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LOGIC MINIMIZATION – I

Akash Kumar

EMBEDDED HARDWARE SYSTEMS DESIGN

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Outline

¨ Logic Synthesis¨ Derivation of logic expressions¨ Minimization using algebraic transformations¨ Minimization using Karnaugh-maps

¤ Upto 5 variables

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Logic synthesis

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Logic Synthesis

¨ It is a process by which an abstract form of desired circuit behavior (typically RTL) is turned into a design implementation in terms of logic gates given a standard-cell library and certain design constraints.

¨ Standard Library or Technology Library¤ It can have basic logic gates like and, or etc.

¤ It can have macro cells like adders, mux, flip flops, etc¤ For FPGAs, it has LUTs, logic elements, etc.

¨ Design constraints can be timing, area, power, and testability

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architecture MLU_DATAFLOW of MLU is

signal A1:STD_LOGIC;signal B1:STD_LOGIC;signal Y1:STD_LOGIC;signal MUX_0, MUX_1, MUX_2, MUX_3: STD_LOGIC;

beginA1<=A when (NEG_A='0') else not A;B1<=B when (NEG_B='0') else not B;Y<=Y1 when (NEG_Y='0') else not Y1;

MUX_0<=A1 and B1;MUX_1<=A1 or B1;MUX_2<=A1 xor B1;MUX_3<=A1 xnor B1;

with (L1 & L0) selectY1<=MUX_0 when "00",

MUX_1 when "01",MUX_2 when "10",MUX_3 when others;

end MLU_DATAFLOW;

HDL descriptionCircuit netlist

Logic Synthesis5

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Steps in Logic Synthesis

1. Parsing and translating the input HDL, and generate internal data structures, e.g. Karnaugh map, prime implicants.

2. Logic minimization: tries to find a minimum cover for the function, i.e. the smallest number of the largest possible circles to cover all the ‘1’. (exploiting the don’t cares condition)

3. Logic optimization: uses a series of factoring, substituting, and elimination steps to simplify the equations.

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4. Technology decomposition: builds a generic network from the optimized logic network. The generic technology independent network is usually simple NAND/INV gates.

5. Finally, technology (logic) mapping: implements the technology independent network by matching pieces of the network with the logic cells that are available in a technology-dependent cell library. While performing technology mapping, the algorithm attempts to minimize area, while meeting other user constraints.

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(b) Minimal-cost realization

FB A

F

(a) Canonical sum-of-products A B

Logic Minimization and Optimization7

BABABAF ... ++=

BAF +=

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Logic optimization

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Logic Optimization

¨ Broadly divided into various categories based on:¨ Circuit representation

¤ Two-level logic optimization¤ Multi-level logic optimization

¨ Circuit characteristics¤ Combinational logic optimization

n Outputs dependent only on current input¤ Sequential logic optimization

n Outputs dependent on both past and presentinputs

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173

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Two-level Optimization

¨ Uses Minimum sum of products (MSOP) or Minimum product of sums (MPOS) expressions

¨ Refers to flattened view of the circuit.¨ Most applicable to PLA (programmable logic array)

implementation – uses MSOP expression.¨ In general, the more optimized the design is, the smaller

is the required area for the design.¨ Forms the basis of most synthesis programs.

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P1

f2

x1 x2 x3

OR plane

Programmable

AND plane

connections

P2

P3

Gate-level Diagram of a PLA11

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Two-level vs Multi-level

¨ If we have two functions F1 and F2

¤ F1 = AB + AC + AD

¤ F2 = A’B + A’C + A’E

¨ The default two-level representation takes 6 AND and 4 OR operations.

¨ A functionally equivalent representation can be

¤ P = B + C¤ F1 = AP + AD¤ F2 = A’P + A’E

¨ This requires only 4 AND and 3 OR operations¤ However, this takes three levels!

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Derivation of logic expressions

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Deriving Logic Expressions From Truth Tables

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¨ What is the Boolean expression for Z?

Light must be ON when both switches A and B are OFF, or when both of them are ON.

Logic FunctionSW.A

SW.BZ (light)

Truth Table:

A B Z

0 0 1

0 1 0

1 0 0

1 1 1

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Deriving Logic Expressions From Truth Tables

¨ Two equivalent logic expressions can be derived from Truth Tables:

1. Sum-of-Products (SOP) expressions:¤ Several AND terms OR’d together, e.g.

2. Product-of-Sum (POS) expressions:¤ Several OR terms AND’d together, e.g.

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ABCCBACBA ++

))(( CBACBA ++++

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Rules for Deriving SOP Expressions

1. Find each row in TT for which output is 1 (rows 1 & 4)

2. For those rows write an AND term of all input variables. For variables with value 0, apply complements: ( and ). These are called minterms (mintermscontain all input variables exactly once).

3. OR together all minterms found in 2): Such an expression is called a Canonical SOP

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ABBAZ +=

BA AB

A B Z

0 0 1

0 1 0

1 0 0

1 1 1

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Rules for Deriving POS Expressions

1. Find each row in TT for which output is 0 (rows 2 & 3)

2. For those rows write an OR factor of all input variables. For variables with value 1, apply complements: & These are called maxterms (maxterms contain all input variables).

3. AND together all maxterms found in 2): Such an expression is called a Canonical POS.

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)( BA+ )( BA+

))(( BABAZ ++=

A B Z

0 0 1

0 1 0

1 0 0

1 1 1

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CSOP and CPOS

¨ Canonical SOP:¨ Canonical POS:¨ Since they represent the same truth table, they should be

identical

¨ CPOS and CSOP expressions for the same TT are logically equivalent. Both represent the same information.

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))(( BABAZ ++=

Verify that ))(( BABAABBAZ ++º+=

ABBAZ +=

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Simplifying Logic Equations

¨ Simplifying logic expressions can lead to using smaller number of gates (parts) to implement the logic expression

¨ Can be done by hand using¤ Boolean Identities (algebraic)

¤ Karnaugh Maps (graphical)

¨ A minimum SOP (MSOP) expression is one that has no more AND terms or variables than any other equivalent SOP expression. (Note: There may be several MSOPs of an expression)

¨ Substitute OR factors for AND terms in above to apply to MPOS

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Minimization using algebraic transformation

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1

23456789101112

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Boolean Identities

¨ Useful for simplifying logic equations.

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Duals

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Boolean Identities

Identities Property

1-5 Single variable, foundations of Boolean manipulation

6 Commutative

7 Associative

8 Distributive

9 De Morgan’s

10 Combining

11 Absorption

13 Consensus

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Boolean Identities

¨ The right side is the dual of the left side

1. Duals formed by replacing

2. The dual of any true statement (axiom or theorem) in Boolean algebra is also a true statement.

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AND OROR AND0 11 0

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Using Boolean Identities – 1

¨ Find an MSOP for

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)( WXYZYWXF +++=

YWX

ZYZWX

WXZYZYWX

+=

+++=

+++=

)1()1(

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Using Boolean Identities – 2

¨ Find an MSOP for

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)!(tricky!

)(

][)(

][))((

)(

VYZXWVY

ZXWVY

BABAAXWZXWVY

XWXWXWZXWVY

ZXWZXWVY

YZXVVWYZXYWVF

+=

+=

+=++=

=+++=

++=

++=

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Simplifying Logic Equations

¨ Remarks:

¨ For complicated expressions, the use of Boolean identities for simplification may not be obvious:¤ Where to begin?

¤ How to proceed?

¤ Do we have a minimum expression?

¨ Better suited for simple expressions

¨ Karnaugh Maps provide a systematic graphical way to minimize expressions – easier to visualize and work with

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Karnaugh maps

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Karnaugh Maps

¨ K-Maps are a convenient way to simplify Boolean Expressions

¨ They can be used for up to 4 or 5 variables¤ For >5 variables, need to use a computer

¨ They are a visual representation of a truth table¨ Expression are most commonly expressed in sum of

products (SOP) form¨ SOP was formed using minterms

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Minterms: Recall

¨ Minterms contain each input variable exactly once¨ A function with n variables has 2n minterms – exactly equal

to the number of rows in truth table¨ A three-variable function, such as f(x, y, z), has 23 = 8

minterms (Note: )

¨ Each minterm is true for exactly one combination of inputs

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x’y’z’ x’y’z x’yz’ x’yzxy’z’ xy’z xyz’ xyz

XX ='

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Minterms: Recall30

¨ All 8 minterms for a 3 variable function

x y z MintermsShortnotation

0 0 0 x’y’z’ m(0)0 0 1 x’y’z m(1)0 1 0 x’yz’ m(2)0 1 1 x’yz m(3)1 0 0 xy’z’ m(4)1 0 1 xy’z m(5)1 1 0 xyz’ m(6)1 1 1 xyz m(7)

x’yz = 1 when x=0, y=1 and z=1

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Karnaugh Map31

¨ A two-variable function has four possible minterms. We can represent these mintermsin a Karnaugh map.

¨ Now we can easily see which minterms contain common literals.¤ Minterms on the left and right sides contain y’ and y respectively.

¤ Minterms in the top and bottom rows contain x’ and x respectively.

x y minterm0 0 x’y’0 1 x’y1 0 xy’1 1 xy

Y

0 10 x’y’ x’y

X1 xy’ xy

Y

0 10 x’y’ x’y

X1 xy’ xy

Y’ YX’ x’y’ x’yX xy’ xy

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Karnaugh Map Simplifications – 1

¨ Imagine a two-variable sum of minterms: x’y’ + x’y¨ Both of these minterms appear in the top row of a

Karnaugh map, which means that they both contain the literal x’.

¨ What happens if you simplify this expression using Boolean algebra?

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x’y’ + x’y= x’(y’ + y) [ Distributive ]= x’ • 1 [ y + y’ = 1 ]= x’ [ x • 1 = x ]

Yx’y’ x’y

X xy’ xy

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Karnaugh Map Simplifications – 2

¨ Another example expression is x’y + xy.¤ Both minterms appear in the right side, where y is

uncomplemented.¤ Thus, we can reduce x’y + xy to just y.

¨ How about x’y’ + x’y + xy?¤ We have x’y’ + x’y in the top row, corresponding to x’.¤ There’s also x’y + xy in the right side, corresponding to y.¤ This whole expression can be reduced to x’ + y.

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Yx’y’ x’y

X xy’ xy

Yx’y’ x’y

X xy’ xy

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Truth Table to Karnaugh Map34

A B P0 0 10 1 11 0 01 1 1

BA 0 1

0 1 1

1 0 1

The expression is:

A.B + A.B + A.B

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2-Variable Groupings

BA 0 1

0 1 11 1

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¨ Adjacent 1’s can be “paired off” ¨ Pairs may be adjacent horizontally or vertically¨ Any variable which is both a 1 and a zero in this pairing

can be eliminated

B is eliminated, leaving A as the term A is eliminated,

leaving B as the term

The expression becomes A + B

a pairanother pair

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2-Variable Groupings36

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3-variable Karnaughmap

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3-Variable Karnaugh Map

¨ Note the ordering for BC – grey coding¤ Adjacent cells differ in exactly one bit

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A B C Z

0 0 0 00 0 1 00 1 0 10 1 1 01 0 0 11 0 1 01 1 0 11 1 1 0

BCA 00 01 11 10

0 0 0 0 1

1 1 0 0 1

A.B.C + A.B.C + A.B.C

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¨ Shorthand notation only use minterms¤ m(i) indicates ith minterm (see slide 21)

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A B C Z

0 0 0 00 0 1 00 1 0 10 1 1 01 0 0 11 0 1 01 1 0 11 1 1 0

A.B.C + A.B.C + A.B.C

å=++=

)6,4,2()6()4()2(

mmmmZ

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BCA 00 01 11 10

0 1

1 1 1

equates to B.C as A is eliminated.

Here, we can “wrap around” and this pair equates to A.C as B is eliminated.

Our truth table simplifies

to A.C + B.C

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n Three Variable K-Map

n Extreme ends of same row considered adjacent

A BC 00 01 11 10

0

1

A.B.C A.B.C A.B.C A.B.C

A.B.C A.B.C A.B.C A.B.C

0010A.B.C

A.B.C

A.B.CA.B.C

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Grouping Rules

¨ Only group adjacent cells¤ Two cells are adjacent if exactly one variable is different

¨ Group-size is always a power of 2 – 2, 4, 8 or 16¨ Grouped squares must be rectangular¨ Simplified expression results by retaining inputs that

don’t change value¤ Drop inputs whose values change

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3-Variable Groupings (4 cells)

¨ Note the gray coding on BC

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CBCCB

ABCBCACBACBAZ

+

+++=

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More 3-Variable Groupings44

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More 3-Variable Groupings

¨ Grouping 8 cells eliminates 3 variables

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Invalid Groupings

¨ The following groupings are invalid

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ABCD

00 01 11 10

00 0 4 12 8

01 1 5 13 9

11 3 7 15 11

10 2 6 14 10

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n Note the gray codes on both axesn The location of minterms are shown in the above tablen All adjacent minterms differ exactly in a power of 2 (WHY?)

Minterms formed using the order ABCD

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ABCD

00 01 11 10

00 0 0 0 0

01 0 1 0 0

11 0 1 0 0

10 0 0 0 0

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n Note the gray codes on both axesn C changes value – therefore it is eliminated

å=+= )7,5(mDCBABCDAZ

BDAZ =

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¨ Input variables arranged so that for any 2 physically adjacent squares, only one of the input variables has a different value.

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Note: If only one variable has a different value, then the 2 squares are adjacent (even if they are not physically adjacent).Figure: Adjacent squares

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¨ Four Variable K-Map

¤ Four corners adjacent

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AB CD 00 01 11 10

00

01

11

10

A.B.C.D A.B.C.D A.B.C.D A.B.C.D

A.B.C.D A.B.C.D A.B.C.D A.B.C.D

A.B.C.D A.B.C.D A.B.C.D A.B.C.D A.B.C.D A.B.C.D A.B.C.D A.B.C.D

A.B.C.D

A.B.C.D

A.B.C.D

A.B.C.D

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Obtaining MSOPs from K-Maps

¨ Circle every 1 at least once¨ Circle a 1 more than once if it helps to make larger

groupings. But do not circle any more times than necessary to circle all 1’s

¨ Make groups as large as possible¨ Use no more groups than necessary

¨ Start with 1’s that can be circled only once. They must occur in MSOP. In general, start with 1’s that are most difficult.

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Obtaining MSOPs from K-Maps54

A B C Z

0 0 0 10 0 1 10 1 0 00 1 1 11 0 0 11 0 1 01 1 0 11 1 1 0

ABC

0 1

00 1 1

01 1

11 1

10 1

CACA ,Essential prime implicants:

BACACA

CBCACAZ

++=

++=

Two groupings are possible for CBA

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Obtaining MSOP Exercises

ABCD

00 01 11 10

00 1 0 0 1

01 1 0 1 1

11 1 1 1 1

10 1 1 1 1

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ABCD

00 01 11 10

00 1 1 0 1

01 1 1 0 0

11 0 0 1 0

10 1 1 0 1

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Obtaining MPOSs from K-Maps

¨ Circle 0’s instead of 1’s¨ Form OR factors instead of AND terms¨ Complement variables that have 1 values, and don’t

complement variables with 0 values¨ Guidelines for forming MPOS’s from K-maps similar to

MSOP’s¨ Which to use, MSOP or MPOS?

¤ solve for both, and use the one that is easily implemented with available gates

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Obtaining MPOSs from K-Maps57

A B C Z

0 0 0 10 0 1 10 1 0 00 1 1 11 0 0 11 0 1 01 1 0 11 1 1 0

ABC

0 1

00

01 0

11 0

10 0

)).(( CBACAZ +++=

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Obtaining MPOS Exercises

ABCD 00 01 11 10

00 1 0 0 1

01 1 0 1 1

11 1 1 1 1

10 1 1 1 1

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ABCD

00 01 11 10

00 1 1 0 1

01 1 1 0 0

11 0 0 1 0

10 1 1 0 1

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Don’t Care Conditions

¨ Sometimes, input variables cannot take values corresponding to all combinations in a truth table¤ For these combinations we don’t care what the output value is¤ For example, input > 9 in a 7-segment display

¨ In a K-map, the don’t care values are denoted by X’s¨ For minimization, we can choose either X = 1 or X = 0,

the value that is most advantageous for minimization

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Don’t Care Conditions60

ABC

0 1

00

01 1 X

11 1

10 X

ABC

0 1

00

01 1 1

11 1

10 0

å å+=

+++=

)5,2()7,1()2()5()7()1(

dmddmmZ

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¨ 5-variable K-map constructed in 3-dimension by placing one 4-variable K-map on top of another

¨ Imagine two different planes¤ The top plane indicates values when A=1¤ The bottom plane corresponds to A=0

¨ Every minterm on the top plane differs by 16 from the corresponding minterm on the bottom plane

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BCDE 00 01 11 10

0016 20 28 24

0 4 12 8

0117 21 29 25

1 5 13 9

1119 23 31 27

3 7 15 11

1018 22 30 26

2 6 14 10

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A

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¨ 2-D representation: divide each square in a 4-variable K-map with a diagonal line

¨ Place terms in the top layer above the line and the terms in the bottom layer below the line

¨ Terms in the bottom or top layer combine just like in a 4-variable K-map

¨ In addition, terms in the same square separated by the diagonal line differ in exactly one variable¤ Therefore, they can be combined

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¨ Some physically adjacent cells may not be grouped, e.g. 0 and 20. ¤ They differ in two variables¤ They appear in a different column and different layer

¨ 5-variables = 5-bits; exactly 1 bit different implies adjacency

¨ Each term can be adjacent to 5 different terms¤ Four in the same layer and one in different layer

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Summary

¨ Derivation of logic expressions¨ Minimization using algebraic transformations¨ Minimization using Karnaugh-maps¨ 5 variables is the maximum that you can easily see using

K-maps¤ For larger variables, a systematic approach needed that is

suitable for computer programs

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© Akash Kumar

Science Pillars of Fame!

¨ Augustus de Morgan¨ Born in India, 1806¨ B.A, from Cambridge in 1828¨ Appointed Mathematics Professor at 22 years of age in

London University¨ Made ample contributions in the field of algebra and

trigonometry

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