State university of New York at New PaltzElectrical and Computer Engineering Department

Logic Synthesis OptimizationLect10: Two-level Logic Minimization

By Dr. Yaser Khalifa

Electrical and Computer Engineering DepartmentState University of New York at New Paltz

Basic Definitions

• Definition:A Boolean circuit is a directed graph C(G,N) where G

are the gates andN ´G is the set of directed edges (nets) connecting

the gates.Some of the vertices are designated:Inputs: I GOutputs: O G, I O = Each gate g is assigned a Boolean function fg which

computes theoutput of the gate in terms of its inputs.

• The fanin FI(g) of a gate g are all predecessor vertices of g:

• FI(g) = {g’ | (g’,g) N}• The fanout FO(g) of a gate g are all successor vertices

of g:• FO(g) = {g’ | (g,g’) N}• The cone CONE(g) of a gate g is the transitive fanin of

g and g itself.• The support SUPPORT(g) of a gate g are all inputs in

its cone:• SUPPORT(g) = CONE(g) I

FI(6) = {2,4}FO(6) = {7,9}CONE(6) = {1,2,4,6}SUPPORT(6) = {1,2}

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Definitions

• Implicant: single element of the ON-set or any group of elements that can be combined together in a K-map

• Prime implicant: implicant that cannot be combined with another implicant to eleminate a term

• Essential prime implicant: if an element of the ON-set is covered by a single prime implicant, it is an essential prime.

A BC D

1 0

11

0 1

0 0

1 0110 10 0

0

00

0

0

1 0

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1

1 11

1

1

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6 Prime Implicants:

A’D, CD, AC, BC’D’, A’BC’, ABD’

Essential Prime Implicants:

A’D, AC

A BC D

1 0

11

0 1

0 0

1 0110 10 0

0

00

0

0

1 0

1

1

1 11

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A BC D

1 0

11

0 1

0 0

1 0110 10 0

0

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1 0

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1 11

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Minimum Cover = A’D, AC, BC’D’

A BC D

1 0

11

0 1

0 0

1 0110 10 0

0

00

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0 0

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1 11

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1 1

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5 Prime Implicants:

A’BC’, A’CD, AC’D, ABC, BD

A BC D

1 0

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0 1

0 0

1 0110 10 0

0

00

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0

0 0

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1 11

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1 1

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Essential Prime Implicants:

A’BC’, A’CD, AC’D, ABC

Minimum Cover = A’BC’, A’CD, AC’D, ABC

Quine-McCluskey Method

• The QM procedure reduces the minterm expansion of a function to obtain the minimum sum of product.

• The procedure consists of two main steps described in the following slide.

1. Eliminate as many litrals as possible from each term by systematically applying the theorem XY + XY’ = X.

2. Use a prime implicant chart to select a minimum set of prime implicants which when ORed together, are equal to the function being simplified and which contain a minimum number of literals.

f (a, b, c, d) = m(0,1,2,5,6,7,8,9,10,14)Is represented by the following minterms

Group 0 0 00001 0001

Group 1 2 00108 10005 0101

Group 2 6 01109 100110 1010

Group 3 7 011114 1110

Colum n I C olum n II C o lum n III

group 0 (0) 00000,1 0 0 0 _0,2 0 0 _ 00,8 _ 0 0 0

0, 1, 8, 9 _ 0 0 _0, 2, 8, 10 _ 0 _ 00, 8, 1, 9 _ 0 0 _0, 8, 2, 10 _ 0 _ 02, 6, 10, 14 _ _ 1 02, 10, 6, 14 _ _ 1 0

group 1 (1) 0001 (2) 0010 (8) 1000

1,5 0 _ 0 11,9 _ 0 0 12,6 0 _ 1 02,10 _ 0 1 08,9 1 0 0 _8,10 1 0 _ 0

group 2 (5) 0101 (6) 0110 (9) 1001 (10) 1010

5,7 0 1 _ 16,7 0 1 1 _6,14 _ 1 1 010,14 1 _ 1 0group 3 (7) 0111

(14) 1110

A BC D

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What are the Prime Implicants?

BD' AB BC' AD B'CD0100 1 11100 1 1 10101 11101 1 1 11001 10011 11111 1 11011 1 10110 11110 1 1

What are the Prime Implicants?

Answer: BD’, AB, BC’, AD, B’CD

What are the Essential Prime Implicants?

Answer: BD’, BC’, AD, B’CD

Cyclic Implicant Chart

• Cyclic Implicant Chart is a chart which has two or more marks in every row/column .

• Example:f (a, b, c)= m(0, 1, 2, 5, 6, 7)

Simplification of Incompletely Specified Functions

• In the process of finding prime implicants, treat the don’t cares terms as if they were required minterms.

• Do not worry if this results in an extra prime implicant

• When forming prime implicant chart do not include don’t care terms.