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Maurice Karnaugh is an American physicist,Maurice Karnaugh is an American physicist,

famous for the Karnaugh map used infamous for the Karnaugh map used in

Boolean algebraBoolean algebra

Karnaugh !or"ed at Bell #abs $1%&2'66(,Karnaugh !or"ed at Bell #abs $1%&2'66(,

de)eloping the Karnaugh map $1%&*( asde)eloping the Karnaugh map $1%&*( as

!ell as patents for +CM encoding!ell as patents for +CM encoding

Karnaugh MapsKarnaugh Maps

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he Karnaugh map comprises a bo- for e)ery line in the truth tablehe Karnaugh map comprises a bo- for e)ery line in the truth table

he binary )alues abo)e the bo-es are those associated !ith the ahe binary )alues abo)e the bo-es are those associated !ith the a

and b inputsand b inputs

.nli"e a truth table, in !hich the input )alues typically follo! a.nli"e a truth table, in !hich the input )alues typically follo! a

binary se/uence, the Karnaugh maps input )alues must bebinary se/uence, the Karnaugh maps input )alues must be

ordered such that the )alues for adacent columns )ary by onlyordered such that the )alues for adacent columns )ary by only

a single bit for e-ample, 00, 01, 11, and 10a single bit for e-ample, 00, 01, 11, and 10

his ordering is "no!n as a 3ray code, and it is a "ey factor !ithhis ordering is "no!n as a 3ray code, and it is a "ey factor !ith

regard to the !ay in !hich Karnaugh maps !or"regard to the !ay in !hich Karnaugh maps !or"

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M55M57A58 .S53 KA9A.3: MA+SM55M57A58 .S53 KA9A.3: MA+S

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3eneric Karnaugh maps for 4' and *'input functions3eneric Karnaugh maps for 4' and *'input functions

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Karnaugh map minimi;ation of e-ample 4'input functionKarnaugh map minimi;ation of e-ample 4'input function

5n the case of a 4'input Karnaugh map, any t!o hori;ontally or5n the case of a 4'input Karnaugh map, any t!o hori;ontally or)ertically adacent minterms, each composed of three )ariables,)ertically adacent minterms, each composed of three )ariables,

can be combined to form a ne! product term composed of onlycan be combined to form a ne! product term composed of only

t!o )ariablest!o )ariables

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he 1s assigned to the Karnaugh maps bo-es represent thehe 1s assigned to the Karnaugh maps bo-es represent the

same minterms as the 1s in the truth tables output columnsame minterms as the 1s in the truth tables output column

As the input )alues associated !ith each ro! andAs the input )alues associated !ith each ro! andcolumn in the map differ by only one bit, any pair of hori;ontallycolumn in the map differ by only one bit, any pair of hori;ontally

or )ertically adacent bo-es corresponds to minterms that differor )ertically adacent bo-es corresponds to minterms that differ

by only a single )ariableby only a single )ariable

Such pairs of minterms can be grouped together and the )ariableSuch pairs of minterms can be grouped together and the )ariablehat differs can be discarded, lea)ing a much'simplified e/uationhat differs can be discarded, lea)ing a much'simplified e/uation

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he Karnaugh map uses the follo!ing rules for the simplification ofhe Karnaugh map uses the follo!ing rules for the simplification of

e-pressions by grouping together adacent cells containing onese-pressions by grouping together adacent cells containing ones

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SummmarySummmary

o ;eros allo!edo ;eros allo!ed

o diagonalso diagonals 8nly po!er of 2 number of cells in each group8nly po!er of 2 number of cells in each group

3roups should be as large as possible3roups should be as large as possible

>)ery one must be in at least one group>)ery one must be in at least one group

8)erlapping allo!ed8)erlapping allo!ed

?rap around allo!ed?rap around allo!ed @e!est number of groups possible@e!est number of groups possible

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.sing the same minterm to form multiple Karnaugh map groups.sing the same minterm to form multiple Karnaugh map groups

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5n the case of a *'input map, any t!o adacent minterms,5n the case of a *'input map, any t!o adacent minterms,

each composed of four )ariables, can be combined to form a ne!each composed of four )ariables, can be combined to form a ne!

product term composed of only three )ariablesproduct term composed of only three )ariables

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Karnaugh map groupings of four adacent mintermsKarnaugh map groupings of four adacent minterms

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Karnaugh map input )alues are ordered so that the )aluesKarnaugh map input )alues are ordered so that the )alues

associated !ith adacent ro!s and columns differ by only a singleassociated !ith adacent ro!s and columns differ by only a single

bitbit

8ne result of this ordering is that the top and bottom ro!s are also8ne result of this ordering is that the top and bottom ro!s are alsoseparated by only a single bit $it may help to )isuali;e the mapseparated by only a single bit $it may help to )isuali;e the map

rolled into a hori;ontal cylinder such that the top and bottom edgesrolled into a hori;ontal cylinder such that the top and bottom edges

are touching(are touching(

Similarly, the left and right columns are separated by only a singleSimilarly, the left and right columns are separated by only a single

bit $in this case it may help to )isuali;e the map rolled into a )erticalbit $in this case it may help to )isuali;e the map rolled into a )ertical

cylinder such that the left and right edges are touching( his leadscylinder such that the left and right edges are touching( his leads

o some additional groupings, a fe! of !hich are sho!no some additional groupings, a fe! of !hich are sho!n

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5C8M+#>># S+>C5@5> @.C58S5C8M+#>># S+>C5@5> @.C58S

5f, for e-ample, the designer "no!s that5f, for e-ample, the designer "no!s that

certain input combinations !ill ne)er occur,certain input combinations !ill ne)er occur,

then the )alue assigned to the output forthen the )alue assigned to the output for

these combinations is irrele)antthese combinations is irrele)ant

Alternati)ely, for some input combinations,Alternati)ely, for some input combinations,

the designer may simply not care about thethe designer may simply not care about the

)alue on the output)alue on the output

he characters indicate dont care states, !hich can behe characters indicate dont care states, !hich can be

considered to represent either 0 or 1 )alues at the designersconsidered to represent either 0 or 1 )alues at the designers

discretiondiscretion

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+opulating Karnaugh maps !ith 0s )ersus 1s+opulating Karnaugh maps !ith 0s )ersus 1s

Karnaugh maps are most often used to represent 4'input andKarnaugh maps are most often used to represent 4'input and*'input functions 5t is possible to create similar maps for &'input*'input functions 5t is possible to create similar maps for &'input

and 6'input functions, but these maps can /uic"ly becomeand 6'input functions, but these maps can /uic"ly become

un!ieldy and difficult to useun!ieldy and difficult to use

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