1. Lecture 15 — CS 2603 Applied Logic for Hardware and Software
CS Applied Logic, University of Oklahoma
1/17/2019
Lecture 15 — CS 2603 Applied Logic for Hardware and Software
Circuit Minimization
using Karnaugh maps
CS2603 Applied Logic for Hardware and Software
Rex Page – University of Oklahoma

2. Boolean Functions new name for same old animal
Boolean function — a special kind of predicate
Domain of discourse: tuples of {True, False} values
Boolean functions  truth tables
F(a, b) — two-variable Boolean function
Domain of discourse: pairs of Boolean values
Specifies a Boolean output for each pair of Boolean inputs
Representations we’ve already seen for 2-var Boolean functions
Truth table with four rows (unique: one per Boolean function)
Propositional WFF with two variables (not a unique rep'n)
Combinational circuit with two input lines, one output line
F(a, b, c) — three-variable Boolean function
Domain of discourse: triples of Boolean values
8-row truth table, 3-variable WFF, 3-input/1-output circuit
F(x1, x2, … xn) — n-variable Boolean function
Domain of discourse: n-tuples of Boolean values
2n-line truth table, n-variable WFF, n-input/1-output circuit
CS2603 Applied Logic for Hardware and Software
Rex Page – University of Oklahoma

3. Example truth-table for 3-variable Boolean functiony z
F(x,y,z) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
CS2603 Applied Logic for Hardware and Software
Rex Page – University of Oklahoma

4. Sum-of-Products Representation constructing a WFF for Boolean function F(x1, x2, … xn)
Focus on places in truth-table where F is True
Boolean tuple (b1, b2, … bn), with F(b1, b2, … bn) =True
That is, (k. (xk = bk))  (F(x1, x2, … xn) = True)
Find a propositional WFF whose value is (k. (xk = bk))
WFF w, variables x1, x2, … xn , value of w = (k. (xk = bk))
That is, w =True if (k. (xk = bk)) =True
w = False if (k. (xk = bk)) = False
w = (y1  y2  …  yn) = (k. (xk = bk))
yk = xk if bk = True
yk = xk if bk = False
known as a “minterm”
Minterms are usually written in EE notation
For example: x1 x2 x3, when b1=1, b2=0, b3=1
A WFF for F(x1, x2, … xn)
F(x1, x2, … xn) = w1 + w2 +  wm — each wk is a minterm
m = number of 1’s in last column of truth table
sum of products
CS2603 Applied Logic for Hardware and Software
Rex Page – University of Oklahoma

5. Example sum-of-products representation for 3-variable function
Truth table representation
of a Boolean function F 1
minterm
F(x,y,z) z y x
Example copied from Shaaban:
CS2603 Applied Logic for Hardware and Software
Rex Page – University of Oklahoma

6. Example sum-of-products representation for 3-variable function
Truth table representation
of a Boolean function F 1
x y z
minterm
F(x,y,z) z y x
Example copied from Shaaban:
CS2603 Applied Logic for Hardware and Software
Rex Page – University of Oklahoma

7. Example sum-of-products representation for 3-variable function
Truth table representation
of a Boolean function F 1
x y z
minterm
F(x,y,z) z y x
Example copied from Shaaban:
CS2603 Applied Logic for Hardware and Software
Rex Page – University of Oklahoma

8. Example sum-of-products representation for 3-variable function
Truth table representation
of a Boolean function F 1
x y z
minterm
F(x,y,z) z y x
Example copied from Shaaban:
CS2603 Applied Logic for Hardware and Software
Rex Page – University of Oklahoma

9. Example sum-of-products representation for 3-variable function
Truth table representation
of a Boolean function F
sum-of-products
representation of F
x y z 1
minterm
F(x,y,z) z y x
Example copied from Shaaban:
CS2603 Applied Logic for Hardware and Software
Rex Page – University of Oklahoma

10. Example sum-of-products representation for 3-variable function
Truth table representation
of a Boolean function F
sum-of-products
representation of F
x y z 1
minterm
F(x,y,z) z y x
F(x,y,z) =
x y z + x y z + x y z + x y z
Example copied from Shaaban:
CS2603 Applied Logic for Hardware and Software
Rex Page – University of Oklahoma

11. 3-Variable Karnaugh Map another representation for 3-variable function
Truth table representation
of a Boolean function F
sum-of-products
representation of F
x y z 1
minterm
F(x,y,z) z y x
F(x,y,z) =
x y z + x y z + x y z + x y z
Karnaugh-map
representation of F
Example copied from Shaaban:
table-based form
of truth table
CS2603 Applied Logic for Hardware and Software
Rex Page – University of Oklahoma

12. 3-Variable Karnaugh Map another representation for 3-variable function
Truth table representation
of a Boolean function F
sum-of-products
representation of F
x y z 1
minterm
F(x,y,z) z y x
F(x,y,z) =
x y z + x y z + x y z + x y z
Karnaugh-map
representation of F
Example copied from Shaaban:
z = 1
x = 1 1
y = 1 z xy
Gray-code arrangement
CS2603 Applied Logic for Hardware and Software
Rex Page – University of Oklahoma

13. Karnaugh-Map Minimization Method 3-variable example
CS Applied Logic, University of Oklahoma
1/17/2019
Karnaugh-Map Minimization Method 3-variable example
sum-of-products
representation of F
Karnaugh-map
representation of F
z = 1
x = 1 1
y = 1 z xy
F(x,y,z) = x y z + x y z + x y z + x y z
x y z
= x y z + x z + y z
minimized
x y z + x y z
= y z (x + x)
= y z
wrap-around
x y z + x y z = x z
Example copied from Shaaban:
Karnaugh-map minimization method
Group together the maximal, contiguous, rectangular regions with 2k adjacent cells containing True (1) values
There is one minterm per cell, so 2k minterms in all for the group
Gray-code ordering arranges it so that (n – k) variables have identical form throughout the group (xj in all terms of group or xj in all of them)
Use the distributive law to factor the 2k minterms into this form:
w = (v1 + v1)(v2 + v2) … (vk + vk) vk+1 vk+2 … vn
= vk+1 vk+2 … vn
Possible because Gray-code ordering puts the other k variables through all possible combinations
CS2603 Applied Logic for Hardware and Software
Rex Page – University of Oklahoma

14. 4-Variable Karnaugh-Map Example00 01 11 10
c = 1
b a
d c
a = 1 1
d = 1
F(a,b,c,d) = a b c d
+ a b c d
= a d
+ a b c
+ b c d
+ a b c
Example copied from Shaaban:
Use the distributive law to factor the 2k minterms into this form: w = (v1 + v1)(v2 + v2) … (vk + vk) vk+1 vk+2 … vn
Group together the maximal, contiguous, rectangular regions with 2k adjacent cells containing True (1) values
There is one minterm per cell, so 2k minterms in all for group
Possible because Gray-code ordering puts the other k variables through all possible combinations
Gray-code ordering arranges it so that (n – k) variables have identical form throughout the group (xj in all terms of group or xj in all of them)
wrap-around
CS2603 Applied Logic for Hardware and Software
Rex Page – University of Oklahoma

15. Another 4-Variable Example00 01 11 10
c = 1
b a
d c
a = 1 1
d = 1
F(a,b,c,d) = a b c d
+ a b c d
= a d
+ a b
+ b c d
+ a c
Use the distributive law to factor the 2k minterms into this form: w = (v1 + v1)(v2 + v2) … (vk + vk) vk+1 vk+2 … vn
Group together the maximal, contiguous, rectangular regions with 2k adjacent cells containing True (1) values
There is one minterm per cell, so 2k minterms in all for group
Possible because Gray-code ordering puts the other k variables through all possible combinations
Gray-code ordering arranges it so that (n – k) variables have identical form throughout the group (xj in all terms of group or xj in all of them)
wrap-around
CS2603 Applied Logic for Hardware and Software
Rex Page – University of Oklahoma

16. Don’t Cares F(a,b,c,d) = a b c d + a b c d = b d + a b00 01 11 10
c = 1
b a
d c
a = 1 1
d = 1 
F(a,b,c,d) = a b c d
+ a b c d
= b d
+ a b
either 1 or 0 ok
+ a d
Use the distributive law to factor the 2k minterms into this form: w = (v1 + v1)(v2 + v2) … (vk + vk) vk+1 vk+2 … vn
Group together the maximal, contiguous, rectangular regions with 2k adjacent cells containing True (1) values
There is one minterm per cell, so 2k minterms in all for group
Possible because Gray-code ordering puts the other k variables through all possible combinations
Gray-code ordering arranges it so that (n – k) variables have identical form throughout the group (xj in all terms of group or xj in all of them)
Use don’t-cares to make groups bigger
reduces number of minterms and/or literals
Leave isolated don’t-cares alone
no point in adding minterms unnecessarily
CS2603 Applied Logic for Hardware and Software
Rex Page – University of Oklahoma

17. 2-Variable Karnaugh-Map Example
 operation b a 1
a = 1
b = 1
F(a,b) = a b + a b + a b
= a + b
Example copied from Shaaban:
Karnaugh-map minimization method
Group together the maximal, contiguous, rectangular regions with 2k adjacent cells containing True (1) values
There is one minterm per cell, so 2k minterms in all for the group
Gray-code ordering arranges it so that (n – k) variables have identical form throughout the group (xj in all terms of group or xj in all of them)
Use the distributive law to factor the 2k minterms into this form:
w = (v1 + v1)(v2 + v2) … (vk + vk) vk+1 vk+2 … vn
Possible because Gray-code ordering puts the other k variables through all possible combinations
CS2603 Applied Logic for Hardware and Software
Rex Page – University of Oklahoma

18. Karnaugh-Map Minimization some observations
Each disjoint group containing 2k cells
Reduces number of minterms by 2k - 1
Reduces number of literals in those minterms to n – k
Overlapping group
Reduces number of literals in some minterms
But not necessarily the the number of minterms
Method produces "minimal" circuit if
Minimum number of maximal groups covers 1s in diagram
Then: minimum number of minterms
And: minimum number of literals in minterms
Not unique … could be several ways to choose groups
Karnaugh maps mostly used for learning
Designing small digital circuits
Circuit design tools do minimization for large circuits
Quine-McClusky algorithm implements Karnaugh method
Invented by Maurice Karnaugh at Bell Labs in early 1950s
CS2603 Applied Logic for Hardware and Software
Rex Page – University of Oklahoma

19. CS2603 - Applied Logic, University of Oklahoma
1/17/2019
The End
CS2603 Applied Logic for Hardware and Software
Rex Page – University of Oklahoma