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6.004 Computation StructuresSpring 2009
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L04 - Logic Synthesis 16.004 Spring 2009 2/12/09

Synhesis of Combinaional Logic

Lab 1 is due Thursday 2/19Quiz 1 is a week from Friday (in secion)

A

B

modified 2/12/09 10:01 L04 - Logic Synthesis 26.004 Spring 2009 2/12/09

Funcional Specificaions

There are many ways of specifying hefuncion of a combinaional device, for

example:

A

BYIf C is 1 hencopy B o Y,

oherwise copyA o YC

Concise alternatives:

truth tablesare a concise description of the combinational

systems function.Boolean expressionsform an algebra in whose operations are

AND (multiplication), OR (addition), and inversion

(overbar).

Any combinational (Boolean) function can be specified as a truthtable or an equivalentsum-of-productsBoolean expression!

Argh Im tired of word games

C B A Y

0 0 0 0

0 0 1 1

0 1 0 0

0 1 1 1

1 0 0 0

1 0 1 0

1 1 0 11 1 1 1

Truh Table

CBAACBBACABCY


Heres a Design Approach

1) Wrie ou our funcional spec as aruh able

2) Wrie down a Boolean expression wiherms covering each 1 in he oupu:

3) Wire up he gaes, call i a day, anddeclare success!

This approach will always give usBoolean expressions in a paricularform: SUM-OF-PRODUCTS

C B A Y

0 0 0 0

0 0 1 10 1 0 0

0 1 1 1

1 0 0 0

1 0 1 0

1 1 0 1

1 1 1 1

Truh Table

-is sysemaic!-i works!

-is easy!-are we done ye???

CBAACBBACABCY


Sraighforward Synhesis

We can implemen

SUM-OF-PRODUCTS

wih jus hree levels of

logic.

INVERTERS/AND/OR

Propagaion delay --

No more han 3 gae delays

(assuming gaes wih an arbirary number of inpus)

AB

CAB

CAB

CAB

C

Y

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Basic Gae Reperoire

Are we sure we have all he gaes we need?

Jus how many wo-inpu gaes are here?

AB Y

00 0

01 0

10 0

11 1

ANDAB Y

00 0

01 1

10 1

11 1

ORAB Y

00 1

01 1

10 1

11 0

NANDAB Y

00 1

01 0

10 0

11 0

NOR

2 = 24 = 1622

Hmmmm all of hese have 2-inpus (no surprise)

each wih 4 combinaions, giving 22 oupu cases

How many ways are here of assigning 4 oupus? ________________


There are only so many gaes

There are only 16 possible 2-inpu gaes some we know already, ohers are jus silly

I

N

P

U

T

AB

Z

E

R

O

A

N

D

A

>

B A

B

>

A B

X

O

R

O

R

N

O

R

X

N

O

R

N

O

T

B

A

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One will do!

NANDs and NORs are universal:

Ah!, bu wha if we wan more han 2-inpus

=

=

=

=

=

=


Supid Gae Tricks

Suppose we have some 2-inpu XOR gaes:

And we wan an N-inpu XOR:

A0011

B0101

C0110

pd = 1cd = 0

pd = O( ___ ) -- WORST CASE.

oupu = 1iff number of 1sinpu is ODD(ODD PARITY)

Can we compue N-inpu XOR faser?

N

A1

A3 A4 AN

A2

A

BC


I hink ha I shall never seea circui lovely as...

N-inpu TREE has O( ______ ) levels...

Signal propagaion akes O( _______ ) gae delays.

Quesion: Can EVERY N-Inpu Boolean funcion be implemened as aree of 2-inpu gaes?

log N

log N

21222

log2N

A1

A2

A4

A3

AN


Are Trees Always Bes?Alernae Plan: Large Fan-in gaes

N pulldowns with complementary pullups

Output HIGH if any input is HIGH = OR

Propagaion delay: O(N) since each

addiional MOSFET adds C

...

N

pd

O(log N)

O(N)

~4

Don be mislead by he big O suffhe consans in his case can be much

smaller so for small N his plan mighbe he bes.

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AB=A+B

Pracical SOP Implemenaion

NAND-NAND

NOR-NOR

CA

B

Y

C

A

B

Y

zyxxyz ++=

C

A

BY

yxyx =+

C

A

B

Y

C

A

BY

AB=A+BPushing Bubbles

C

A

BY

You migh hink all hese exrainverers would make his srucureless atracive. However, quie heopposie is rue.


Logic Simplificaion

Can we implemen he same funcion wih fewer gaes?

Before rying well add a few more ricks in our bag.BOOLEAN ALGEBRA:

OR rules: a + 1 = 1, a + 0 = a, a + a = aAND rules: a1 = a, aO = 0, aa = aCommuaive: a + b = b + a, ab = baAssociaive: (a + b) + c = a + (b + c), (ab)c = a(bc)Disribuive: a(b+c) = ab + ac, a + bc = (a+b)(a+c)Complemens:

Absorpion:

Reducion:DeMorgans Law:

0aa1,aa ==+

babaaa,aba+=+=+

abb)aa(a,b)a(a =+=+

bb)ab)((ab,baab =++=+

baba,abba +==+


Boolean Minimizaion:An Algebraic Approach

BACCBAACBABCY +++=

Les (again!) simplify

Using he ideniy

=+ AA

BACCBAACBABCY +++=

CBACY +=

BACCBABCY ++=

Can he come upwih a new example???

For any expression and variable A:Hey, I could wrieA programo do

Tha!


A Case for Non-Minimal SOP

ABCBACY ++=

AC

B

Y

NOTE: The seady sae behavior ofhese circuis is idenical. Theydiffer in heir ransien behavior.

Y(1)C(1)

Y = CA +CB

A(1)

B(1)

CD = 1 nSPD = 2nS

00

1

C B A Y

0 0 0 0

0 0 1 1

0 1 0 0

0 1 1 1

1 0 0 0

1 0 1 0

1 1 0 1

1 1 1 1

CA

CB

BA

A

BC

Y

Thas whawe call a

glich orhazard

A

BC

YNow isLENIENT!

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Truh Tables wih Don Cares

C B A Y

0 0 0 0

0 0 1 1

0 1 0 0

0 1 1 1

1 0 0 0

1 0 1 01 1 0 1

1 1 1 1

CA

CB

BA

C B A Y

0 -- 0 0

0 -- 1 1

1 0 -- 0

1 1 -- 1

-- 0 0 0

-- 1 1 1

One way o reveal he opporuniies for a more compac implemenaion iso rewrie he ruh able using don cares (--) o indicae when he

value of a paricular inpu is irrelevan in deermining he value of heoupu.


Weve been designing a mux

D0

D1

S

0

1

0101S

0101S

0101S

D00D01

D10D11

S0 S1

and implemened as aree of smaller MUXes:

00

01

10

11

D00

D10D11

S0S1

YD01

MUXes can be generalized o 2k daa

inpus and k selec inpus

2-inpu Muliplexer

YC B A Y

0 0 0 0

0 0 1 10 1 0 0

0 1 1 1

1 0 0 0

1 0 1 0

1 1 0 1

1 1 1 1

Truh Table

Y


Sysemaic Implemenaionsof Combinaional Logic

Consider implemenaion of some arbirary Booleanfuncion, F(A,B,C) ... using a MULTIPLEXERas he only circui elemen:

A B Cin Cout

0 0 0 0

0 0 1 0

0 1 0 0

0 1 1 1

1 0 0 0

1 0 1 1

1 1 0 11 1 1 1

Full-Adder

Carry Ou Logic

0123456

7A,B,Cin

Cou

00010111


General Table Lookup Synhesis

MUX

Logic

A B

Fn(A,B)

Generalizing:In heory, we can build any 1-oupu combinaionallogic block wih muliplexers.

For an N-inpu funcion we need a _____ inpu mux.

BIG Muliplexers?How abou 10-inpu funcion? 20-inpu?

AB Fn(A,B)

00 0

01 1

10 111 0

2N

Muxes are UNIVERSAL!

In fuure echnologiesmuxes migh be he

naural gae.

0101S

10

A

Y A Y=

0

1

0

1S

0

B

A

Y

0101S

B1

A

Y

=

=

A

B

Y

AB

Y

0101S

BB

A

Y

Wha does

ha one do?

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A New Combinaional Device

k

D1D2

DN

DECODER:

k SELECT inpus,

N = 2k DATA OUTPUTs.

Seleced Dj HIGH;all ohers LOW.

NOW, we are well on our way o building a general

purpose able-lookup device.

We can build a 2-dimensional ARRAY of decoders andselecors as follows ...

Have Imenioned

ha HIGHis a synonym

for 1 andLOW means

he same

as 0


Read-only memories (ROMs)

COUTS

000

001

010

011

100

101

110

111AB

CIN

A B Ci S Co

0 0 0 0 0

0 0 1 1 0

0 1 0 1 0

0 1 1 0 1

1 0 0 1 01 0 1 0 1

1 1 0 0 1

1 1 1 1 1

FA

A B

Co Ci

S

Full Adder

For K inpus, decoder

produces 2K signals,only 1 of which is

assered a a ime --hink of i as one signalfor each possible

produc erm.

Each column is large fan-in OR as describedon slide #12. Noe locaion of pulldownscorrespond o a 1 oupu in he ruh able!

Shareddecoder

One selecor foreach oupu


Read-only memories (ROMs)

A B Ci S Co

0 0 0 0 0

0 0 1 1 0

0 1 0 1 0

0 1 1 0 1

1 0 0 1 0

1 0 1 0 1

1 1 0 0 1

1 1 1 1 1

FA

A B

Co Ci

S

Full Adder LONG LINES slow down propagaion imes

The bes way o improve his is o buildsquare arrays, using some inpus o driveoupu selecors (MUXes):

00

01

10

11

0 1 0 1AB

CIN

COUTS

2D Addressing: Sandard for ROMs, RAMs, logic arrays


Logic According o ROMs

ROMs ignorehe srucure of combinaional funcions ... Size, layou, and design are independen of funcion Any Truh able can be programmed byminor reconfiguraion:

- Meal layer (masked ROMs)- Fuses (Field-programmable PROMs)- Charge on floaing gaes (EPROMs)... ec.

Model: LOOK UP value of funcion in ruh able...Inpus: ADDRESS of a T.T. enryROM SIZE = # TT enries...

... for an N-inpu boolean funcion, size = __________

2N x #oupus

ROMs end ogenerae glichyoupus. WHY?

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Summary

Sum of producs Any funcion ha can be specified by a ruh able or, equivalenly,

in erms of AND/OR/NOT (Boolean expression)

3-level implemenaion of any logic funcion Limiaions on number of inpus (fan-in) increases deph

SOP implemenaion mehods NAND-NAND, NOR-NOR

Muxes used o build able-lookup implemenaions Easy o change implemened funcion -- jus change consans

ROMs Decoder logic generaes all possible produc erms Selecor logic deermines which perms are ored ogeher

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