1. ECB2212-Digital Electronics
Ms.K.Indra Gandhi
Asst Prof (Sr.Gr) /ECE

2. Two-Level Minimization I
Logic Synthesis
Two-Level Minimization I

3. Quine-McCluskey Procedure (Exact)
Given G’ and D (covers for F = (f,d,r) and d), find a minimum cover G of primes where:
f  G  f+d
(G is a prime cover of F )
Q-M Procedure:
1. Generate all the primes of F , {Pj} (i.e. primes of (f+d)=G’+D)
2. Generate all the minterms of f=G’D, {mi}
3. Build Boolean matrix B where
Bij = 1 if mi Pj
= 0 otherwise
4. Solve the minimum column covering problem for B (unate covering problem)

4. Difficulty Note: Can be ~ 2n minterms ~ 3n/n primes
Thus O(2n) rows and O(3n/n ) columns AND minimum covering problem is NP-complete. (Hence can probably be double exponential in size of input, i.e. difficulty is O(23n)
primes
3n/n 1
minterms 2n 1

5. Example Primes: y + w + xz Covering Table
Solution: {1,2}  y + w is minimum prime
cover. (also w+ xz)
xz
xy
xy xy
xy y
zw w
Karnaugh map
zw y w
xz
xyzw
xyzw
xyzw
xyzw zw
zw

6. Covering Table
Definition: An essential prime is any prime that uniquely covers a minterm of f.
Primes of f+d y w
xz
xyzw
xyzw
xyzw
xyzw
Minterms of f
Row singleton
(essential minterm)
Essential prime

7. Covering Table Row Equality:
In practice, many rows are identical. That is there exist minterms that are contained in the same set of primes. m m
Any row can be associated with a cube -- called the signature cube.
e.g. m1  m2  P2P4P5P7

8. Row and Column Dominance
Definition: A row i1 whose set of primes is contained in the set of primes of row i2 is said to dominate i2.
Example: i i
i1 dominates i2
We can remove row i2, because we have to choose a prime to cover i1, and any such prime also covers i2. So i2 is automatically covered.

9. Row and Column Dominance
Definition: A column j1 whose rows are a superset of another column j2 is said to dominate j2.
Example:
j1 dominates j2
We can remove column j2 since j1 covers all those rows and more. We would never choose j2 in a minimum cover since it can always be replaced by j1.
j j2
1 0
0 0
1 1

10. Pruning the Covering Table
1. Remove all rows covered by essential primes (columns in row singletons). Put these primes in the cover G.
2. Group identical rows together and remove dominated rows.
3. Remove dominated columns. For equal columns, keep one prime to represent them.
4. Newly formed row singletons define n-ary essential primes.
5. Go to 1 if covering table decreased.
The resulting reduced covering table is called the cyclic core. This has to
be solved (unate covering problem). A minimum solution is added to G -
the set of n-ary essential primes. The resulting G is a minimum cover.

11. Example Essential Prime and Column Dominance G=P1
n-ary Essential Prime
and
Column Dominance
G=P1 + P3
Row dominance
Cyclic
Core

12. Solving the Cyclic Core
Best known method (for unate covering) is branch and bound with some
clever bounding heuristics.
Independent Set Heuristic:
Find a maximum set of “independent” rows I. Two rows Bi1 ,Bi2 are independent if j such that Bi1j =Bi2j=1. (They have no column in common)
Example: Covering matrix B rearranged with independent sets first. 1 1 1 B=
Independent set = I
of rows 1 1 1 1 1 1 C A

13. Solving the Cyclic Core
Lemma:
|Solution of Covering|  |I| 1 1 1 1 1 1 1 1 1 C A

14. Heuristic Let I={I1, I2, …, Ik} be the independent set of rows
choose j  Ii which covers the most rows of A. Put j  J
eliminate all rows covered by column j
I  I\{Ii}
go to 1 if |I|  0
If B is empty, then done (in this case we have the guaranteed minimum solution - IMPORTANT)
If B is not empty, choose an independent set of B and go to 1 1 A C

15. Generating Primes - single output func.
Tabular method
(based on consensus operation):
Start with all minterm canonical form of F
Group pairs of adjacent minterms into cubes
Repeat merging cubes until no more merging possible; mark () + remove all covered cubes.
Result: set of primes of f.
Example:
F = x’ y’ + w x y + x’ y z’ + w y’ z
F = x’ y’ + w x y + x’ y z’ + w y’ z
w’ x’ y’ z’ 
w’ x’ y’ z 
w’ x’ y z’ 
w x’ y’ z’ 
w x’ y’ z 
w x’ y z’ 
w x y z’ 
w x y’ z 
w x y z 
w’ x’ y’ 
w’ x’ z’ 
x’ y’ z’ 
x’ y’ z 
x’ y z’ 
w x’ y’ 
w x’ z’ 
w y’ z
w y z’
w x y
w x z
x’ y’
x’ z’
Courtesy: Maciej Ciesielski, UMASS

16. Generating Primes – multiple outputs
Procedure similar to single-output function, except:
include also the primes of the products of individual functions
Example:
x y z
0 – 0
0 1 1
1 – 1
f1 f2
0 1
1 1
1 0 x y z
000
100
110
010
111
011 f1 f2
101
Can also represent it as:
x y z
0 – 0
0 1 –
– 1 1
1 – 1
f1 f2
0 1
1 0

17. Generating Primes - example
Modification (w.r.t single output function):
When two adjacent implicants are merged, the output parts are intersected
000 | 
010 | 
011 | 11
101 | 
111 | 
0 – 0 | 01
0 1 – | 01
– 1 1 | 10
1 – 1 | 10
x y z
0 – 0
0 1 1
1 – 1
f1 f2
0 1
1 1
1 0 f2
000
100
110
010
111
011 f1
101
There are five primes listed for
this two-output function.
- What is the min cover ?

18. Minimize multiple-output cover - example
List multiple-output primes
000 | 01
010 | 01
011 | 01
011 | 10
101 | 10
111 | 10
listed twice
p1 p2 p3 p4 p5
Create a covering table, solve
p1 = | 11
p2 = 0 – 0 | 01
p3 = 0 1 – | 01
p4 = – 1 1 | 10
p5 = 1 – 1 | 10 f1 f2
000
100
110
010
111
011
101
Min cover has 3 primes:
F = { p1, p2, p5 }

19. Generating Primes
We use the unate recursive paradigm. The following is how the merge step
is done. (Assumption: we have just generated all primes of
and )
Theorem: p is a prime of f iff p is maximal among the set consisting of
P = xiq, q is a prime of ,
P = xir, r is a prime of ,
P = qr, q is a prime of , r is a prime of

20. Generating Primes Example: Assume q = abc is a prime of .
Form p=xiabc. Suppose r=ab is a prime of
Then is an implicant of f
Thus abc and xiab are implicants, so xiabc is not prime.
Note: abc is prime because if not, ab  fx (or ac or bc) contradicting abc prime of
Note: xiab is prime, since if not then either ab  f or xia  f . The first contradicts abc prime of and the second contradicts ab prime of

21. Quine-McCluskey Summary
Q-M:
1. Generate cover of all primes 2. Make G irredundant (in optimum way)
Note: Q-M is exact i.e. it gives an exact minimum
Heuristic Methods:
Generate (somehow) a cover of  using some of the primes
Make G irredundant (maybe not optimally)
Keep best result - try again (i.e. go to 1)

22. Generalized Cofactor
Definition: Let f, g be completely specified functions. The generalized cofactor of f with respect to g is the incompletely specified function:
Definition: Let  = (f, d, r) and g be given. Then

23. Relation with Shannon Cofactor
Let g=xi . Shannon cofactor is
Generalized cofactor with respect to g=xi is
Note that
In fact fxi is the unique cover of co(f, xi ) independent of the variable xi .

24. Shannon Cofactor on
off
Don’t care

25. Shannon Cofactor (Cont.)So

26. Properties of Generalized Cofactor
Shannon Cofactor
Generalized Cofactor
We will get back to use of generalized cofactor later