Logic Circuits and Computer Architecture

RLAC (2008-09) by Luciano Gualà 1A1 -

Logic Circuits and Computer Architecture

Appendix ADigital Logic Circuits

Part 1: Combinational Circuitsand Minimization


Digital System• Takes a set of discrete information inputs and

discrete internal information (system state) and generates a set of discrete information outputs

System State

DiscreteInformationProcessingSystem

DiscreteInputs Discrete

Outputs


Basic circuits• Combinational

without memory, stateless Output = Function(Input)

• Sequential with memory, state dependent behaviour State = Function (State, Input) Output = Function (State) or Function (State, Input) It can be:

• Synchronous: state updated at discrete times• Asynchronous: State updated at any times


Digital System Example:

Combinational Circuit

A B

A + B Inputs: Representation of A and B

Outputs: Representation of A+ B


Digital System Example:

A Digital Counter (e. g., odometer):

1 30 0 5 6 4Count Up

Reset

Inputs: Count Up, Reset Outputs: Visual Display State: "Value" of stored digits


Information Representation - Signals• Information variables represented by physical

quantities (signals) • For digital systems, the variables take on discrete

values • Two level, or binary values are the most

prevalent values in digital systems • Binary values are represented abstractly by:

digits 0 and 1 words (symbols) False (F) and True (T) words (symbols) Low (L) and High (H) and words On and Off

• Binary values are represented by values or ranges of values of physical quantities


Signal Example – Physical Quantity: Voltage

Threshold Region


Signal Examples Over Time

Analog

Asynchronous

Synchronous

TimeContinuous in value &

time

Discrete in value & continuous

in timeDiscrete in

value & time

Digital


• What are other physical quantities represent 0 and 1? CPU Disk CD Dynamic RAM

Binary Values: Other Physical Quantities

Magnetic Field DirectionSurface Pits/Light

Electrical Charge

voltage


Information processing

• by means of boolean gates• a boolean gate implements simple

boolean operations (see later)• basic gates:

AND, OR, NOT• gates are used to build circuits


Number Systems – Representation Positive radix, positional number systems

• A number with radix r is rapresented by a string of digits: A=an-1an-2…a1a0 .a-1,a-2,…,a-m

0 ai < r “.” is the radix point

• Its (decimal) value is:

11

0 mjj

jn

ii

i ararA


General Decimal BinaryRadix (Base)

r 10 2

Digits 0 => r - 1 0 => 9 0 => 10123

4 5

-1-2-3-4-5

r0

r1

r2

r3

r4

r5

r -1

r -2

r -3

r -4

r -5

110100

100010,000

100,0000.1

0.010.001

0.00010.00001

1248

16320.5

0.250.125

0.06250.03125

Number Systems – Examples

powers of

radix


Case r =2

• Let A=an-1an-2…a1a0 .a-1,a-2,…,a-m be a string of bits (binary digit)• Its (decimal) value is:

11

0

22mj

jj

n

ii

i aaA

Example:1001.1011 = 23 + 20 +2-1 + 2-3 + 2-4 =9.6875

2-1 2-2 2-3 2-4 2-5

0.5 0.250 0.125 0.0625 0.03125


00010203040506070809101112131415

0000000100100011010001010110011110001001101010111100110111101111

00010203040506071011121314151617

0123456789ABCDEF

Decimalr =10

Binaryr =2

Octalr =8

Hexadecimalr =16


Decimal to binary conversion

• Represent (111.6875)10 in pure binary

Idea:• Convert the integer part into • Convert the fractional part into • Result: .


Conversion of integer part• Represent (111)10 in pure binary• Repeat division by 2

111:2= 55 remainder 1 55:2= 27 remainder 1 27:2= 13 remainder 1 13:2= 6 remainder 1 6:2= 3 remainder 0 3:2= 1 remainder 1 1:2= 0 remainder 1

• RESULT: 1101111• N.B.: Result bits are produced in reverse order

Least Significant Bit(LSB)

Most Significant Bit(MSB)


• Iterate multiplication taking the integer parts• Binary representation for 0.6875 ?

0.6875 x 2 = 1.375 take 10.375 x 2 = 0.75 take 00.750 x 2 = 1.5 take 1

0.5 x 2 = 1.0 take 1• Result: 0.1011• N.B.: Result bits here are produced MSB to LSB• the procedure sometimes does not converge

stop when the desired precision is reached

(111.6875)10 = (1101111.1011)2

Conversion of fractional part

LSB

MSB


Boolean Algebra

•A useful mathematical system for specifying and transforming logic functions

•We study Boolean algebra as a foundation for designing and analyzing digital systems!

•Named from George Boole


George Boole (1815-1864)

An Investigation of the Laws of Thought, on Which are founded the Mathematical Theories of Logic and Probabilities (1854)


Claude Shannon(1916-2001)

A Symbolic Analysis of Relay and Switching Circuits (1938)

ENIAC (Electronic Numerical Integrator And Calculator) (1946)


Boolean Algebra

• Boolean Algebra deals with Binary variables take on one of two values. Logical operators operate on binary values

and binary variables• the two binary values have different names:

True/False On/Off Yes/No 1/0

• Basic logical operators are the logic functions AND, OR and NOT


Logical Operations

• The three basic logical operations are: AND , OR, NOT

• AND is denoted by a dot (·). • OR is denoted by a plus (+).• NOT is denoted by an overbar ( ¯ ), a single

quote mark (') after, or (~) before the variable• Intended meaning (for humans - Laws of

Thought) AND: both inputs are true OR: at least one input is true NOT: negate the input


• Examples: Y = AB is read “Y is equal to A AND B.” z = x+y is read “z is equal to x OR y.” X=A’ is read “X is equal to NOT A.”

Note: The statement: 1 + 1 = 2 (read “one plus one equals two”)

is not the same as1 + 1 = 1 (read “1 or 1 equals 1”).

Notation Examples


Operator Definitions

Operations are defined on the values "0" and "1" for each operator:

AND

0 · 0 = 00 · 1 = 01 · 0 = 01 · 1 = 1

OR

0 + 0 = 00 + 1 = 11 + 0 = 11 + 1 = 1

NOT

1001


Boolean functions• Basic logical operators are the boolean functions

f(x1,…,xn): {0,1}n {0,1}

arguments domain codomain


Formal definition of functions (1)

• By means of “truth tables” Explicit representation of the output for all

possible combinations of values on its arguments

A B AND0 0 00 1 01 0 01 1 1

A B OR0 0 00 1 11 0 11 1 1

A NOT0 11 0


…truth table for a 3-variable function…

f(A,B,C)= 1 if and only if at least 2 variables are equal to 1

A B C f(A,B,C)

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


…number of rows of a truth table…

• 1-variable function 2



…• n-variable function

2n


Formal definition of functions (2)• By means of “boolean equation”: the function is

specified as boolean expression of its arguments• boolean equation consists of:

variables constants 0 and 1 boolean operations (AND, OR, NOT) parentheses

M(A,B) = (((A)’(B)’) + (AB)) M = (((A)’(B)’) + (AB))


Boolean Operator Precedence• The order of evaluation in boolean expression is:

1. Parentheses2. NOT3. AND4. OR

• Consequence: parantheses appear around OR expressions

• Example: F=A(B+C)(C+D’)

M = (((A)’(B)’) + (AB))

M = A’B’ + AB


From the boolean formula to truth table

• Explicit case-by-case evaluation

• Example: F = X + Y’Z

X Y Z F

0 0 0 0

0 0 1 1

0 1 0 0

0 1 1 0

1 0 0 1

1 0 1 1

1 1 0 1

1 1 1 1


Logic gates

• A logic gate implements simple boolean operation

• basic gates: AND, OR, NOT

• gates are used to build circuits


NOT gate - the simplest one• NOT gate - inverts the signal

If A is 0, X is 1 If A is 1, X is 0

• A NOT gate is also called an inverter


AND gate

• Output is 1 if all inputs are 1 In general, if the AND gate has N inputs, input

1 AND input 2 AND … AND input N must be 1 for the output to be 1

• 2-input AND gate


OR gate

• Output is 1 if at least one input is 1 In general, if the OR gate has N inputs, input 1

OR input 2 OR … OR input N must be 1 for the output to be 1

• 2-input OR gate


A more complex example

• 2-input “equivalence” circuit• The output is 1 if the inputs are the same

(i.e., both 0 or both 1) Truth table

• Boolean function:M = A’B’ + AB

A B M0 0 10 1 01 0 01 1 1


Formal definition of functions (3)

• By means of logic circuits Combination of logic gates joined by wires


Conventions for logic circuits


From boolean formula to logic circuit

• Idea: top-down or bottom-up construction

• Example: write the logic circuit for F = X + Y’Z


A more complex example

• Exercise: bluid the logic circuit of the following function

F = (A+BC)D + E


From logic circuit to…

• …truth table explicit case-by-case computation

• …boolean formula left-to-right inspection


Example

• Write the boolean function and its truth table for the following logic circuit

X’YZ’

XY

F =Y’+X’YZ’+XY


Function and Truth Table

• F = Y’ + X’YZ’ + XYX Y Z F0 0 0 10 0 1 10 1 0 10 1 1 01 0 0 11 0 1 11 1 0 11 1 1 1


One more example…

• Write the boolean function and its truth table for the following logic circuit

X’

Z’

X’YZ

X’YZ’

XZ

X’YZ+X’YZ’+XZ=


Function and Truth Table

• F = X’YZ + X’YZ’ + XZ


Conversion between representations• Circuit ->

-> Boolean formula (left-to-right inspection)-> Truth table (explicit case-by-case computation)

• Boolean formula ->-> Circuit (top-down/bottom-up construction)-> Truth table (explicit case-by-case evaluation)

• Truth table -> -> Circuit (through boolean formula)-> Boolean formula (through canonical form – see

later)


Boolean Identities1A = A 0+A = A Identity0A = 0 1+A = 1 NullAA = A A+A = A IdempotentAA’ = 0A+A’ = 1 InverseAB = BA A+B = B+ACommutative(AB)C = A(BC) (A+B)+C = A+(B+C) AssociativeA+BC = (A+B)(A+C) A(B+C) = AB+AC DistributiveA(A+B) = A A+AB = A Absorption(AB)’ = A’+B’ (A+B)’ = A’B’ De Morgan

duality principle: any algebraic equality remains true when the operators OR and AND, and the elements 0 and 1 are interchanged


Truth tables to verify De Morgan’s theorem


RemarkEach equality remains true if you sobstitute any variable withany expression

Examples

(A+B)(A+CD’) = A + BCD’ (distributive)

((A+BC)(D+A))’ = (A+BC)’ + (D+A)’ (De Morgan) = A’ (BC)’ + D’A’ (De Morgan) = A’(B’+C’) + D’A’


Generalized De Morgan’s theorems

X1+X2+…+Xn = X1 X2 … Xn

X1 X2… Xn = X1+X2+…+Xn


F = X’YZ + X’YZ’ + XZ (distributive) = X’Y(Z + Z’) + XZ (inverse) = X’Y 1 + XZ (identity) = X’Y + XZ

… algebraic manipulation…


Boolean Algebra Vs Switching Algebra (1)A Boolean Algebra is a structure A = <A, +, · , ’, 0, 1> where

• A is a set• + and · are binary operations• ‘ is a unary operation• 0, 1 A

satisfying the following axioms

(i) + and · are commutative (ii) 0 and 1 satisfy: a·1=a and a+0=a, a A

(iii) + and · distribute over each other

(iv) for each element a A, there exists an element a’ A such that a + a’= 1 and a·a’=0


Boolean Algebra Vs Switching algebra (2)

Switching Algebra is the following boolean algebra

A = <{0,1}, +, · , ’, 0, 1>


…one more example of boolean algebra…

Let U be a finite set

A = <2U, , , , , U>


ObservationAxioms (i)-(iv) can be used to prove all the other identities

An example: Idempotent X + X = X

X + X = (X + X)·1 (ii) = (X + X)(X + X’) (iv) = X + (X·X’) (iii) = X + 0 (iv) = X (ii)


NAND gate - the negation of AND

• The opposite of the AND gate is the NAND gate (output is 0 if all inputs are 1)

Truth table• Logic diagram A B NAN

D0 0 10 1 11 0 11 1 0


Alternative NAND representations


NOR gate - the negation of OR

• The opposite of the OR gate is the NOR gate (output is 0 if any input is 1) Truth table

• Logic diagram A B NOR0 0 10 1 01 0 01 1 0


Alternative NOR representations


Exercise

• Write the truth table for: a 3 input NAND gate a 3 input NOR gate


XOR gate - the exclusive OR

• For a 2-input gate Output is 1 if exactly one of the inputs is 1

Truth table• Logic diagram

• A B = A’B + AB’

A B XOR0 0 00 1 11 0 11 1 0


Properties of XOR operator

X 0 = X X 1 = X’

X X = 0 X X’ = 1

X Y’ = (X Y)’

X Y = Y X(X Y) Z = X (Y Z)


(X Y)’ = (XY’ + X’Y)’ = (XY’)’ (X’Y)’ = (X’+Y) (X+Y’) = X’X + X’Y’ + XY + YY’ = X’Y’ + XY

X Y Z = (X Y)Z’ + (X Y)’Z = (XY’ + X’Y)Z’+ (X’Y’ + XY)Z = XY’Z’+X’YZ’+X’Y’Z+XYZ

Notice: X Y Z = 1 if an odd number of arguments is 1

X1 … Xn = 1 if an odd number of arguments is 1

…some interesting algebraic manipulations…


Universal Gates

• Universal gate: a gate type that can implement any Boolean function

• How many logical functions with n input?• With n inputs there are 2(2n ) possible logical

functionsA B 0 1 2 3 4 5 6 7 8 9 1

011

12

13

14

15

0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 10 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 11 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 11 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1


Universal Gates (2)• AND, OR, NOT can generate all possible boolean

functions (see later)

• Is it possible to use fewer basic operations?

• AND, NOT are enough !

• OR, NOT are enough !

• Even NAND alone or NOR alone are enough !


• Simulation of NOT ???

How NAND simulates AND, OR


• Simulation of NOT ???

How NOR simulates AND, OR


Gate equivalence• Any AND, OR, NOT gate can be obtained using just

NAND gates or just NOR gates

• Consequence: any circuit can be constructed using just NAND gates or just NOR gates (easier to build)


ExerciseTransform the following circuit into a circuit using only NAND gates by replacing each gate with its equivalent NAND only circuit


Equivalence modificationson any wire, one can introduce a bubble at the beginning and at the end


Exercise

• Write a NAND only logic circuit for the exclusive OR function (XOR)

A B = A’B + AB’


Solution (1)

• Truth table initial circuit

A B XOR0 0 00 1 11 0 11 1 0


Solution (2): equivalence transform.A’BAB’

A’BAB’

A’BAB’


Transforming OR, AND to NAND

• Transform the following circuit


Solution


Exercise

• Write a NOR only logic circuit equivalent to the following circuit


SOP and POS representations• It is useful to specify boolean function in a

particular form• A literal is a variable in the positive form or in the

negative• sum of products (SOP) rapresentation

it is an OR of AND combinations of literals F1=A’B + BC it is a SOP rapresentation F2= A’B + (BC)’ it is not a SOP rapresentation

• product of sums (POS) rapresentation it is an AND of OR combiations of its literals F1=A’(B + C’)(B+A) it is a POS rapresentation F2=A’B + BC it is not a POS rapresentation


Canonical Form for boolean functions• It is a “standard” way of expressing SOP or POS • It is:

a sum of minterms, for canonical SOP a product of maxterms for canonical POS

• A minterm is a product containing all variables, either in the positive form or in the negative form

• A maxterm is a sum containing all variables, either in the positive or in the negative form.

• Examples:F = (A’+B+C) (B’+C) is not in a POS canonical formM = AB + A’BC is not in a SOP canonical form


Minterms

• Given that each variable may appear normal (e.g. X) or complemented (e.g. X’), there are 2n minterms for n variables

• Example: Two variables (X and Y) produce 4 combinations: XY X’Y XY’ X’Y’

• Thus there are 4 minterms of 2 variables


Maxterms

• Given that each variable may appear normal (e.g. X) or complemented (e.g. X’), there are 2n maxterms for n variables

• Example: Two variables (X and Y) produce 4 combinations: X+Y X+Y’ X’+Y X’+Y’

• Thus there are 4 maxterms of 2 variables


Maxterms and Minterms• Examples: 2 variable minterms and maxterms

• The index above is important for describing which variables in the terms are true and which are complemented

Index Minterm Maxterm0 X’Y’ X+Y1 X’Y X+Y’2 XY’ X’+Y3 XY X’+Y’


Standard Order• Minterms and Maxterms are designated with subscript• The subscript is a numer, corresponding to a binary pattern• The bits in the pattern represent the complemented or

normal state of each variable listed in a standard order• All the variables will be present in a minterm or maxterm

and will be listed in the same order (usually alphabetically)• Example: For variables A, B, C:

Maxterms: (A + B + C’), (A + B + C) Terms: (B + A + C), AC’B, and (C + B + A) are NOT in

standard order Minterms: AB’C, ABC, AB’C’ Terms: (A+C), B’C, and (A+B’) do not contain all

variables


Purpose of the Index

• The index for the minterm or maxterm, expressed as a binary number, is used to determine whether the variable is shown in the true form or complemented form.

• For Minterms: “1” means the variable is “Not Complemented” and “0” means the variable is “Complemented”

• For Maxterms: “0” means the variable is “Not Complemented” and “1” means the variable is “Complemented”


X Y Z Minterm

Symbol m0 m1

m2

m3

m4

m5

m6

m7

0 0 0 X’Y’Z’ m0 1 0 0 0 0 0 0 00 0 1 X’Y’Z m1 0 1 0 0 0 0 0 00 1 0 X’YZ’ m2 0 0 1 0 0 0 0 00 1 1 X’YZ m3 0 0 0 1 0 0 0 01 0 0 XY’Z’ m4 0 0 0 0 1 0 0 01 0 1 XY’Z m5 0 0 0 0 0 1 0 01 1 0 XYZ’ m6 0 0 0 0 0 0 1 01 1 1 XYZ m7 0 0 0 0 0 0 0 1


X Y Z Maxterm

Symbol M0 M1 M2 M3 M4 M5 M6 M7

0 0 0 X+Y+Z M0 0 1 1 1 1 1 1 10 0 1 X+Y+Z

’M1 1 0 1 1 1 1 1 1

0 1 0 X+Y’+Z

M2 1 1 0 1 1 1 1 1

0 1 1 X+Y’+Z’

M3 1 1 1 0 1 1 1 1

1 0 0 X’+Y+Z

M4 1 1 1 1 0 1 1 1

1 0 1 X’+Y+Z’

M5 1 1 1 1 1 0 1 1

1 1 0 X’+Y’+Z

M6 1 1 1 1 1 1 0 1

1 1 1 X’+Y’+Z’

M7 1 1 1 1 1 1 1 0


Minterm and Maxterm Relationship

• DeMorgan's Theorem:

• Two-variable example:

• Thus M2 is the complement of m2 and vice-versa.

• Since DeMorgan's Theorem holds for n variables, the above holds for terms of n variables

• giving:

• Thus Mi is the complement of mi.

yx y· x yxyx

y x M2 yx· m2

i mM i ii Mm


Boolean function implementation

• Any function can be implemented as the OR of the AND combinations of its inputs canonical SOP

• Start from the truth table For each 1 in the output Write its corresponding minterm Write these in OR


Example: canonical SOPX Y Z F

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

m1

m3

m6

m7

0 0 0 01 0 0 00 0 0 00 1 0 00 0 0 00 0 0 00 0 1 00 0 0 1

F = m1 + m3+ m6 + m7

= X’Y’Z + X’YZ + XYZ’ +XYZ

canonical SOP


Boolean function implem. (2)

• Any function can be implemented as the AND of the OR combinations of its inputs canonical POS

• Start from the truth table For each 0 in the output Write its corresponding maxterm Write these in AND


Example: canonical POSX Y Z F

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

M0

M2

M4

M5

0 1 1 11 1 1 11 0 1 11 1 1 11 1 0 11 1 1 01 1 1 11 1 1 1

F = M0 · M2 · M4 · M5

= (X+Y+Z) · (X+Y’+Z) (X’+Y+Z) · (X’+Y+Z’)

canonical POS


Boolean function implem. (4)

• Alternative procedure for POS form Complement the table by substituting

everywhere a 0 with a 1 and a 1 with a 0 Write a SOP form for the complemented table Complement the formula by substituting

everywhere and AND with an OR and an OR with an AND

Why does it work ?


F = X’YZ + X’YZ’ + XZ = X’Y(Z + Z’) + XZ = X’Y 1 + XZ = X’Y + XZ

… algebraic manipulation…

…a simpler SOP representation leads to a simpler circuit…


Circuit Optimization• Goal: To obtain the best implementation for a

given function what means best?

speed vs costmeasured by

number of levelsmeasured by

some cost criterion


Circuit Optimization• Optimization is a more formal approach to

simplification that is performed using a specific procedure or algorithm

• Optimization requires a cost criterion to measure the quality of a circuit

• Distinct cost criteria we will use: Literal cost (L) Gate input cost (G) Gate input cost with NOTs (GN)


Literal cost• Literal cost – the number of literal

occurrences in a Boolean expression corresponding to the logic circuit diagram

• it is equal to the number of inputs of the circuit

• Examples: F = BD + AB’C + AC’D’ F = BD + AB’C + AB’D’ + ABC’ F = (A + B)(A + D)(B + C + D’)(B’+C’+D) Which solution is the best?

L=8L=11L=10


Gate input cost

• Gate input cost: the number of inputs to the gates in the circuit corresponding exactly to the furmula G - inverters not counted, GN - inverters counted

• For SOP and POS equations, it can be found from the equation(s) by finding the sum of: all literal occurrences the number of terms excluding single literal

terms (G) optionally, the number of distinct complemented

single literals (GN)


• F = A + B C +

Cost Criteria Example

A

BC

F

B C L = 5

L (literal count) counts the AND inputs and the single literal OR input.

G = L + 2 = 7

G (gate input count) adds the remaining OR gate inputs

GN = G + 2 = 9

GN(gate input count with NOTs) adds the inverter inputs


• F = A B C +• L = 6 G = 8 GN = 11• F = (A + )( + C)( + B)• L = 6 G = 9 GN = 12• Same function and same

literal cost• But first circuit has better

gate input count and bettergate input count with NOTs

• Select it!

Cost Criteria

B C

AC B

F

ABC

A

ABC

F


Minimal SOP and POS

• Optimization for two-level (SOP and POS) circuits: minimal SOP

• minimum number of pruduct terms• minimum number of literals for each product term

minimal POS• minimum number of sum terms• minimum number of literals for each sum term


Minimization procedures

• Karnaugh’s maps (by hand) Used to minimize boolean functions of up to 4

input variables• Quine-McKluskey (programmable)

For more variables• Practical Optimization: Espresso

sub-optimal heuristic method


Karnaugh’s Maps (KM)• Grid-like representation for boolean functions• Each cell represents a minterm• The K-map can be viewed as a reorganized version

of the truth table• Minterms with just one variable different occupies

adjacent cells• Alternative algebraic expressions for the same

function are derived by recognizing patterns of squares

• Consider only 1s in the representation (focusing on a SOP representation)

• IDEA: if 2 adjacent cells have a 1 the function can be simplified


A KM for 2-variable functions• The generic KM

• Function F = XY Function F = X + Y


A KM for 3-variable functions

labels indicate the area of the map where the corresponding variables are true


example

X

Y

Z

0

1

00 01 11 10XYZ X Y Z F=

Z0 0 0 00 0 1 10 1 0 00 1 1 11 0 0 01 0 1 11 1 0 01 1 1 1

1

1 1

1


X

Y

Z

0

1

00 01 11 10XYZ X Y Z F=

Z’0 0 0 10 0 1 00 1 0 10 1 1 01 0 0 11 0 1 01 1 0 11 1 1 0

1

11

1


X

Y

Z

0

1

00 01 11 10XYZ X Y Z F=Y

Z0 0 0 00 0 1 00 1 0 00 1 1 11 0 0 01 0 1 01 1 0 01 1 1 1

1

1


An example• F = X’YZ + X’YZ’ + XY’Z + XY’Z’

Idea: • we want to cover all 1s by using rectangles of adjacent cells• each rectangle of 2k adjacent cells (for some k) represents a literal product term• bigger rectangles correspond to simpler product terms

X Y Z F0 0 0 00 0 1 00 1 0 10 1 1 11 0 0 11 0 1 11 1 0 01 1 1 0

= X’Y(Z+Z’) + XY’(Z+Z’)= X’Y + XY’


Combining Squares• By combining squares, we reduce

number of literals in a product term• On a 3-variable K-Map:

One square represents a minterm with three variables

Two adjacent squares represent a product term with two variables

Four “adjacent” terms represent a product term with one variable

Eight “adjacent” terms is the function of all ones (no variables) = 1.


Circular adjacencies for 3 variables

labels are useful to get the expressioncorresponding to a given rectangle


Three-Variable Maps

• Example Shapes of 2-cell Rectangles:y

0 1 3 2

5 64 7xz

X’Y’

YZ

X’Z’


Three-Variable Maps

• Example Shapes of 4-cell Rectangles:y

0 1 3 2

5 64 7xz


Four 1 adjacents

F = X’Y’Z’ + XY’Z’ + X’YZ’ + XYZ’ = Y’Z’(X’+X) + YZ’ (X’+X) = Y’Z’ + YZ’ = Z’(Y’+Y) = Z’


An example• F = X’YZ + X’YZ’ + XY’Z + XY’Z’

Idea: • we want to cover all 1s by using rectangles of adjacent cells• each rectangle of 2k adjacent cells (for some k) represents a literal product term• bigger rectangles correspond to simpler product terms

X Y Z F0 0 0 00 0 1 00 1 0 10 1 1 11 0 0 11 0 1 11 1 0 01 1 1 0

= X’Y(Z+Z’) + XY’(Z+Z’)= X’Y + XY’


Exercise (1)

• Which is the minimal SOP expression for function F1=m3+m4+m6+m7?


Solution

• F1 = YZ + XZ’


Exercise (2)

• Which is the minimal SOP expression for function F2=m0+m2+m4+m5+m6?


Solution

• F2 = Z’ + XY’


k-cube of 1s

• Is a set of 2k adjacent cells• 0-cube, 1 cell, a minterm• 1-cube, 2 adjacent cells• 2-cube, 4 adjacent cells• 3-cube, 8 adjacent cells• ….


Prime implicants

• a product term P is said to be an implicant for a function F if P implies F, i.e. if P is true then F is true

• The product term corresponding to a k-cube is an implicant

• An implicant is said to be a prime implicant for F if it does not imply any other implicant of F

• A prime implicant can be chosen by selecting a maximal k-cube, i.e. a k-cube in the KM which is not contained in any larger h-cube (h>k)


Minimal representation

F = P1 + P2 + P3 + ... has a minimal SOP representation if:

1. Each Pi is a prime implicant 2. There is a minimum number of them


Minimality procedure

1. Find maximal k-cubes (prime implicants)2. If a 1 is covered by only one maximal k-

cube this has to be chosen (essential prime implicants)

3. Select a minimum number of the remaining k-cubes so to cover all 1s not covered by essential prime implicants


Exercise (3)• Find the minimal SOP expression for F=m1+m3+m4+m5+m6

X

Y

Z

0

1

00 01 11 10XYZ

1 1

1 11

F=X’Z+XZ’+XY’


Exercise (3)• Find the minimal SOP expression for F=m1+m2+m3+m5+m7

X

Y

Z

0

1

00 01 11 10XYZ

1 1

1

1

1

F=Z+X’Y


A KM for 4-variable functions


Circular adjacenciesfor 4 variables


Four Variable Terms Four variable maps can have rectangles

corresponding to:• A single 1 = 4 variables, (i.e. Minterm)• Two 1s = 3 variables,• Four 1s = 2 variables• Eight 1s = 1 variable,• Sixteen 1s = zero variables (i.e. Constant "1")


Four-Variable Maps• Example Shapes of Rectangles:

8 9 1011

12 13 1415

0 1 3 2

5 64 7

X

Y

Z

W

XZ

X’Z’YW’


Four-Variable Maps• Example Shapes of Rectangles:

X

Y

Z

8 9 1011

12 13 1415

0 1 3 2

5 64 7

W


Simplify F(A, B, C, D) given on the K-map. Example:

1

1

1

1 1

1

1

B

D

A

C

1

1

1

1

1

1 1

1

1

B

D

A

C

1

1

Essential

Minterms covered by essential prime implicants

Selected

minimal SOP: A’B+A’CD+AC’D+B’C’D’


One more example

DB

CB

1 1

1 1

1 1

B

D

A

1 1

1 1

1

ESSENTIAL Prime Implicants

C

BD

CD

BD

Minterms covered by single prime implicant

DB

1 1

1 1

1 1

B

C

D

A

1 1

1 1

1

AD

BA

CD

BA

minimal SOP: BD+B’D’+CD+AB’


Five Variable or More K-Maps• For five variable problems, we use two adjacent K-maps. It becomes

harder to visualize adjacent minterms for selecting k-cubes.

• You can extend the problem to six variables by using four K-Maps.

X

Y

Z

W

V = 0

X

Z

W

V = 1Y


The KM method for POS

• Which is the POS expression of function F represented by this KM?

• Use the same method used for build POS canonical form from truth tables Find the minimal SOP for F’ apply DeMorgan


Example

F = (CD+BD’+AB)’ = (CD)’ . (BD’)’ . (AB)’ = (C’+D’) . (B’+D’’) . (A’ + B’) = (C’+D’) . (B’+D) . (A’ + B’)

B

D

A

C

0 0

0

00

0

0

0 0

1 11

1

1 1 1


Don't Cares in K-Maps• Sometimes a function table or map contains

entries for which it is known: the input values for the minterm will never occur, or the output value for the minterm is not used

• In these cases, the output value does not need to be defined

• Instead, the output value is defined as a “don't care”

• By placing “don't cares” ( an “x” entry) in the function table or map, the cost of the logic circuit may be lowered.


Example 1• A logic function having

the binary codes for the binary-coded decimal (BCD) digits as its inputs.

• Only the codes for 0 through 9 are used.

• The six codes, 1010 through 1111 never occur, so the output values for these codes are “x” to represent “don’t cares.”

A B C D f(A,B,C,D)

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

0 0 0 1 f(0,0,0,1)

0 0 1 0 f(0,0,1,0)

0 0 1 1 f(0,0,1,1)

0 1 0 0 f(0,1,0,0)

0 1 0 1 f(0,1,0,1)

0 1 1 0 f(0,1,1,0)

0 1 1 1 f(0,1,1,1)

1 0 0 0 f(1,0,0,0)

1 0 0 1 f(1,0,0,1)

1 0 1 0 x1 0 1 1 x1 1 0 0 x1 1 0 1 x1 1 1 0 x1 1 1 1 x


Example 2• Consider the following function f(A,B)

f(A,B)=1 if A=B=0 f(A,B)=0 if A B

A B f0 0 10 1 01 0 01 1 x

A B f0 0 10 1 01 0 01 1 0

A B f0 0 10 1 01 0 01 1 1

Truth table on the left may be substitued by anyone on the right


Example 3• Consider the following function f(A,B)

f(A,B)=1 if A=B=0 f(A,B)=0 otherwise f(A,B) is used only as input for another function

g(f,A,B)=(A’+B) f(A,B)A B f0 0 10 1 01 0 x1 1 0

Notice that:for A=1 and B=0g(f,A,B) = (1’+0) f(A,B) = 0


Example

X

Y

Z

W

X

X

X1 1

1

1

1

f(W,X,Y,Z)= YZ + X’W’

Simplify the choice, since each X can be

considered a 0 or a 1


X

Y

Z

W

X

X

X1 1

1

1

1

f(W,X,Y,Z)= YZ + ZW’

Example

a different choise

is possible


Multiple-Level Optimization

• Multiple-level circuits - circuits that are not two-level

• Multiple-level circuits can have reduced gate input cost compared to two-level (SOP and POS) circuits

• Multiple-level optimization is performed by applying transformations to circuits represented by equations while evaluating cost


An Example

• G=ABC + ABD + E + ACF + ADF gate input cost: 17

• G=AB(C+D) + E + AF(C+D) gate input cost: 13

• G=(AB + AF)(C+D) + E gate input cost: 11

• G=A(B+F)(C+D)+E gate input cost: 9


Logic Circuits and Computer Architecture

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