Boolean Algebra. ENEE 6442 > A Boolean algebra =A set of operators (e.g. the binary operators: +,, INV) =A set of axioms or postulates Boolean Algebra

Boolean AlgebraBoolean Algebra

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A A Boolean algebraBoolean algebra A set of operators (e.g. the binary operators: +, A set of operators (e.g. the binary operators: +, •, •,

INV)INV) A set of axioms or postulatesA set of axioms or postulates

Boolean AlgebraBoolean Algebra

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CommutativeCommutative x+y = y+xx+y = y+xxx•y=y•x•y=y•x

DistributiveDistributive x+(y•z)=(x+y)•(x+z)x+(y•z)=(x+y)•(x+z)x•(y+z)=x•y+x•zx•(y+z)=x•y+x•z

IdentitiesIdentities x+0=xx+0=xx•1=xx•1=x

ComplementComplement x+x’=1x+x’=1 x•x’=0x•x’=0

ClosureClosure x+yx+y x•y x•y

AssociativeAssociative (x+y)+z=x+(y+z) (x+y)+z=x+(y+z) (x•y)•z=x•(y•z)(x•y)•z=x•(y•z)

PostulatesPostulates

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Complement of a variable is unique.Complement of a variable is unique. (x’)’ = x (x’)’ = x -- involution -- involution x+x = xx+x = x --idempotent --idempotent x+1=1x+1=1 x+xx+x•y=x•y=x -- absorption -- absorption (x+y)’=x’(x+y)’=x’•y’•y’ -- -- DeMorgan’s LawDeMorgan’s Law xy+x’z+yz = xy + x’zxy+x’z+yz = xy + x’z

-- consensus -- consensus

Complement of a variable is unique.Complement of a variable is unique. (x’)’ = x (x’)’ = x -- --

involutioninvolution x+x = xx+x = x xx•x=x•x=x -- --

idempotentidempotent x+1=1x+1=1 xx•0=0•0=0 x+xx+x•y=x•y=x x•(x+y)=xx•(x+y)=x -- --

absorptionabsorption (x+y)’=x’(x+y)’=x’•y’•y’ (x•y)’=x’+y’(x•y)’=x’+y’ -- -- DeMorgan’s LawDeMorgan’s Law xy+x’z+yz = xy + x’zxy+x’z+yz = xy + x’z

(x+y) (x+y) •(x’+z) •(y+z) = (x+y) •(x’+z)•(x’+z) •(y+z) = (x+y) •(x’+z) -- consensus -- consensus

Duality:Duality: + + • and 0 • and 0 1 1

Properties of Boolean AlgebraProperties of Boolean Algebra

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Proof of ConsensusProof of Consensus

a <= b essentially implies that if a =1 then b =1 and if a = 0 then b a <= b essentially implies that if a =1 then b =1 and if a = 0 then b could be anythingcould be anything

Theorem: In Consensus xy + x’z + yz = xy + x’z (How?)Theorem: In Consensus xy + x’z + yz = xy + x’z (How?)We Prove that xy + x’z + yz >= xy + x’z and xy + x’z + yz <= xy + x’z We Prove that xy + x’z + yz >= xy + x’z and xy + x’z + yz <= xy + x’z

(This would imply equality and prove the theorem)(This would imply equality and prove the theorem)

Proof: First: xy + x’z + yz >= xy + x’z. Let xy + x’z = a and yz = b;Proof: First: xy + x’z + yz >= xy + x’z. Let xy + x’z = a and yz = b;So we want to prove that a + b >= a which is true by definition of the So we want to prove that a + b >= a which is true by definition of the inequalityinequality

Second: xy + x’z + yz <= xy + x’z. This inequality could be split into xy + Second: xy + x’z + yz <= xy + x’z. This inequality could be split into xy + x’z <= xy + x’z and yz <= xy + x’zx’z <= xy + x’z and yz <= xy + x’z

All we need to do is to prove yz <= xy + x’zAll we need to do is to prove yz <= xy + x’z

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Proof of ConsensusProof of Consensus

yz <= xy + x’zyz <= xy + x’z

We know that a <= b iff ab’ = 0 (How?)We know that a <= b iff ab’ = 0 (How?)

So if the above inequality is true yz(xy + x’z)’ = 0 So if the above inequality is true yz(xy + x’z)’ = 0 must be true. Which is always true.must be true. Which is always true.

Hence Proved.Hence Proved.

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A A Boolean functionBoolean function is a mapping is a mapping f(x): Bf(x): Bnn B B.. Constant function: Constant function: f(xf(x11,…,x,…,xnn) = b.) = b.

Projection (to the Projection (to the ii-th axis):-th axis): f(xf(x11,…,x,…,xnn) = x) = xii..

A Boolean function is A Boolean function is completecomplete if if f(x)f(x) is defined is defined for all for all x xBBnn. Otherwise the point . Otherwise the point xx that that f(x)f(x) is not is not defined is called a defined is called a don’t caredon’t care condition. condition.

Operations on Boolean functions:Operations on Boolean functions: Sum:Sum: ((f+g)(x) = f(x) + g(x)f+g)(x) = f(x) + g(x) Product:Product: (f(f••g)(x) = f(x)g)(x) = f(x)••g(x)g(x) Complement:Complement: (f’)(x) = (f(x))’(f’)(x) = (f(x))’

Boolean FunctionsBoolean Functions

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Algebraic expressionsAlgebraic expressions f(x,y,z) = xy+zf(x,y,z) = xy+z

Tabular formsTabular forms Venn diagramsVenn diagrams Cubical Cubical

representationsrepresentations Binary decision Binary decision

diagrams (BDD)diagrams (BDD)

xx yy zz ff

00 00 00 00

00 00 11 11

00 11 00 00

00 11 11 11

11 00 00 00

11 00 11 11

11 11 00 11

11 11 11 11

x

zy

x

z

y

Representations of Boolean FunctionsRepresentations of Boolean Functions



Tabular formsTabular forms


Tabular formsTabular forms Venn diagramsVenn diagrams



representationsrepresentations

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representationsrepresentations Binary decision Binary decision

diagrams (BDD)diagrams (BDD)

xx yy zz ff

00 00 00 00

00 00 11 11

00 11 00 00

00 11 11 11

11 00 00 00

11 00 11 11

11 11 00 11

11 11 11 11

x

zy

x

z

y

Representations of Boolean FunctionsRepresentations of Boolean Functions

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The The cofactorcofactor of of f(xf(x11,…,x,…,xnn)) w.r.t. w.r.t. xxii (or (or x’x’ii) is a ) is a

Boolean functionBoolean function , s.t. , s.t.

Boole’s Expansion Theorem:Boole’s Expansion Theorem:

Boole’s Expansion TheoremBoole’s Expansion Theorem

BBforf n

xxii

1:)( '

),,,0/1,,,(

),,,,,(

111

111/ '

nii

niixx

xxxxf

xxxxfii

)()(

),,,,,,(,,

111

,,iiii xixixixi

niii

fxfxfxfx

xxxxxf

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Boole’s Expansion: ExamplesBoole’s Expansion: Examples

f(x) = x f(x) = x • f(1) + x’ • f(0) • f(1) + x’ • f(0) for single-variable for single-variable xx

= (x + f(0)) • (x’+f(1)) = (x + f(0)) • (x’+f(1)) (from duality)(from duality) f(x+y) • f(x’+y) = f(1) • f(y)f(x+y) • f(x’+y) = f(1) • f(y)

Define Define g(x,y) = f(x+y)f(x’+y)g(x,y) = f(x+y)f(x’+y)

g(x,y) g(x,y) = xg(1,y) + x’g(0,y) = xg(1,y) + x’g(0,y) = xf(1)f(y) + x’f(y)f(1) = xf(1)f(y) + x’f(y)f(1) = f(1)f(y)= f(1)f(y)

What if expand w.r.t variable y?What if expand w.r.t variable y?

f(x+y) • f(x’+y) f(x+y) • f(x’+y) =yf(1)f(1) + y’f(x)f(x’) =yf(1)f(1) + y’f(x)f(x’) = yf(1)+y’f(1)f(0) = f(1)(yf(1) + y’f(0)) = yf(1)+y’f(1)f(0) = f(1)(yf(1) + y’f(0))

=f(1)f(y)=f(1)f(y)

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Boole’s Expansion: ExamplesBoole’s Expansion: Examples

ff ff

11 yy

ff ff

yyyyxx x’x’Hardware? Hardware? Delay?Delay?

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nn

nn

nnnn

xxxf

xxxf

xxxfxxxf

11

'1

'1

''1

'111

)1,1,,1(

)1,0,,0(

)0,0,,0(),,,(

discriminants

minterms

Complete ExpansionComplete Expansion

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A form is called A form is called canonicalcanonical if the representation of if the representation of the function in that form is unique.the function in that form is unique.

Minterm Canonical FormMinterm Canonical Form AND-OR circuitsAND-OR circuits

Pro and Cons of Canonical FormsPro and Cons of Canonical Forms Unique up to permutationUnique up to permutation InefficientInefficient

Canonical FormsCanonical Forms

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SOP (SOP (Disjunctive Normal FormDisjunctive Normal Form)) A disjunction of product termsA disjunction of product terms A product term (1 is also considered a product term)A product term (1 is also considered a product term) 00

The Primary Objective During Logic Minimization The Primary Objective During Logic Minimization is to Remove the Redundancy in the is to Remove the Redundancy in the Representation. Representation.

Normal (Standard) FormsNormal (Standard) Forms

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An An implicantimplicant of a function is a product term that is of a function is a product term that is included in the function.included in the function.

An implicant is An implicant is primeprime if it cannot be included in if it cannot be included in any other implicants.any other implicants.

A prime implicant is A prime implicant is essentialessential if it is the only one if it is the only one that includes a minterm.that includes a minterm.

Example: Example: f(x,y,z) = xy’ + yzf(x,y,z) = xy’ + yz xy(not I),xyz(I, not PI), xy(not I),xyz(I, not PI), xz(PI,not EPI), yz(EPI)xz(PI,not EPI), yz(EPI)

ImplicantsImplicants

x

z

y

xy’z’

xy’z

xyzx’yz

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Specification for IncompletenessSpecification for Incompleteness

A Boolean function is A Boolean function is incompleteincomplete if if f(x)f(x) is NOT is NOT defined for some defined for some x xBBnn. Such point . Such point xx is called a is called a don’t caredon’t care condition. condition.

Tabular representationTabular representation {1-set, 0-set, don’t care set}{1-set, 0-set, don’t care set}

1-set = {xy}1-set = {xy} 0-set = {x’y’}0-set = {x’y’} Don’t care set = {x’y, xy’}Don’t care set = {x’y, xy’}

xx yy ff

00 00 00

00 11 --

11 00 --

11 11 11

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Don’t Care ConditionsDon’t Care Conditions

Satisfiability don’t caresSatisfiability don’t cares of a subcircuit consist of of a subcircuit consist of all input patterns that will never occur.all input patterns that will never occur.

Observability don’t caresObservability don’t cares of a subcircuit are the of a subcircuit are the input patterns that represent situations when an input patterns that represent situations when an output is not observed.output is not observed.

Example:Example:

x

y

{xy,x’y}{xy,x’y} {xy’,x’y’}{xy’,x’y’}

y=0y=0

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Boolean Algebra. ENEE 6442 > A Boolean algebra =A set of operators (e.g. the binary operators: +,, INV) =A set of axioms or postulates Boolean Algebra