“Minimization Techniques”;
1.1 Minimization Techniques
The minimization technique is a process or a simplification procedure that are used to minimize the number of literals in a Boolean logic expression. The main objective of all minimization techniques is to obtain an expression that has the minimum number of terms. If we have more than one possible solution with the same number of terms, the one having the minimum number of literals is the choice. The process is usually called logic minimization and the goal is to form a result which is efficient.
There are several methods for simplification of Boolean logic expressions. The most commonly used techniques are:
•Boolean algebra postulates and theorems
•Karnaugh maps
•Quine-McCluskey method
•Petrick's method
•Generalized concensus algorithm
For very complicated problems, the Algebraic Minimization method can be done using special software analysis programs. K-maps are also limited to problems with upto 4 binary inputs. For more than 4 binary inputs, the Quine–McCluskey tabular method is used.
1.2 Boolean postulates
The algebraic system usually used to work with binary logic expressions
Postulates:
Closure: Any defined operation on (0, 1) gives (0,1)
Identity: 0 + x = x+0 ; 1. x = x
Commutative: x + y = y + x ; x.y = y.x .
Distributive: x (y + z) = xy + xz; x + (yz) = (x + y)(x + z).
For every element x B, there exists an element x' B (complement of x) such that:
a. x + x' = 1
b. x . x' = 0
There exists at least two elements x, y B such that x= y.
Precedence rule: (1) parentheses (2) NOT (3) AND (4) OR
1.3 De-Morgan‟s Theorem
De-Morgan was a great logician and mathematician. Among his contribution, the following two theorems are important.
De-Morgan‘s First Theorem
It States that "the complement of the sum of the variables is equal to the product of the complement of each variable".
This theorem also may be expressed by the following Boolean expression.
De-Morgan‘s Second Theorem
It states that the "Complement of the product of variables is equal to the sum of complements of each individual variables".
Boolean expression for this theorem is Boolean Function.
Canonical Form of Boolean Expressions
It is the expanded form of Boolean expression, where each term contains all Boolean variables in their true or complemented form, is also known as the canonical form of the expression.
De Morgan suggested two theorems that form important part of Boolean algebra. They are:
1) The complement of a product is equal to the sum of the complements.
(AB)' = A' + B'
2) The complement of a sum term is equal to the product of the complements.
(A + B)' = A'B'
Problems
Here we will discuss the methods to reduce the expression
Solution:
Let us prove that (a) a + āb = a + b and
Solution:
(a) a + āb = a + b
a + āb = a + ab + āb = a + b(a + ā)(∴a + ā = l)
= a + b
L.H.S. = R.H.S.
1.4 Principle of Duality
Duality Theorem:
The Duality Theorem states that starting with a Boolean relation we can derive another Boolean relation by
(i)changing OR (operation) i.e. + (Plus) sign to an AND operation.
(ii)complement any 0 or 1 appearing in the expression.
1.5 Boolean expression – Minimisation of Boolean expressions, MinTerm, Maxterm, Sum of products(SOP), Product of sums(POS)
Boolean expressions:
The basic properties of Boolean algebra are commutative property, associative Property and distributive property.
Associative Property:
The associative property of Boolean algebra states that the OR ing of several variables.
results in the same regardless of the grouping of the variables. The associative property is stated as follows:
A+ (B+C) = (A+B) +C
Commutative Property:
The commutative property states that the order in which the variables are OR ed makes no difference. The commutative property is:
A+B=B+A
Distributive Property:
The distributive property states that AND ing several variables and OR ing the result with a single variable is equivalent to OR ing the single variable with each of the the several variables and then AND ing the sums.
The distributive property is:
A+BC= (A+B) (A+C)
Absorption Law:
The absorption law of Boolean algebra is given by X+XY=X, X(X+Y) =X.
Minterm:
A product term containing all the variables of the function in either complemented or un complemented form is called a minterm.
Maxterm :
A sum term containing all the variables of the function in either complemented or un complemented form is called a maxterm.
Sum of Products (SOP) :
A Sum-Of-Products Boolean expression is literally a set of Boolean terms added (summed) together each term being a multiplicative (product) combination of Boolean variables.
sum-of-products-expression = term1 + term2 … + term n
Product terms that include all of the input variables (or their inverses) are called minterms.
In a sum-of-products expression, we form a product of all the input variables (or their inverses) for each row of the truth table for which the result is logic 1.
The output is the logical "sum" of these minterms.
Sum-Of-Products expressions are easy to generate by determining which rows of the table have an output of 1, writing one product term for each row and finally summing all the product terms.
This creates a Boolean expression representing the truth table as a whole.
Sum-Of-Products expressions lend themselves well to implementation as a set of AND gates (products) feeding into a single OR gate (sum).
Truth table:
The output can be expressed as:
Product of Sums (POS)
An alternative to a Sum-Of-Products expression to account for all the "high" (1) output conditions in the truth table is to generate a Product-Of-Sums, or POS expression to account for all the "low" (0) output conditions instead.
POS Boolean expressions can be generated from truth tables quite easily, by determining which rows of the table have an output of 0, writing one sum term for each row and finally multiplying all the sum terms.
This creates a Boolean expression representing the truth table as a whole.
These "sum" terms that include all of the input variables (or their inverses) are called maxterms.
For POS implementation, the output variable is the logical product of maxterms.
Product-Of-Sums expressions lend themselves well to implementation as a set of OR gates (sums) feeding into a single AND gate (product).
Truth table:
The output can be expressed as:
Problems
Lets we reduce the given expressions using Boolean algebra.
x'y'z' +x'y'z' + x'yz + xy'z + xyz
= x'y' (z' + z) + yz (x' + x)+ xy'z
= x'y' + yz + xy'z
= y'(x' + z)+yz
= y'x' + y'z+yz
= y'x' + z(y'+y)
= y'x' + z.
abc' + ab'c + a'bc + abc
= ab (c' + c) + ab'c + a'bc
= ab + ab'c + a'bc = a (b + b'c) + a'bc
= ab + ac + a'bc = b(a + c) + ac = ab+ac+bc
p'q'r + p'qr' + p'qr + pqr' + pq'r'
= p'r (q' + q) + qr' (p' + p) + pq'r'
= p'r + qr' + pq'r'
= r' (q + q'p) + p'r
= qr' + pr' + p'r
= qr' + p r.
Here we show that a function expressed as a sum of its minterms is equivalent to a function expressed as a product of its maximum terms.
Solution:
Let us assume a function
SOP:
Let us show that a function expressed as a sum of its minterms is equivalent to a function expressed as a product of its maximum terms.
Solution:
POS:
Lets we see an example to express the function in Canonical SOP form.
Solution:
Sum of Product (SOP)
Y = m7 + m6 + m5 + m3 + m5 + m1
Y = Σ(1, 4, 5, 6, 7)
Here we see an another example to express the function in Canonical POS form.
Solution:
Product of Sum (POS)
Y = M2 M3 M0 = M0 M2 M3
Y = π(0, 2, 3)
Here we shall minimize and implement the following multiple output functions in SOP form.
f1 = Σm (0,2,6,10,11,12,13) + d (3,4,5,14,15)
f2 = Σm (1,2,6,7,8,13,14,15) + d (3,5,12)
Solution:
The K maps are filled ones and don't cares using the expression.
After reduction we find that CD occurs both in f1 , and f2. So it can be shared.
Fig: Logic Circuit Implementation
Here we will discuss about converting the given expression in canonical SOP form Y = AC + AB + BC.
Solution:
Y = AC + AB + BC
=AC (B + B’) + AB (C + C’) + (A + A') BC
=ABC + ABC' + AB'C + AB'C' + ABC + ABC' + ABC
=ABC + ABC' +AB'C + AB'C' [A + A =1]