boolean algibra

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Fall 2002 EECS150 Lec4-bool1 Page 1, 9/5 9am

EECS150 - Digital Design

Lecture 4 - Boolean Algebra I

(Representations of Combinational

Logic Circuits)September 5, 2002

John Wawrzynek


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Outline

Review of three representations for combinational logic: truth tables,

graphical (logic gates), and

algebraic equations Relationship among the three

Adder example

Formal description of Boolean algebra Laws of Boolean algebra


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Combinational Logic (CL) Defined

yi = fi(x0 , . . . . , xn-1), where x, y are {0,1}.

Y is a function of only X.

If we change X, Y will change immediately (well almost!).

There is an implementation dependent delay from X to Y.


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CL Block Example #1

Truth Table Description:

Boolean Equation:y0 = (x0 AND not(x1))

OR (not(x0) AND x1)

y0 = x0x1' + x0'x1

Gate Representation:

x0 x1

y0 0 00

00

1 111

1 1

How would weprove that all three representations are equivalent?


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Boolean Algebra/Logic Circuits

Why are they called logic circuits? Logic: The study of the principles of reasoning.

The 19th Century Mathematician, George Boole, developed a

math. system (algebra) involving logic, Boolean Algebra.

His variables took on TRUE, FALSE

Later Claude Shannon (father of information theory) showed

(in his Masters thesis!) how to map Boolean Algebra to digital

circuits: Primitive functions of Boolean Algebra:


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Relationship Among Representations

Truth Table

BooleanExpression

gate

representation(schematic)

? ?

unique

notunique

notunique

[convenient for

manipulation]

[close to

implementaton]

* Theorem: Any Boolean function that can be expressed as a truth table canbe written as an expression in Boolean Algebra using AND, OR, NOT.

How do we convert from one to the other?


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Notes on Example #1

The example is the standardfunction called exclusive-or

(XOR,EXOR)

Has a standard algebraic

symbol:

And a standard gate symbol:

x0 x1 y

0 0 00

00

1 1

111 1


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CL Block Example #2

4-bit adder:

R = A + B,

c is carry out

Truth Table Representation:

In general: 2n rows for n inputs.

Is there a more efficient (compact) way to specify this function?

256 rows!


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4-bit Adder Example

Motivate the adder circuit designby hand addition:

Add a0 and b0 as follows:

Add a1 and b1 as follows:

carry to

next stage

r = a XOR b

c = a AND b = ab r = a XOR b XOR ci

co = ab + aci + bci


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4-bit Adder Example

In general:ri = ai XOR bi XOR cin

cout = aicin + aibi + bicin = cin(ai + bi) + aibi

Now, the 4-bit adder: Full adder cell

ripple adder


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4-bit Adder Example

Graphical Representation of FA-cellri = ai XOR bi XOR cin

cout = aicin + aibi + bicin

Alternative Implementation(with 2-input gates):

ri = (ai XOR bi) XOR cin

cout = cin(ai + bi) + aibi


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Boolean Algebra

Defined as: { }{ }

.0,1

:Complement.6

.)(),()()(

:lawsveDistributi5.

.1,0

in1,0:Identities.4

.,

:lawseCommunitiv3.

.in,in,in

,in:Closure2.

.such thatelementsleast twoatcontains1.

:holdaxiomsfollowingthesuch that,operationunary,,operatorsbinary,elementsofSet

==+

+=+++=+

==+

=+=+

+

+

aaaa

cabacbacabacba

aaaa

B

abbaabba

BaBbaBba

Ba,b

baa,bB

B


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Logic Functions

Do the axioms hold?

Ex: communitive law: 0+1 = 1+0?

Algebra.Booleanvalidais

NOTAND,OR,},1,0{ ===+=B

00 001 110 111 1

00 001 010 011 1

0 11 0


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Theorem: Any Boolean function that can be expressed as a truth

table can be expressed using NAND and NOR.

Proof sketch:

How would you show that either NAND or NOR is sufficient?

Other logic functions of 2 variables (x,y)x y f0 f1

00 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 101 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 110 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 111 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

0 AND X Y + OR NOR XNOR NAND 1

Look at NOR and NAND:

= NOT = AND

= OR


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Laws of Boolean AlgebraDuality: A dual of a Boolean expression is derived by interchanging OR

and AND operations, and 0s and 1s (literals are left unchanged).

)},,0,1,,...,,({)},,1,0,,...,,({ 2121 +=+ nD

nxxxFxxxF

Any law that is true for an expression is also true for its dual.

Operations with 0 and 1:

1. x + 0 = x x * 1 = x

2. x + 1 = 1 x * 0 = 0

Idempotent Law:3. x + x = x x x = x

Involution Law:

4. (x) = x

Laws of Complementarity:5. x + x = 1 x x = 0

Commutative Law:

6. x + y = y + x x y = y x


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Laws of Boolean Algebra (cont.)

Associative Laws:

(x + y) + z = x + (y + z) x y z = x (y z)

Distributive Laws:

x (y + z) = (x y) + (x z) x +(y z) = (x + y)(x + z)

Simplification Theorems:x y + x y = x (x + y) (x + y) = xx + x y = x x (x + y) = x

DeMorgans Law:

(x + y + z + ) = xyz (x y z ) = x + y +z

Theorem for Multiplying and Factoring:(x + y) (x + z) = x z + x y

Consensus Theorem:x y + y z + x z = (x + y) (y + z) (x + z)

x y + x z = (x + y) (x + z)


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Proving Theorems via axioms of Boolean

Algebra

Ex: prove the theorem: x y + x y = x

x y + x y = x (y + y) distributive law

x (y + y) = x (1) complementary law

x (1) = x identity

Ex: prove the theorem: x + x y = x

x + x y = x 1 + x y identity

x 1 + x y = x (1 + y) distributive law

x (1 + y) = x (1) identity

x (1) = x identity


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DeMorgans Law

* DeMorgans Law can be used to convert AND/OR expressions to

OR/AND expressions:Example: z = abc + abc + abc + abc

z = abc + abc + abc + abc

x y x' y' (x + y)' x'y'

0 0 1 1 1 10 1 1 0 0 01 0 0 1 0 01 1 0 0 0 0

x y x' y' (x y)' x' + y'

0 0 1 1 1 10 1 1 0 1 1

1 0 0 1 1 11 1 0 0 0 0

(x + y) = x y

=

(x y) = x + y

=

Exhaustive

Proof

Exhaustive

Proof


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Algebraic Simplification

Ex: full adder (FA) carry out function

Cout = abc + abc + abc + abc


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Algebraic Simplification

Ex: full adder (FA) carry out functionCout = abc + abc + abc + abc

= abc + abc + abc + abc + abc

= abc + abc + abc + abc + abc

= (a + a)bc + abc + abc + abc

= (1)bc + abc + abc + abc

= bc + abc + abc + abc + abc

= bc + abc + abc + abc + abc

= bc + a(b +b)c + abc +abc

= bc + a(1)c + abc + abc

= bc + ac + ab(c + c)= bc + ac + ab(1)

= bc + ac + ab

boolean algibra

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