KU College of Engineering Elec 204: Digital Systems Design 1 Optimization Algorithm Find all prime implicants. Include all essential prime implicants in

KU College of EngineeringElec 204: Digital Systems Design

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Optimization Algorithm

• Find all prime implicants.

• Include all essential prime implicants in the solution

• Select a minimum cost set of non-essential prime implicants to cover all minterms not yet covered:– Obtaining a good simplified solution

• Minimize the overlap among prime implicants as much as possible.

• In particular, in the final solution, make sure that each prime implicant selected includes at least one minterm not included in any other prime implicant selected.


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Selection Rule Example

• Simplify F(A, B, C, D) given on the K-map.

1

1

1

1 1

1

1

B

D

A

C

1

1

1

1

1

1 1

1

1

B

D

A

C

1

1

Essential

Minterms covered by essential prime implicants

Selected


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Four-Variable Map Simplification

• )8,10,13,152,4,5,6,7, (0, Z)Y,X,F(W, m


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3,14,15

Four-Variable Map Simplification

• )(3,4,5,7,9,1 Z)Y,X,F(W, m


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Prime-number detector (again)


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• When we solved algebraically, we missed one simplification -- the circuit below has three less gate inputs.


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Another example


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Yet another example

• Distinguished 1 cells

• Essential prime implicants


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Product-of-Sums Simplification

• F(A,B,C,D) =

• F’ = CD+AB+BD’

• F = (C’+D’)(A’+B’)(B’+D)

)10,9,8,5,2,1,0(m


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Five Variable or More K-Maps• For five variable problems, we use two adjacent K-maps. It

becomes harder to visualize adjacent minterms for selecting PIs. You can extend the problem to six variables by using four K-Maps.

X

Y

Z

W

V = 0

X

Z

W

V = 1Y


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• Sometimes a function table or map contains entries for which it is known:

– the input values for the minterm will never occur, or

– The output value for the minterm is not used

• In these cases, the output value need not be defined

• Instead, the output value is defined as a “don't care”

• By placing “don't cares” ( an “x” entry) in the function table or map, the cost of the logic circuit may be lowered.

• Example 1: A logic function having the binary codes for the BCD digits as its inputs. Only the codes for 0 through 9 are used. The six codes, 1010 through 1111 never occur, so the output values for these codes are “x” to represent “don’t cares.”

Don't Cares in K-Maps


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Selection Rule Example with Don't Cares

• Simplify F(A, B, C, D) given on the K-map.

Selected

Minterms covered by essential prime implicants

1

1

x

x

x x

x

1

B

D

A

C

1

1 1

1

x

x

x x

x

1

B

D

A

C

1

1

Essential


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Don’t Care Conditions• F(A,B,C,D) =• d(A,B,C,D) =

)15,11,7,3,1(m

)5,2,0(m


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Examples:

• F(A,B,C,D) = m(4,5,8,12,13); d=m(3,7,10,11,14,15);

• F(A,B,C,D) = m(0,2,5,8,10,13); d=m(7,15);


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Multiple-Level Optimization

• Multiple-level circuits - circuits that are not two-level (with or without input and/or output inverters)

• Multiple-level circuits can have reduced gate input cost compared to two-level (SOP and POS) circuits

• Multiple-level optimization is performed by applying transformations to circuits represented by equations while evaluating cost


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Transformations

• Factoring - finding a factored form from SOP or POS expression– Algebraic – Without using of axioms specific to

Boolean algebra such as complements or idempotence

– Boolean - Uses axioms unique to Boolean algebra

• Decomposition - expression of a function as a set of new functions


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Transformations (continued)

• Substitution of G into F - expression function F as a function of G and some or all of its original variables

• Elimination - Inverse of substitution• Extraction - decomposition applied to multiple

functions simultaneously


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Transformation Examples

• Algebraic Factoring F = A’C’D’ + A’BC’ + ABC + ACD’ G = 16

– Factoring: F = A’(C’D’ + BC’) + A (BC + CD’) G = 16

– Factoring again: F = A’C’( B +D’) + AC (B +D’) G = 12

– Factoring again: F = (A’C’+ AC) (B +D’) G = 10


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• Decomposition– The terms B +D’ and A’C’+ AC can be defined as new

functions E and H respectively, decomposing F:

F = E H, E = B +D’ , and H = A’C’+ AC G = 10

• This series of transformations has reduced G from 16 to 10, a substantial savings. The resulting circuit has three levels plus input inverters.


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• Substitution of E into F– Returning to F just before the final factoring step:

F = A’C’( B +D’) + AC (B +D’) G = 12– Defining E = B + D’, and substituting in F:

F = A’C’E + ACE G = 10– This substitution has resulted in the same cost as the

decomposition


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• Elimination– Beginning with a new set of functions:

X = B + C

Y = A + B

Z = A’X + C Y G = 10 – Eliminating X and Y from Z:

Z = A’(B + C) + C (A + B) G = 10– “Flattening” (Converting to SOP expression):

Z = A’B + A’C + AC + BC G = 12– This has increased the cost, but has provided an new SOP

expression for two-level optimization.


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• Two-level Optimization– The result of 2-level optimization is:

Z = A’B + C G = 4

• This example illustrates that:– Optimization can begin with any set of equations, not just with

minterms or a truth table

– Increasing gate input count G temporarily during a series of transformations can result in a final solution with a smaller G


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• Extraction– Beginning with two functions:

E = A’B’D’+ A’BD

H = B’CD’ + BCD G = 16 – Finding a common factor and defining it as a function:

F = B’D’+ BD– We perform extraction by expressing E and H as the three

functions:

F = B’D’+ BD, E = A’F, H = CF G = 10– The reduced cost G results from the sharing of logic between

the two output functions


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Example:

• Multiple Output Minimization:– F(A,B,C) = m(3,6,7) and G(A,B,C) = m(0,1,3)

– Implement F and G using minimum number of gates.

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KU College of Engineering Elec 204: Digital Systems Design 1 Optimization Algorithm Find all prime implicants. Include all essential prime implicants in