (cont.) Eliminate all columns covered by
0,4(0-00)
0,8(\000)
8,9(100-)
8,10(10-0)
9,13(1-01)
4,5,6,7(01--)
5,7,13,15(-1-1) 4 X 5 6 8 9 10 13
0,4(0-00)
0,8(\000)
8,9(100-)
8,10(10-0)
9,13(1-01)
4,5,6,7(01--)
5,7,13,15(-1-1) 4 X 5 6 8 9 10 13
Eliminate all columns covered by
essential primes
Find minimum set of rows that
cover the remaining columns

17. Algorithm
Step 1: Gather all minterms & don't care terms (if there are any) and convert them to binary form, then sort them in groups and list them in a list.
Step 2: Compare group 1 with group 2 & group 2 with group 3 & group 3 with group 4, the comparison is achieved by comparing each term in the first group with all terms in the next group. The comparison concept is that if two terms have only 1 different bit, then this bit must be replaced with (-).
Step 3: After the comparison is done in (List 1), move to List 2 and do the same comparison, but we will find a new element, the dash (-) that we must handle it in the comparison. When we compare two dashed terms, the comparison is legal only if the dash position is the same in the two terms, otherwise we can't compare those terms like: (00- , 01-) ==> (0--), but (00- , 0-1) is illegal.
If two terms compared successfully, a check character is put next to the compared terms (say the check chr is 't'), if not, I mean that the two terms have more different bits or the dash position is not the same in the two terms, then a check character must be put next to the uncompared terms (say check chr is '*')
The terms with ‘*’ indicates prime implicants.

18. Algorithm(cont.) Step 4: Coverage table.( Essential prime implicants)
we will put the minterms horizontally & the prime implicants vertically, then we should determine which terms a prime implicant cover like (10-0) covers 8,10. and put an indicator (say 'X') in the cells corresponding to the row of prime implicant and its terms.
Step 5: Traversing the columns for single ‘X’ variable. If yes identify the PI.
Step 6: Traverse the row of that PI and if any ‘X’ variable is found, Nullify the element ‘X’ in that rows.
Step 7: Traversing the rows which contains more no’s of ‘X’. This row represents essential PI. Identify and nullify the particular column where the ‘X’ represents.

24. Application
It is used as a logic synthesis(in CAD) tool to minimize the Boolean expression. It finally gives out a simple expression to design circuit.
CAD: VERILOG
VHDL

25. Advantages
The Quine-McCluskey (Q-M) method is a computer-based technique for simplification and has mainly two advantages over the K-map method.
Firstly, it is systematic for producing a minimal function. Secondly, it is good in handling a large number of variables.

26. Conclusion
1. By using this technique we can get the minimized SOP or POS forms.

27. References
[1]. Digital Design Fundamentals, 2nd Ed [Kenneth J. Breeding]
[2]. algorithm
[3].
[4]. Logic Design By D.A.Godse A.P.Godse
[5]. V.P. Nelson e.a., Digital Circuit Analysis and Design, Prentice Hall, 1995, pag. 234