Cellular bulk transfer system
by Kalyan Kumar Roy
A thesis submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY in Electrical Engineering
Montana State University
© Copyright by Kalyan Kumar Roy (1970)
Abstract:
In this thesis the results of investigations on a cellular bulk transfer system from the viewpoint of its
logical capabilities have been presented. The model adopted for the bulk transfer system consists of an
input array, a mapping device, an output array and an output logic. The influence of such factors as
flexibility of the mapping device, flexibility of output logic and parallelism of operation has been
determined. The main results obtained are: the bulk transfer system can be made logically universal
with a proper combination of output logic and maps. In realizing arbitrary logic, a trade-off among the
number of mapping operations, number of independent maps and amount of logical flexibility in the
output logic is possible. A least upper bound on the number of necessary transposition maps is derived
for an output logic consisting of a flexible cellular cascade. The possibility of a set of bulk transfer
units operating in parallel has been studied and the functions realizable in this manner have been
characterized. An algorithm for test-synthesis of realizable functions has been presented.
CELLULAR BULK TRANSFER SYSTEM
by
. KALYAN KUMAR ROY
A t h e s i s s u b m itte d to th e G ra d u a te F a c u lty in p a r tia l
fu lf illm e n t of th e re q u ire m e n ts fo r th e d e g re e
of
DOCTOR OF PHILOSOPHY
in
E le c tr ic a l E n g in e e rin g
A pproved:
zH e a d , M a jo r D e p a rtm e n t
C h a irm a n , E x am in in g C o im n ittee
GradiQi Ite D ean
MONTANA STATE UNIVERSITY
B ozem an, M o n ta n a
M a rc h , 1970
iii
'
ACKNOWLEDGEMENT
The a u th o r w is h e s to o ffe r g r a te f u l th a n k s to D r. Amar M u k h o p a d h y a y
fo r m any d i s c u s s i o n s an d s u g g e s tio n s d u rin g th e c o u rs e o f th is w o rk .
H e i s a l s o g ra te fu l to P r o f e s s o r R. C ..M in n ick fo r m any h e lp f u l s u g g e s ­
tio n s d u rin g t h i s p e rio d .
The f in a n c ia l s u p p o rt fo r th e g ra d u a te s tu d y .th ro u g h th e a w ard of
a r e s e a r c h a s s i s t a n t s h i p b y th e D e p a rtm e n t o f E le c tr ic a l E n g in e e rin g ,
M o n ta n a S ta te U n iv e r s ity , a te a c h in g a s s i s t a n t s h i p b y th e D e p a rtm e n t
o f C o m p u ter S c ie n c e , U n iv e r s ity of Io w a a n d th ro u g h N a tio n a l S c ie n c e
F o u n d a tio n G ra n t n o s . GJ 158 a n d GJ 723 i s th a n k fu lly a c k n o w le d g e d .
•
rX
IV
TABLE OF CONTENTS
C h a p te r I :
■
IN T R O D U C T IO N .............................. ■..........................
.1
1 .1
In tro d u c tio n . .....................................................................................
2
1 .2
The Bulk T ra n s fe r S y ste m in R e la tio n to Some ■
P a r a lle l P r o c e s s o r s . ..................................................................... .4
.1 .3
O rg a n iz a tio n o f th e R em ain in g C h a p t e r s ...............................
9.
LOGICAL CAPABILITY OF A BULK TRANSFER SYSTEM . .
11
2 .1
A Bulk T ra n s fe r S y s t e m ....................................................... . .
12
2 .2
L o g ic a l C a p a b ility of th e Bulk T ra n s fe r S y stem . . . . .
16
2 .3
A S im ple D e sig n o f th e Bulk T ra n s fe r S y stem .
24
2 .4
D e te rm in a tio n o f th e N e c e s s a r y M ap p in g O p e ra tio n s
C h a p te r 2:
C h a p te r 3:
. . . .
T r a n s p o s itio n M a p s ................................................................
3 .2
O u tp u t L ogic w ith M a itra C a s c a d e . . . . . . . . .
3 .4
C h a p te r 4:
28
BULK TRANSFER WITH CELLULAR C A S C A D E S ........................... 35
3 .1
3 .3 .
.
D e te rm in a tio n o f N e c e s s a r y T r a n s p o s itio n s
. . . . . .
•
36
36
46
Bulk T ra n s fe r in C a s c a d e s ................................................................. 51
PARALLEL BULK TRANSFER SYSTEM
. . . . . . . . .
60
4 .1
P a r a lle l T r a n s f e r s .................................................
61
4 .2
D is jo in t D e c o m p o s itio n ...................................................................... 65
4 .3
N o n - d is jo in t D e c o m p o s itio n .............................
97
V
C h a p te r 5:
PARALLEL BULK TRANSFER SYSTEM WITH FLEXIBLE
INPUT D O M AIN.............................
109
5 .1
The P roblem o f V a ria b le G ro u p in g . . . . . . . . .
1.10
5 .2
U n a te L ogic N e t w o r k .................................................
Ill
5 .3
An A lgorithm fo r G e n e ra l D is ju n c tiv e N etw o rk
S y n t h e s i s ............................................................................................... 118
5 .4
S y n th e s is o f N o n - d is ju n c tiv e N e tw o r k .................................... 127
C h a p te r 6: - C O N C L U S IO N S .............................
132
6 .1
S um m ary .............................
6 .2
S cope of F u rth e r R e s e a r c h .............................................................135
APPENDIX:
.................................................................' .............................. ...
133
138
A p p en d ix A ...................................................... ......................................................
139
A ppen d ix B .......................... . ..........................................................................
142
LITERATURE CITED
147
vi
LIST OF TABLES
T ab le 2 .3
T h r e e -v a ria b le M in term s-....................................................... 25
T a b le 4 . 2 . 1
D e c o m p o s itio n T a b l e ............................................................ 66
T ab le 4 . 2 . 2
T ypes of Prim e Im p lic a n ts in T w o -in p u t
ULM c a s e
...............................................................................83
T a b le 4 . 2 . 3
D e c o m p o s itio n T ab le fo r F =
+ x^Xg
+ X 1X4 + XgXg + X 2X4 ............................................................84
T a b le 4 . 2 . 4
A T ruth T ab le fo r U 3 in te rm s o f In te rm e d ia te
L e v e l F u n c tio n s U ^ , U 3 ....................* ............................. 85
T a b le 4 . 2 . 5
T y p e s of Prim e Im p lic a n ts in T h re e -in p u t
ULM c a s e ..........................................................................................87
T a b le 4 . 2 . 6
A D e c o m p o s itio n T a b le fo r th e F u n c tio n
F in E xam ple 4 . 2 . 2 .................... .... .. .............................. 93
T a b le 4 . 2 . 7
A D e c o m p o s itio n T a b le fo r F = x^X gX . + X4 X4
+ X g X g + X g X g .......................................
97
T a b le 4 . 3 . 1
T y p e s o f Prime Im p lic a n ts in Sim ple N o n D is ju n c tiv e c a s e .
...................................99
T a b le 4 . 3 . 2
A D e c o m p o s itio n T a b le fo r th e F u n c tio n F
in E x am p le 4 . 3 . 1 ..................................................................... 103
T a b le 4 . 3 . 3
A T ruth T a b le of Ug in te rm s of In te rm e d ia te
le v e l F u n c tio n s U^ a n d U g .................................................. 104
T a b le 4 . 3 . 4
A D e c o m p o s itio n T a b le fo r F = x , x . + XgX4
+ XgX4 + X 5 + X g X 7 + XgX9 .................................................. 106
vii
LIST OF FIGURES
F ig u re 1 . 2 . 1
T he U n g er M a c h in e ................................................................
5
F ig u re 1 . 2 . 2
O p tic a l S um m ation S y s te m ................................................
8
F ig u re 2 . 1 . 1
A B ulk T ra n s fe r S y s t e m ......................................................
12
F ig u re 2 . 1 . 2
M ap p in g D e v ic e w ith L o g i c ............................................
13
F ig u re 2 . 3 . 1
The C irc u it for G e n e ra tin g th e P e rm u ta tio n
26
$
F ig u re 2 . 3 . 2
T he C irc u it fo r G e n e ra tin g th e M ap
.........................
27
F ig u re 2 . 3 . 3
The C irc u it for G e n e ra tin g th e P e rm u tatio n
(m j , mg) ....................................................................................
29
O ne d im e n s io n a l G raph Form at fo r T h ree
V a ria b le F u n c tio n s . ............................................
37
F ig u re 3 . 2 . 2
The M a itra -C a s c a d e .............................................................
38
F ig ure 3 . 3 . 1
P lo t of F = Sg (x^ , x 3 ,Xg , Xj ) in O ne D im e n s io n a l
G r a p h ........................................................................................
47
T he F u n c tio n F = Sg (x^ ,X 3 ,Xg , Xj ) a fte r
A p p lic a tio n of S u ita b le T r a n s p o s i t i o n s ....................
48
F le x ib le M ap p in g E lem en t fo r T r a n s p o s itio n
M ap s (3- v a r i a b l e ) ................................................................
50
F ig u re 3 . 4 . 1
(a) In p u t C a s c a d e
(b)
O u tp u t C a s c a d e
. . . .
52
F ig u re 3 . 4 . 2
(a) In p u t C a s c a d e
(b) O u tp u t C a s c a d e
. . . .
55
F ig u re 3 . 4 . 3
A rray C o n fig u ra tio n on M ap p in g for th e R e a liz e tio n of F =
(xg + X3) + (xg +X g + x^) Xg
F ig u re 3 . 2 . 1
F ig ure 3 . 3 . 2
F ig u re 3 . 3 . 3
+ X2 X3 X 6
.............................................................................................................. 57
viii
F ig u re 4 .1
M u lti- le v e l Bulk T ra n s fe r S y s t e m .............................. 61
F ig u re 4 . 2 . 1
T w o - le v e l N etw o rk w ith T w o -in p u t ULM a t
th e L a s t L e v e l ........................................................................... 65
F ig ure 4 . 2 . 2
T w o - le v e l N etw o rk w ith T h r e e -in p u t ULM a t
th e L a s t L e v e l ............................................
F ig u re 4 . 2 . 3
86
T he S tru c tu re of th e N e tw o rk for T e s t - r e a l i z a t i o n
of th e F u n c tio n of E xam ple 4 . 2 . 2 .................................... 92
F ig u re 4 . 2 . 4
S p e c if ic a tio n of th e N etw o rk to R e a liz e
' th e F u n c tio n of E xam ple 4 . 2 . 2 ................................ 96
F ig u re 4 . 3 . 1
L in e a rly A rranged N o n - d is jo in t S u b -a rra y s . . .
F ig u re 4 . 3 . 2
D e c o m p o s itio n in to Two. S u b - a r r a y s ........................99
F ig u re 4 . 3 . 3
A N e tw o rk to R e a liz e F in E xam ple 4 . 3 . 1
. .
. . 102
F ig ure 4 . 3 . 4
The N etw o rk to R e a liz e F in E xam ple 4 . 3 . 2 .
. . 105
F ig u re 5 . 2 . 1
A T ree N e t w o r k ...................................................................112
F ig u re 5 . 2 . 2
A T ree w ith k - in p u t L ogic E l e m e n t s ...................... 115
F ig ure 5 . 2 . 3
F ig u re 5 . 4 . 1
'
98
A N e tw o rk to R e a liz e F = a b + c d e f + g ................ 116
A N e tw o rk to R e a liz e F - x^Xg + XgX^Xg
+ X 3X4X6
+ X gXgXg
+ X g X y X g ..........................
130
ABSTRACT
In th is , th e s i s the. r e s u l t s of in v e s t ig a ti o n s on a c e l l u l a r b u lk ■
tr a n s f e r s y s te m from th e v ie w p o in t o f i t s lo g i c a l .c a p a b i l i t i e s h a v e
b e e n p r e s e n t e d T h e m o d e l a d o p te d fd'r th e b u lk tr a n s f e r s y s t e m .
c o n s i s t s o f a n in p u t a r r a y , a m ap p in g d e v i c e , a n o u tp u t a rra y a n d a n
o u tp u t lo g ic .. TKe in flu e n c e o f s u c h f a c to r s a s T le x ib ilIty of th e '
m a p p in g d e v ic e , f le x ib ili ty of. o u tp u t lo g ic a n d p a r a lle lis m o f o p e ra tio n
h a s b e e n d e te r m in e d .
The m ain r e s u l t s o b ta in e d are:, th e .b u lk
tr a n s f e r s y s te m c a n be m ade lo g ic a lly u n iv e r s a l w ith a p ro p e r
c o m b in a tio n o f o u tp u t lo g ic a n d m a p s . In r e a liz in g a r b itr a r y lo g ic ,
a tr a d e - o f f am ong th e n u m b e r.o f m ap p in g o p e r a tio n s , n u m b er of
in d e p e n d e n t m ap s a n d a m o u n t o f lo g ic a l f le x ib ility in th e o u tp u t lo g ic
i s p o s s i b l e . A l e a s t u p p e r b o und on th e n u m b er o f n e c e s s a r y
tr a n s p o s it io n m ap s i s d e riv e d fo r a n o u tp u t lo g ic c o n s is tin g .o f a fle x ib le
c e ll u la r c a s c a d e . The p o s s ib il ity o f a s e t o f b u lk tr a n s f e r u n its
o p e ra tin g in' p a r a lle l h a s b e e n s tu d ie d a n d th e fu n c tio n s r e a liz a b le
in t h i s m a n n er h a v e b e e n c h a r a c t e r i z e d . An a lg o rith m fo r t e s t s y n th e s is o f r e a liz a b le fu n c tio n s h a s b e e n p r e s e n te d .
C h a p te r I
INTRODUCTION
-
1 .1
2
-
Introduction
A c e ll u la r
a rra y i s som e g e o m e tric a l a rra n g e m e n t o f c e ll s
in o n e , tw o or th re e d im e n s io n s .
E a c h c e l l in a c e llu la r .a r r a y h a s som e,
lo g ic a l p ro p e rty a n d i t m ay a ls o h a v e so m e s to ra g e c a p a b ility .
of a n a rra y h a v e a uniform in te r c o n n e c tio n s tr u c tu r e .
The c e ll s
B e c a u se of t h i s ,
th e lo g ic d e s ig n e r i s f a c e d w ith a n ew k in d o f c o n s tr a in t in r e a liz in g
a r b itr a r y lo g ic f u n c tio n s .
W ith e x is tin g te c h n iq u e s in c e ll u la r lo g ic
( H)
i t i s n e c e s s a r y to u s e e ith e r a c o m p lic a te d in te rc o n n e c tio n p a tte rn am ong
c e l l s or a v e ry la rg e n u m b er of c e ll s to r e a liz e a rb itra ry lo g ic .
The id e a of a b u lk tr a n s f e r o f d a ta o rig in a te d from th e p ro b lem s' .
o f r e a liz in g a rb itra ry lo g ic fu n c tio n s w ith c e ll u la r a r r a y s . W ith a v ie w
to a v o id in g th e c o m p lic a te d in te r c o n n e c tio n p a t t e r n , i t h a s b e e n p ro p o se d
to tr a n s f e r th e d a ta from o n e c e llu la r a rra y to a n o th e r by som e k in d of
tra n s fe rrin g d e v ic e .
The a d v a n ta g e in d o in g th is i s th a t n e c e s s a r y
in te r c o n n e c tio n s c a n be r e a liz e d by s u ita b ly tra n s fe rrin g th e in p u ts in
th e s e c o n d a r r a y .
P u rsu in g th is l i n e , a c e ll u la r b u lk tr a n s f e r sy ste m
c a n b e c o n c e iv e d w h ic h c o n s i s t s of tw o c e ll u la r a rra y s w h ic h m ay be
re fe rre d to a s in p u t a n d o u tp u t a r r a y s ;
E a c h of th e s e a r r a y s is c a p a b le
o f c o n ta in in g d a ta a n d p o s s ib ly p erfo rm in g lo g ic on i t .
B etw een th e tw o
a r r a y s th e r e i s a d a t a - t r a n s f e r d e v ic e w h ic h m ay tr a n s f e r d a t a .
-3-
a .cc o m p a n ied by som e k in d of tra n s fo rm a tio n .
An e x am p le of a sim p le ty p e
o f tra n s fo rm a tio n i s a p e rm u ta tio n o f th e v a r ia b le s on th e a r r a y .
If n e c e s ­
s a r y , th e d e v ic e m ay h a v e the. c a p a b ility fo r m ore c o m p le x tra n s fo rm a tio n s
T h is d e v ic e m ay be g iv e n th e g e n e r a l n am e 'm ap p in g d e v i c e '.
The m a p p in g m ay be a p p lie d in r e v e r s e d ir e c tio n a n d c a n b e ite r a te d a
lim ite d n u m b er o f tim e s..
D ata m ay b e lo g ic a lly p r o c e s s e d u s in g th e
• -
b u i l t - i n lo g ic in th e c e ll u la r a r r a y s , m a p s in th e m ap p in g d e v ic e an d
if r e q u ir e d , b y a s e p a r a te lo g ic d e v i c e .
A s tu d y o f th e c h a r a c t e r i s t i c s of th is s y s te m r e v e a ls th a t lo g ic a l
u n iv e r s a li ty c a n be a c h ie v e d by a s u ita b le c o m b in a tio n of m ap a n d lo g ic . •
T h is le a d s to m any in te r e s t in g q u e s tio n s re g a rd in g i t s e f f ic ie n c y ,
e f f e c t o f v a r ia tio n in s tr u c tu r e a n d c o n s tr u c tio n , c a p a b ility , an d
p r a c tic a lity a s a co m p u tin g s y s te m an d so o n .
The m o d e l o f th e
b u lk tr a n s f e r s y s te m i s r e la te d to th e p e rc e p tro n s y s te m s tu d ie d by
M in s k y an d P a p e r t ^ ) b u t th e m a jo r i n t e r e s t in th e s y s te m i s b a s e d
/ on a r e a liz a tio n t h a t i t m ay h a v e a n im p o rta n t p la c e in fu tu re d ig ita l
s y s te m s b e c a u s e of a p o s s ib il ity th a t-im p ro v e m e n t in th e c o m p u tin g
p o w e r m ay be a c h ie v e d th ro u g h a m ode o f o p e ra tio n in w h ic h th e
a d v a n ta g e s of in c r e a s e d p a r a lle lis m o f o p e r a tio n s , g r e a te r
h o m o g e n e ity o f th e h a rd w a re s t r u c t u r e , in te rm ix tu re o f s to r a g e .
-4-
a r ith m e tic an d c o n tro l o p e ra tio n s h a v e b e e n c o m b in e d .
The g e n e r a lity o f th e s y s te m a n d i t s p o te n tia l c a p a b iliti e s m ay be
v is u a l iz e d by c o n s id e rin g th e b a s ic s c h e m e s o f th e p a r a lle l p r o c e s s o r s
s u g g e s te d r e c e n tly a n d a tte m p tin g to m irro r th e ir fu n c tio n s in to th e b u lk
tr a n s f e r s y s te m .
M o s t o f th e s e m a c h in e s h a v e u tiliz e d th e b a s ic c o n c e p ts
o f tw o d i s t i n c t p a r a lle l p r o c e s s o r o rg a n iz a tio n p ro p o s e d b y lin g er^ ^
H o lla n d
( 7)
an d
. T h e s e m a c h in e s , w ill b e b rie fly d is c u s s e d in o rd e r to b rin g
o u t fu n c tio n a l s im ila r itie s b e tw e e n t h e s e an d th e b u lk tr a n s f e r s y s te m .
1 .2
The Bulk T ra n s fe r S y stem in. R e la tio n to Some. P a r a lle l
■
P r o c e s s o rs
U n g e r M a c h in e
In 1958 U n g e r d e s c r ib e d a s to re d program c o m p u te r o rie n te d
to w a rd s p a t i a l p ro b le m s . The p a r tic u la r pro b lem il l u s t r a t e d w ith
th is m a c h in e w a s p a tte rn d e te c tio n .
The c o m p u te r c o n s i s t s of a m a s te r
c o n tro l a n d a r e c ta n g u la r a rra y of lo g ic a l m o d u le s e a c h o f w h ic h c an
c o m m u n ic a te w ith i t s n e a r e s t n e ig h b o rs (F ig u re 1 . 2 . 1 ) .
The m a s te r •
c o n tro l c o n ta in s a c lo c k , d e c o d in g c ir c u its a n d a random a c c e s s
m em ory fo r s to rin g in s t r u c t i o n s .
It re a d s o u t in s tr u c tio n s from m em ory, .
d e c o d e s them a n d s e n d s o u t a p p ro p ria te com m ands w h ic h go s im u lta ­
n e o u s ly to a l l th e m o d u le s .
E a c h m o d u le c o n ta in s a o n e - b i t
—5
—
II
M odule
M a s te r
C o n tro l
F ig u re 1 . 2 . 1
The U n g e r M a c h in e
a c c u m u la to r , som e s to ra g e a n d a s s o c i a t e d lo g ic an d w o rk s in p a r a lle l
w ith th e re m a in in g m o d u le s . W ith th e u s e o f a n e le m e n ta ry in s tr u c tio n
s e t U n g e r h a s d e s c r ib e d p ro g ram s to d e t e c t c e r ta in lo c a l a n d g lo b a l
f e a tu r e s o f p a tte r n s on th e tw o d im e n s io n a l f i e l d .
To r e f le c t th e s e f e a tu r e s o f th e U n g e r m a ch in e in th e b u lk tra n s fe r
s y s te m , l e t u s re g a rd th e f ie ld c o n s is tin g of th e m o d u le s a s th e in p u t
a rra y a n d a s s u m e th a t th e r e s u ltin g c o n fig u ra tio n of th e f ie ld on
e x e c u tio n o f o n e in s tr u c tio n w ill be on th e o u tp u t a r r a y .
The in s tr u c tio n
c a n b e d e s c r ib e d by a p ro p e r c o m b in a tio n o f m apping w ith lo g ic .
T hus,
a n ite r a te d p ro c e d u re o f m ap p in g w ith lo g ic m ay c o n s titu te a program fo r
th e d e te r m in a tio n of som e p a r tic u la r fe a tu re of a p a t t e r n . I t a p p e a rs
t h a t th e b u lk tr a n s f e r s y s te m c a n be p o te n tia lly m ore p o w e rfu l in som e
- 6r e s p e c t s th a n th e U n g e r m a ch in e in v ie w o f th e f a c t t h a t .s u ita b le .c h o ic e
o f m ap s in th e m ap p in g d e v ic e m ay e n a b le th e b u lk tr a n s f e r s y s te m to
p r o c e s s a la r g e r p a rt of th e f ie ld th a n th e g ro u p of im m e d ia te n e ig h b o rs .
F u rth e rm o re , so m e o p e ra tio n s c a n b e c o n v e n ie n tly e x e c u te d b y a sim p le
m ap p in g w h e re i t m ay ta k e s e v e r a l in s tr u c tio n s in th e U n g e r m a c h in e to.
do th e s a m e .
On th e o th e r h a n d , th e re m ay be c e r ta in f e a tu r e s in th e ■
U n g e r m a c h in e w h ic h m ay be d if f ic u lt to. im p le m e n t b y m ap p in g in th e
b u lk tr a n s f e r s y s te m , b u t c an be in c o rp o ra te d th ro u g h th e b u i l t - i n
lo g ic o f th e in p u t a n d o u tp u t a r r a y s .
H o lla n d M a c h in e '
The H o lla n d m a c h in e c o n s i s t s o f a tw o -d im e n s io n a l a rra y of
m o d u le s . E a c h m odule i s a s m a ll g e n e r a l p u rp o s e c o m p u te r a n d c a n
b e a c ti v e o r in a c tiv e a t a g iv e n tim e .
W h en a c t i v e , a m o d u le tr e a ts
th e c o n te n ts of i t s s to ra g e r e g is te r a s - a n in s tr u c tio n .
A fter d e te rm in in g
th e lo c a tio n o f th e o p e r a n d , a p a th is b u ilt to a c c e s s th e d a t a .
The
in s tr u c tio n in th e m o d u le i s th e n e x e c u te d a n d th e a c tiv e s t a t u s is
tr a n s f e r r e d to one o f th e fo u r n e a r e s t n e ig h b o rin g m o d u le s in th e a rra y . .
In s tru c tio n s , c a n b e a rra n g e d s p a ti a lly th ro u g h o u t th e a rra y o f m o d u les
w ith a n a r b itr a r y n u m b er o f t h e s e e x e c u te d a t th e sam e t i m e .
-
7
-
R e fle c tin g on th e B. T. s y s te m , it a p p e a r s t h a t , th o u g h a n y d a ta
tra n s fo rm a tio n p o s s ib le in th e H o lla n d m a c h in e c a n b e d o n e b y p erfo rm in g .
s u ita b le m a p p in g , a m o d ific a tio n o f th e s tru c tu re o f th e m a p p in g d e v ic e
m ay be m ore s u ite d to perform th e k in d o f p a r a lle l c o m p u ta tio n th a t th e
H o lla n d m a c h in e i s a b le to d o .
L et th e in p u t a rra y b e th e p ro p e r
c o n fig u ra tio n c o n ta in in g d a ta a n d in s t r u c t i o n s .
The m a p p in g d e v ic e
m ay be d iv id e d in to m any s e p a r a te e le m e n ts , e a c h h a v in g .th e c a p a b ility
o f s c a n n in g a lim ite d a re a of a n y re g io n in th e in p u t a r r a y .
The
e x e c u tio n of a n in s tr u c tio n m ay c o n s i s t of a tra n s fo rm a tio n of d a ta , f i r s t b y
m a p p in g a n d th e n by th e b u i l t - i n lo g ic of th e a r r a y s .
If th e o p e ra n d *
o f a n in s tr u c tio n b e lo n g s to a n .a r e a d iff e r e n t from th a t s c a n n e d b y a
m ap p in g e le m e n t, a s e q u e n c e of m a p p in g -o p e ra tio n s m ay b e u n d e rta k e n to
b rin g th e d a ta in to th e p ro p e r a r e a , th u s s im u la tin g th e p a th - b u ild in g an d
d a t a - a c c e s s p ro c e d u re in th e H o lla n d m a c h in e . T here c o u ld b e o th e r
v a r ia tio n s o f th is s im u la tio n p ro c e d u re in c lu d in g a g lo b a l m a p p in g .
T w o -d im e n s io n a l C o m p u tin g T e c h n iq u e
The b u lk tr a n s f e r s y s te m i s s tr u c tu r a lly o rie n te d to w a rd p a r a lle l
c o m p u ta tio n in th e sa m e m a n n er a s th e tw o -d im e n s io n a l ite r a tiv e n e tw o rk
c o m p u tin g te c h n iq u e o f H a w k in s a n d M u n sey ^ ? ) .
The co m p u tin g
m a c h in e d e v is e d by t h e s e a u th o rs c o n s i s t s of tw o d a ta p l a n e s ,
-8“
c a lle d th e in p u t p la n e a n d r e s u lta n t p la n e (F igure 1 . 2 . 2 ) .
in te rv e n in g p la n e c a lle d th e m ap p in g m a s k .
T here is a n
D ata on th e in p u t p la n e c an
b e p r o c e s s e d th ro u g h th e u s e of th e m a sk an d p ro je c te d o n to th e r e s u lta n t
p la n e .
U sin g o p tic a l te c h n iq u e s a n d lin e a r th re s h o ld lo g ic to p ro c e s s
s p a ti a lly d is tr ib u te d d a ta on th e in p u t p la n e , th e a u th o rs h a v e show n
th a t th e s y s te m i s c a p a b le o f re c o g n iz in g c e r ta in tin y o b je c ts a g a in s t
th e b a c k g ro u n d of o th e r la rg e o b je c t s .
In p u t
M ask
R e s u lta n t
F ig u re 1 . 2 . 2 O p tic a l Sum m ation S y stem
( x 's a re in p u t v a r i a b l e s , g 's a re tr a n s m itta n c e s )
-9-
From th e a b o v e d i s c u s s i o n s i t s h o u ld b e a p p a r e n t th a t g iv e n th e
p ro p e r te c h n o lo g ic a l b a c k g ro u n d , th e b u lk tr a n s f e r s y s te m h a s th e
p o t e n tia litie s o f b e c o m in g a n im p ro v e d , e c o n o m ic a n d v e r s a t i l e d a t a p r o c e s s in g d e v ic e .
The te c h n o lo g ic a l a d v a n c e in th e f ie ld o f b a tc h
f a b r ic a tio n te c h n iq u e h a s m ade i t p o s s ib le to p ro d u c e r e lia b le an d
e c o n o m ic a l a rra y s o f lo g ic c e l l s on a m a s s s c a le
. C u rre n t r e s e a r c h ( 1 )
on th e u s e of o p tic a l te c h n iq u e s fo r d a ta tr a n s f e r on a la rg e s c a le p o in ts
to a d ir e c tio n in w h ic h th e p r a c tic a l r e a liz a tio n of a b u lk tr a n s f e r
s y s te m m ay se e m f e a s i b l e .
W ith t h e s e a s p e c t s in v ie w , a s tu d y on th e lo g ic a l c a p a b iliti e s
o f th e b u lk tr a n s f e r s y s te m h a s b e e n p ro p o s e d in th is t h e s i s . .The
c o n te n ts a n d o r g a n iz a tio n of th e t h e s i s i s g iv e n in th e fo llo w in g s e c tio n .
1 .3
O rg a n iz a tio n of th e R em aining C h a p te rs
C h a p te r 2 d i s c u s s e s th e lo g ic a l c a p a b ility of a sim p le
b u lk tr a n s f e r s y s te m .
The r e s u lts o b ta in e d in th is c h a p te r form th e
b a s i s of fu rth e r d e v e lo p m e n t on th is to p ic in C h a p te r 3 .
—
/
In th is
c h a p te r th e e f f e c t of lim ite d f le x ib ility in th e o u tp u t lo g ic i s s tu d ie d
a n d r e s u l t s o n b u lk tr a n s f e r w ith c e ll u la r c a s c a d e s a re r e p o r te d .
-
10
-
C h a p te r 4 d e a ls w ith th e c h a r a c te r iz a tio n o f r e a liz a b le fu n c tio n s
u s in g p a r a lle l b u lk tr a n s f e r te c h n iq u e .
It c o n ta in s a n a lg o rith m fo r th e
t e s t - s y n t h e s i s of fu n c tio n s u s in g p a r a lle l b u lk tr a n s f e r on th e a s s u m p tio n
th a t th e p a rtio n in g o f th e in p u t a rra y is f ix e d .
In C h a p te r 5 a s im ila r
a lg o rith m on th e a s s u m p tio n o f f le x ib ility in th e p a ritio n in g of th e in p u t
a r r a y h a s b e e n d e v e lo p e d .
C h a p te r 6 g iv e s a sum m ary o f th e fo re g o in g c h a p te r s fo llo w e d by
a d i s c u s s i o n a b o u t th e s c o p e of fu rth e r r e s e a r c h w ork in th e a r e a .
C h a p te r 2
LOGICAL CAPABILITY OF A BULK TRANSFER SYSTEM
“ 122 .1
A Bulk T ra n s fe r S ystem
An id e a o f th e p o s s ib le fu n c tio n of a b u lk tr a n s f e r s y s te m h a s
b e e n g iv e n in th e p re v io u s c h a p te r .
S u ch a s y ste m w ill b e d e s c r ib e d in
th is c h a p te r an d th e n in v e s tig a tio n s in to i t s lo g ic a l c a p a b iliti e s w ill
be m ade.
A b lo c k d ia g ra m o f th e s y s te m is sh o w n in F ig u re 2 . 1 . 1 .
The
b lo c k c a lle d In p u t A rray m ay be c o n c e iv e d a s a r e c ta n g u la r p la n e
d iv id e d in to m any r e c ta n g u la r c e l l s . E a c h c e ll i s a s to ra g e d e v ic e
w h ic h c a n ta k e on one o f th e tw o in p u t v a l u e s , I o r 0 .
C o rre sp o n d in g
to e a c h in p u t c e l l th e re i s a b in a ry v a r ia b le a n d a n a s s ig n m e n t of
in p u t v a lu e s to a l l th e c e l l s r e p r e s e n ts o n e of 2n p o s s ib le in p u t
c o n f ig u r a tio n s , w h e re n i s th e n u m b er of c e l l s in th e in p u t a rra y .
In p u t A rray
M ap p in g D e v ic e O u tp u t Array
F ig u re 2 . 1 . 1
O u tp u t L ogic
A Bulk T ra n s fe r S ystem
—13
—
A s e c o n d c o m p o n e n t o f th e b u lk tr a n s f e r s y s te m i s th e b lo c k
c a lle d th e Ou tp u t A rray : i t i s a p la n e id e n tic a l to th e in p u t a rra y .
The c o n te n ts o f th e in p u t c e ll s a re m a p p ed (tra n sfe rre d ) o n to th e c e lls
o f th e o u tp u t a r r a y .
F or e x a m p le , a s to re d b it in a c e ll (i , j ) in th e
in p u t a rra y m ay b e m a p p ed in to a c e ll (m, k) in th e o u tp u t a r r a y .
The th ir d c o m p o n e n t i s c a lle d th e M ap p in g D e v ic e : i t im p le m e n ts '
th e m ap p in g o p e ra tio n b e tw e e n th e in p u t a n d o u tp u t a r r a y s . The
m a p p in g m ay in c lu d e lo g ic (F igure 2 . 1 . 2 ) o r m ay n o t.
A m ap p in g is
c a lle d lo g ic - f r e e if th e v a lu e of a n in p u t b it is n o t c h a n g e d d u rin g
m a p p in g .
F o r th e p r e s e n t d i s c u s s i o n , s u c h a m ap p in g i s a lw a y s
o n e - to - o n e ;
th a t i s , e a c h o u tp u t c e ll g e ts a s ig n a l from a t m o st o ne
L o g ic a l E lem en t
In p u t A rray •
F ig u re 2 . 1 . 2
M ap p in g D e v ic e •
O u tp u t A rray
M ap p in g D e v ic e w ith L ogic
-
in p u t c e l l .
14
-
(N ote t h a t a l l m a p p in g s h a v e b e e n a s s u m e d to b e o n to , t h a t
i s , a n o u tp u t c e l l g e ts a s ig n a l from a t l e a s t one in p u t c e l l . )
A
lo g ic a l m ap p in g i s o n e t h a t in v o lv e s tra n s fo rm a tio n o f th e in p u t b its
d u rin g m a p p in g (for e x a m p le , c o m p le m e n ta tio n ).
B e s id e s b e in g o n e -
to - o n e , a lo g ic a l m a p p in g m ay be m a n v - t o - o n e . w h e re a n u m b er of
in p u t b its co m b in e w ith so m e lo g ic to p ro d u c e a b in a ry s ig n a l w h ic h
i s th e n m ap p ed in to a n o u tp u t c e l l .
A fter o n e m a p p in g , th e o u tp u t a rra y c a n be id e n tic a ll y m apped
b a c k in to th e in p u t a rra y fo r a s e c o n d m a p p in g from in p u t a rra y to
o u tp u t a r r a y .
A ssum e t h a t th e p re v io u s a s s ig n m e n t of s w itc h in g
v a lu e s to th e c e l l s in a n a rra y i s a u to m a tic a lly c le a r e d w h e n a new
m ap p in g to th e a rra y i s p e rfo rm e d . ' A ls o , a s s u m e th a t a m ap p in g from
th e in p u t a rra y to th e o u tp u t a rra y c a n b e re p e a te d a n y n u m b e r of tim e s
by id e n tic a ll y m ap p in g b a c k from th e o u tp u t a rra y to th e in p u t a rra y
a f te r e a c h m ap p in g from th e in p u t a rra y to th e o u tp u t a r r a y .
H ence-
' f o r th , by "m a p p in g " th e m ap p in g from in p u t a rra y to o u tp u t a rra y w ill
b e m e a n t, u n l e s s o th e rw is e s p e c if ie d , b e c a u s e th e r e v e r s e m ap p in g
i s a s s u m e d a lw a y s to be th e id e n tity m a p p in g .
.The m ap p in g d e v ic e i s c a p a b le of p ro d u c in g a lim ite d n um ber
of in d e p e n d e n t m a p p in g s , n o n e of w h ic h c a n b e p ro d u c e d b y a n y
c o m p o s itio n of o th e r m a p p in g s th a t th e d e v ic e im p le m e n ts .
.
-
15
-
The f in a l c o m p o n e n t o f th e b u lk tr a n s f e r s y s te m i s c a lle d th e
O u tp u t L o g ic : i t i s a d e v ic e .that p erfo rm s a lo g ic a l fu n c tio n on th e
‘ I
'
o u tp u t a rra y to p ro d u c e a n o u tp u t v a lu e .
A ssum e th a t d u rin g c o m p o site
m ap p in g th i s d e v ic e d o e s n o t o p e ra te u n til th e en d o f c o m p o s itio n .
The lo g ic of th e d e v ic e m ay be f le x ib le so th a t d iff e r e n t fu n c tio n s
o f i t s in p u ts c a n b e o b ta in e d .
N ote th a t a n a s s ig n m e n t of v a lu e s to th e in p u t a rra y of n c e ll s
r e p r e s e n ts a m interm of n v a r ia b le s w h ic h i s c o n v e rte d in to a n o th e r
m interm in th e o u tp u t a rra y a f te r m a p p in g .
'm in te rm s o f n v a r i a b l e s .
C o n s id e r th e s e t M o f a ll
C le a r ly , th e m ap p in g d e v ic e c a n p ro d u c e
a m ap p in g from M in to M .
L et a m interm m^ a f te r a m ap p in g #
c h a n g e d to a n o th e r m interm r e p r e s e n te d b y m^a
w h e n s u b je c te d to th e m ap
• re p re s e n te d a s m ^ a
2
a
. If m ^ a
be
. The sa m e m interm
tw ic e , i s c h a n g e d to .a m interm
2
i s d iff e r e n t from m ^h
a n d th e
v a lu e s o f th e o u tp u t fu n c tio n fo r th e tw o m in te rm s a re d if f e r e n t, th e n
e v id e n t ly , ■a n ew fu n c tio n ( i . e . , a fu n c tio n d iff e r e n t from th a t
o b ta in a b le a f te r s in g le m apping) i s g e n e r a te d a t th e o u tp u t by one
r e p e titio n o f th e
a
-m a p .
T he b u lk tr a n s f e r sy stem , m ay b e lo o k e d u p o n a s a .d e v ic e fo r
p ro d u c in g d if f e r e n t lo g ic a l fu n c tio n s o f a s e t of in p u t v a r i a b l e s ,
—16"-
u s in g a s e t o f m ap s
{a
^ , a £/ • • • a a n d
a n o u tp u t lo g ic d e v ic e .
We
s h a ll s tu d y th e lo g ic a l c a p a b i l i t i e s o f th e s y s te m u n d e r d iff e r e n t ty p e s
of m a p p in g a n d o u tp u t lo g ic .
In th is c o n n e c tio n , som e o f th e b a s ic
id e a s a n d te rm in o lo g y o f G roup T h e o r y ^ ) w ill o fte n b e u s e d .
2 .2
L o g ic a l C a p a b ility o f th e Bulk T ra n s fe r System -
P e rm u ta tio n M ap s
L et th e s e t o f 2n m in te rm s b e a rra n g e d in som e o rd e r a s
(m, ,m , . . .m J . If i t w e re p o s s ib le to p ro d u c e a l l p o s s ib le p e rm u ta 1 2
2
tio n s of th e s e t of m in term s w ith th e h e lp o f th e lim ite d n u m b er of
in d e p e n d e n t m aps o f th e b u lk tr a n s f e r ( B. T. ) s y s te m , th e n , w ith an o u tp u t
lo g ic d e v ic e of v e ry lim ite d f le x ib ili ty , a n e x tre m e ly la rg e n um ber of
fu n c tio n s c o u ld be g e n e r a te d , b e c a u s e , by p erm u tin g th e s e t of m in term s
a d if f e r e n t fu n c tio n c o u ld be o b ta in e d a t th e o u tp u t w ith o u t c h a n g in g
th e o u tp u t fu n c tio n o f th e o u tp u t d e v ic e .
i s d ir e c te d to w a rd th is g o a l.
So th e a tte m p t in th e fo llo w in g
T heorem 2 . 2 . 1 i s a r e s u l t o f th e fo re ­
g o in g d i s c u s s i o n , an d so is g iv e n w ith o u t p ro o f.
Theorem 2 . 2 . 1:
A o n e - to - o n e m ap o f th e in p u t a rra y in a b u lk
tr a n s f e r s y s te m p ro d u c e s a p e rm u ta tio n o f th e s e t o f m in te rm s .
A lo g ic - f r e e m ap p in g of th e in p u t a rra y i s e f f e c tiv e ly a g e o m e tric a l
re p o s itio n in g of th e in p u t b i t s .
T h is ty p e o f m ap p in g h a s lim ita tio n s
-
17
-
in th e m a tte r of p ro d u c in g a r b itr a r y p e rm u ta tio n s of th e s e t o f m in term s
a s th e fo llo w in g th e o re m w ill sho w :
I
,
Theorem 2 . 2 . 2 :
'
.
'
W ith a l l p o s s ib le lo g ic - f r e e o n e - to - o n e m a p p in g s
of th e in p u t a r r a y , i t i s n o t p o s s ib le to p ro d u c e a l l p e rm u ta tio n s of th e
s e t o f m in te rm s .
P ro o f: A lo g ic - f r e e m ap p in g c a n n o t m ap a m interm of in d e x n u m b er i
(num ber o f I ' s in th e m interm ) in to a m interm o f in d e x n u m b e r j , w h e re
i
j.
T h e re fo re , s u c h a m ap p in g c a n n o t p ro d u c e a l l p o s s ib le
p e rm u ta tio n s .
Q .E . D .
The ty p e s a n d th e t o t a l n u m b er of p e rm u ta tio n s t h a t a l l lo g ic fre e o n e - to - o n e m a p s p ro d u c e w ill b e g iv e n u n d e r T heorem 2 . 2 . 5 .
S u p p o se th e s e t of m in te rm s is i d e n tic a ll y m ap p ed an d th e fu n c tio n
p ro d u c e d by th e o u tp u t lo g ic (fixed) i s f .
C an a d iff e r e n t fu n c tio n
b e p ro d u c e d a t th e o u tp u t by re s o r tin g to a d iffe re n t p e rm u ta tio n of th e
s e t of m in te rm s ?
C o n s id e r a p e rm u ta tio n o f th e s e t of m in te rm s
w h e re tw o m in te rm s m. a n d m. a re m a p p ed o n e in to th e o th e r w h ile
i
I
th e r e s t a re m ap p ed i d e n t i c a l l y .
If th e o u tp u t lo g ic is s u c h th a t b o th
m^ a n d my a re tru e or b o th f a l s e , th e n , w ith th is p e rm u ta tio n , no new
fu n c tio n c a n be p ro d u c e d .
The c o n d itio n fo r p ro d u c in g a n ew fu n c tio n
i s g iv e n in th e fo llo w in g th e o re m .
I
—
T heorem 2 . 2 . 3 :
18
-
L et f be th e fu n c tio n p ro d u c e d w ith id e n tity
m ap p in g b y a b u lk tr a n s f e r s y s te m h a v in g a fix e d o u tp u t lo g ic .
Then
w ith a n y m ap p in g o f th e in p u t a rra y w h ic h p ro d u c e s a p e rm u ta tio n o f
th e s e t o f m in te rm s , i t i s p o s s ib le to p ro d u c e a new fu n c tio n a t th e
o u tp u t if an d o n ly if th e c o rre s p o n d in g p e rm u ta tio n o f th e s e t of
m in te rm s m ap s a t l e a s t o n e tru e (fa ls e ) m interm in to a f a l s e (true)
m in te rm .
P roof: The p ro o f fo llo w s e a s i l y from e a r l i e r d i s c u s s io n s a n d s o is
o m itte d .
Theorem 2 . 2 . 4 g iv e s th e m axim um n u m b e r o f fu n c tio n s r e a liz a b le by
a b u lk tr a n s f e r s y s te m w ith a fix e d m ap a n d a fix e d o u tp u t lo g ic .
T heorem 2 . 2 . 4 :
A b u lk tr a n s f e r s y s te m w ith a s in g le o n e - to - o n e
m ap a n d w ith fix e d o u tp u t lo g ic c a n r e a liz e a t m o st k f u n c tio n s w h e re
k i s th e o rd e r o f th e c o rre s p o n d in g p e rm u ta tio n of th e s e t o f m in te rm s.
P ro o f: S in c e w ith r e p e a te d a p p lic a tio n s o f th e m a p , o n ly k d iffe re n t
p e rm u ta tio n s of th e s e t o f m in te rm s a re p ro d u c e d , h e n c e , if e v e ry
p e rm u ta tio n g e n e r a te d a new f u n c tio n , a t m o st k fu n c tio n s c o u ld be
p ro d u c e d .
Q .E . D .
-19-
A cco rd in g to th is th e o re m , o u t o f 25 6 fu n c tio n s o f th re e v a r i a b l e s ,
a t m o s t 15 fu n c tio n s m ay be r e a liz e d by a b u lk tr a n s f e r s y s te m w ith a
s in g le m ap a n d o u tp u t lo g ic .
If th e r e i s no r e s tr ic tio n a b o u t th e n u m b er
of lo g ic - f r e e m ap s th a t th e m ap p in g d e v ic e c a n im p le m e n t, th e n th e
m axim um n u m b er o f f u n c tio n s th a t a b u lk tr a n s f e r .s y ste m w ith fix e d
o u tp u t lo g ic c a n r e a l i z e , i s g iv e n b y th e fo llo w in g th e o re m :
T heorem 2 . 2 . 5 :
A b u lk tr a n s f e r s y s te m w ith a fix e d o u tp u t lo g ic
an d c a p a b le of p ro d u c in g a l l lo g ic - f r e e o n e - to - o n e m a p s c a n r e a liz e
a t m o st
n
G)
■ fu n c tio n s
i=0
w h e re
n = n u m b e r o f in p u t c e l l s ,
i = in d e x n u m b er of a m in te rm ,
Uj= n u m b er o f tru e m in te rm s in th e
o u tp u t lo g ic fu n c tio n th a t h a v e
in d e x n u m b e r i , w h e re Og a^gf
P ro o f: L et th e s e t o f 2n m in te rm s b e d iv id e d in to s u b s e t s s u c h th a t
a s u b s e t Gi c o n ta in s a l l m in term s of in d e x n u m b er i . T h e re w ill be
(n+1) -su ch s u b s e t s , i ra n g in g from 0 to n .
A s u b s e t Gi w ill c o n ta in
m e m b e rs . If th e o u tp u t lo g ic i s s u c h th a t o n ly Oi m in te rm s of
-20-
m in te rm s 9 re tr u e , th e n th e n u m b er of p o s s ib le c o m b in a tio n s of
th e s e
t h e ^e
m in te rm s is
.
N ow , in lo g ic - f r e e m a p p in g , th e
p e rm u ta tio n of th e s e t of m in term s is r e s tr ic te d s o th a t a m interm
o f in d e x n u m b er i i s m ap p ed to a n o th e r m interm of sam e in d e x
n u m b e r.
If w e a s s u m e a l l p o s s ib le p e rm u ta tio n s w ith in th e s u b s e t
of m in te rm s of sa m e in d e x n u m b er th e n th e nu m b er o f d iff e r e n t
fu n c tio n s th a t c a n b e g e n e r a te d by p e rm u ta tio n s o f th e s u b s e t
.
is
Now e a c h o f t h e s e w a y s of g e n e ra tin g a fu n c tio n c a n be
lin k e d w ith e a c h w a y of g e n e ra tin g a fu n c tio n by th e m em b ers of a
d iff e r e n t s u b s e t G j .
is
So th e m axim um n u m b er of fu n c tio n s p o s s ib le
. Q . E . D ..
' I t h a s b e e n s e e n t h a t th e s e t o f a l l lo g ic - f r e e o n e - to - o n e m ap s
c a n n o t p ro d u c e a l l p e rm u ta tio n s o f th e s e t of m in te rm s .
W ith a s e t of
lo g ic a l m a p s , h o w e v e r, i t i s p o s s ib le to p ro d u c e a l l p e rm u ta tio n s .
G iv e n a l l p o s s ib le p e rm u ta tio n s o f M , i t i s s t i l l n o t p o s s ib le to r e a liz e
a l l f u n c tio n s o f n v a r ia b le s u n le s s th e o u tp u t lo g ic i s s u f f ic ie n tly
*
f l e x i b l e . The f le x ib ili ty of th e o u tp u t lo g ic i s a n e s s e n t i a l f a c to r in
'
-21th e r e a liz a tio n of a r b itr a r y lo g ic w ith a b u lk tr a n s f e r s y s te m c a p a b le
o f p ro d u c in g o n ly p e rm u ta tio n m a p s .
The fo llo w in g th e o re m s t a t e s
t h is :
T heorem 2 . 2 . 6 :
In o rd e r to p ro d u c e a rb itra ry fu n c tio n s o f in p u t
v a r i a b l e s , a b u lk tr a n s f e r s y s te m h a v in g th e m e an s to p ro d u c e a l l
p e rm u ta tio n s of th e s e t o f m in te rm s m u s t have, a n o u tp u t lo g ic of
s u f f ic ie n t f le x ib ili ty so th a t a t l e a s t (2n + I) d iff e r e n t fu n c tio n s c a n
b e p ro d u c e d by th e o u tp u t lo g ic d e v i c e .
P ro o f: Let f be th e fu n c tio n p ro d u c e d b y th e fix e d o u tp u t lo g ic
d e v ic e o f a B. T . s y s te m u n d e r id e n tity m ap w h e re r is n u m b er o f tru e
m in te rm s . .W ith a l l p o s s ib le p e rm u ta tio n s of M , th e t o t a l nu m b er o f
f u n c tio n s r e a liz e d i s
(2 | . W ith th e fix e d o u tp u t lo g ic an d a s e t
\r j
o f m ap s to p ro d u c e a l l p e rm u ta tio n s o f M , i t is n o t p o s s ib le to c h a n g e
r , th e n u m b er o f tru e m in te rm s .
So in o rd e r to r e a liz e a rb itra ry f u n c t i o n s , th e o u tp u t lo g ic m u st
b e f le x ib le a n d c a p a b le o f p ro d u c in g e a c h fu n c tio n f^, i = 0 , . I , . . . 2 n .
T h e se fu n c tio n s a re l i s t e d b e lo w :
(1)
fy = f a l s e fo r a l l m in te rm s
(2)
f j = tru e fo r o n ly o n e m in term
r
-
(3)
' (2
+1)
(2n + I)
f
22
-
= tru e fo r o n ly tw o m in term s
f^ r = tru e fo r o n ly 2r m in te rm s
f^ n = tru e fo r a l l m in te rm s
The to ta l n u m b er o f d i s t i n c t fu n c tio n s t h a t c a n b e r e a liz e d by th e B. T.
s y s te m in th is c a s e i s
+ ..+
+.. +
on
= 2Z
Q.E.D.
A cco rd in g to th is th e o re m , th e n u m b er o f f u n c tio n s re q u ire d o f th e
o u tp u t lo g ic h a s b e e n re d u c e d to (2n + I)
from 22n fo r s y s te m s w ith o u t
u s in g b u lk tr a n s f e r , a v a lu e p ro p o rtio n a l to th e lo g a rith m o f th e
o rig in a l n u m b e r.
W e c a n fu rth e r im p ro v e u p o n th is an d d r a s t i c a l l y
,
re d u c e th e n u m b er o f re q u ire d f u n c tio n s of th e o u tp u t lo g ic w ith th e
:
u s e of a s p e c i a l m ap to b e d i s c u s s e d h o w .
!
'
M a n y - to - o n e M ap
'
j
The p re c e d in g d is c u s s io n w a s c o n c e rn e d w ith p e rm u ta tio n s of
i
th e s e t of m in te rm s a n d , th e r e f o r e , in v o lv e d o n ly th o s e m a p p in g s o f
?
th e in p u t a rra y t h a t p ro d u c e d p e r m u ta tio n s .
C o n s id e r th e m a n y -to -o n e
lo g ic a l m ap s th a t m ap th e s e t o f m in te rm s o n to o n e of i t s p ro p e r s u b ­
s e ts .
In a s p e c ia l c a s e , a s u f f ic ie n t n u m b e r of r e p e titio n s o f s u c h
,
)
—23 —
' a m ap m ay r e s u l t in a s itu a tio n in w h ic h a l l th e m in te rm s w ill b e m ap p ed
o n to a s in g le m in te rm .
L et JEf b e a m ap p in g s u c h t h a t a m interm rrn i s
m a p p ed in to m^+ -^ e x c e p t th e l a s t m in term m n w h ic h i s m ap p ed in to
its e lf.
The m ap p in g
0^,
k g. (2n - l ) ( i . e . ,
0 is
r e p e a te d k - tim e s )
m ap s (k+T) m in te rm s o n to m n .
2
If th e m ap p in g d e v ic e of th e b u lk tr a n s f e r s y s te m i s .c ap a b le of
p ro d u c in g
0 a lo n g
w ith a r b itr a r y p e rm u ta tio n , an d th e o u tp u t lo g ic is
fix e d a n d p ro d u c e s o n ly th e fu n c tio n f^ w h e re f^ = tr u e , o n ly w h en m^n
i s t r u e , th e n e v e ry fu n c tio n l i s t e d in th e p ro o f o f T heorem 2 . 2 . 6
e x c e p t th e z e ro fu n c tio n c a n be p ro d u c e d w ith th e u s e o f a s u ita b le
n u m b er of r e p e titio n s of th e 0 - m a p .
F o r e x a m p le ,
0 a p p lie d
th ric e
w ill g e n e r a te th e fu n c tio n f^ (the fu n c tio n i s tru e fo r fo u r m in te rm s ).
T h e re fo re , th e re q u ire d f le x ib ili ty of th e o u tp u t lo g ic c o m e s dow n to
tw o fu n c tio n s :
1) fg (z e ro -fu n c tio n )
2) f^ (fu n c tio n i s tru e fo r o n ly o n e m in te rm , .m^n) •
I t h a s b e e n s e e n t h a t to p ro d u c e a l l p o s s ib le p e rm u ta tio n s o f th e
s e t o f m in te rm s , lo g ic a l m ap s m u s t b e u s e d .
The fo llo w in g lem m a
g iv e s th e m inim um n u m b er o f in d e p e n d e n t lo g ic a l m a p s n e c e s s a r y to
a c h ie v e t h i s .
-24-
Lemma 2 , 2 . 1 :
The tw o p e rm u ta tio n s (m ^ m ^ )# (m^ ,m ^ , . . .m ^, , . . m ^n )
c a n g e n e r a te a l l p e m u ta tio n s o f th e s e t of m in te rm s .
P ro o f: It i s a w e ll-k n o w n r e s u lt in G ro u p T h e o ry .
T e x ts on G roup Theory^ 6 )
m ay be s e e n fo r p r o o f .
In c o rp o ra tin g th e p re c e d in g r e s u l t s , w e h a v e :
T heorem 2 . 2 . 7 :
A b u lk tr a n s f e r s y s te m w ith a m ap p in g d e v ic e
c a p a b le of p ro d u c in g tw o p e rm u ta tio n s (m^ , m g ) ,
a n d th e ma p
0,
(m^ , rr^ , . . . nn , . . . m
2
a n d a f le x ib le o u tp u t lo g ic c a p a b le of p ro d u c in g tw o
f u n c t i o n s , f^ a n d f ^ , c a n r e a liz e a n y fu n c tio n of n v a r i a b l e s .
P ro o f: The p ro o f fo llo w s from p re c e d in g th e o re m s an d r e m a rk s .
N o te t h a t th e f le x ib ili ty of th e o u tp u t lo g ic i s n o t n e c e s s a r y
h e re ; th e o u tp u t lo g ic m ig h t b e m ad e fix e d to p ro d u c e o n ly th e fu n c tio n
f^.
In th is c a s e , a n o th e r
0 m ap,
m a p p in g e a c h m interm o n to a f a ls e
m interm i s n e c e s s a r y fo r r e a liz in g th e z e ro fu n c tio n .
2 .3
A S im ple D e sig n of th e Bulk T ra n s fe r S y stem
A d e s ig n fo r a b u lk tr a n s f e r s y s te m c a p a b le o f a rb itra ry
lo g ic r e a liz a tio n w ill b e g iv e n h e r e .
s e t o f m in te rm s in som e o rd e r.
The f i r s t s te p i s to a rra n g e th e
The fo llo w in g o rd e r (T able 2 . 3 ) i s
-25-
c o n v e n ie n t fo r th e c a s c a d e d c ir c u it w e u s e h e r e .
F o r c o n v e n ie n c e ,
th e n u m b e r o f v a r ia b le s i s c o n fin e d to t h r e e .
x
*
X
I
m
2
0
I
0
0
0
I
0 .
0
0
4
m
m
6
r c ij
3
m
5
I
0
I
0
I
I
0
0
I
I
0
I
I
I
I
CO
m
T h r e e - v a r ia b le M in term s
3
CO
T a b le 2 .3
I
.
2
9
The d e s ig n d e p e n d s on th e c h o s e n o rd e r of m in te rm s .
sequence
H e n c e th e
{ m ^ , m ^ , . . - m^n I ^ a lw a y s w ith r e s p e c t to a p a r tic u la r s y s te m
T h is s e q u e n c e is re fe rre d to a s min te rm s e q u e n c e .
F ig u re 2 . 3 . 1
s h o w s th e c ir c u it fo r g e n e ra tin g th e p e rm u ta tio n
c y c le (m. , m 9 , .'. . m ).
I
z
2n
E ach lo g ic a l u n it i s a 2 - i n p u t , 2 - o u tp u t
d e v i c e . The lo g ic u s e d fo r c o m p u tin g th e n e x t s ta t e o f a v a ria b le
• Xfi, d e s ig n a te d b y
is g iv e n by
X1
X1;
x 2)
For " S 3 ,
(x ^ ©
x j + X1n ^ x ji
The c ir c u it to g e n e r a te
0 uses
som e a d d itio n a l lo g ic w ith th e p re v io u s
c irc u it.
F ig u re 2 . 3 . 2 sh o w s th e c i r c u i t . The lo g ic e q u a tio n s a re ;
—2 6 -
xI
x3
xn
F ig u re 2 . 3 . 1
The C irc u it fo r G e n e ra tin g th e P erm u tatio n
. . .m n )
-27-
C ir c u it a s in F ig . 2 . 3 . 1
*3
F ig u re 2 . 3 . 2
The C irc u it fo r G e n e ra tin g th e M ap
0
-28-
x i = x X + x Xx 2 - - x V x IIi x 2 ■ ( x i ® x 2> - t
F o r HS 3 , x ; = X ^ 1 (Xn ^i e x n) +
x I x 2 - - x X - - x H+
X ft.
.X1. .Xn
The d e r iv a tio n o f t h e s e e q u a tio n s a re g iv e n in A p pendix A.
The c ir c u it fo r g e n e r a tin g th e p e rm u ta tio n (m ^ ,rr^) i s g iv e n in
F ig u re 2 . 3 . 3 . The o u tp u t lo g ic i s b u ilt u p of a c a s c a d e of AND g a te s
(2- in p u t , o n e o u tp u t) e x c e p t th e l a s t g a te w h ic h i s f le x ib le a n d c a n
p ro d u c e th e z e ro fu n c tio n b e s id e s th e A N D -operation .
2 .4
D e te rm in a tio n o f th e N e c e s s a r y M ap p in g O p e ra tio n s
In o rd e r to r e a liz e a n a r b itr a r y s w itc h in g f u n c tio n , th e
m ap p in g o p e r a tio n s on th e in p u t th a t w ill b e n e c e s s a r y to g e n e ra te
th e fu n c tio n m u st b e k n o w n .
The fo llo w in g i s a s y s te m a tic
p ro c e d u re fo r d e te rm in in g th e n e c e s s a r y m a p s .
I t i s a s s u m e d h e re
t h a t th e fu n c tio n i s a v a il a b le in tru th ta b le form:
S te p I :
N ote th e n u m b er of tru e m in te rm s .
If i t i s th e z ero
fu n c tio n ,- th e n s e t th e o u tp u t lo g ic .to p ro d u c e th e z e ro fu n c tio n (fQ)
a n d do no m o re .
In t h i s c a s e , a n y m a p p in g of in p u t a rra y w ill
p ro d u c e th e z e ro fu n c tio n .
.S te p 2: If th e n u m b e r of tru e m in te rm s is k (^O ), th e n th e m ap
0
f c - l ) ( a p p lic a tio n o f
0
(k -1 )- t i m e s , in s e q u e n c e ) i s to b e e m p lo y e d
-29-
x
n
X11 = (X1G )X 1X2 . . . Xi . . . x n ) + X j X2 . . x . . . x n . For n > I , x ' n = x n
F ig u re 2 . 3 . 3
The C irc u it fo r G e n e ra tin g th e P e rm u tatio n (m - ^ m J
( o i n d ic a te s c o m p le m e n te d in p u t) .
-30-
a t th e en d of a l l o th e r m ap p in g o p e r a tio n s .
8 t e p '3 : N ote th e s e q u e n c e o f m in te rm s w h ic h h a s b e e n u s e d in
d e s ig n in g th e s y s te m .
L ist th e tru th v a lu e s o f th e fu n c tio n to be
r e a liz e d a g a i n s t e a c h m interm in th e s e q u e n c e .
W ith th e h e lp o f th e
a v a ila b le m a p s , th e p e rm u ta tio n th a t w ill m ap th e s u b s e t o f k tru e
m in te rm s in to th e l a s t k m in term s of th e m interm s e q u e n c e , w ill
h a v e to b e p r o d u c e d . T h is p e rm u ta tio n c a n b e d e c o m p o s e d in to
a v a ila b le .m a p s a s fo llo w s :
N ote how m an y of th e m in te rm s in th e
a re
tr u n c a te d m interm s e q u e n c e
tru e .
a.
If a ll of th em a re t r u e , th e n a p p ly th e
0
^ m ap of
s te p 2 a n d th e n go on to S te p 4 .
b.
If som e o f th e m in term s in th e tr u n c a te d s e q u e n c e a re
n o t tr u e , c h e c k if th e s u b s e t o f k tru e m in te rm s form a
c o n s e c u tiv e s e q u e n c e w ith in th e m interm s e q u e n c e
■ a s s u m in g th e l a s t m interm to b e a d ja c e n t to th e f i r s t .
If s o , th e n u s e th e m ap fo r c y c li c p e rm u ta tio n
(m ,m , . . .m ); th e n u m b er of tim e s i t s h o u ld b e
1 2
2n
e m p lo y e d i s e q u a l to th e d is ta n c e (m e a su re d b y th e n um ber
o f m in te rm s in b e tw e e n ) b e tw e e n th e l a s t m in term in th e
-31-
m interm s e q u e n c e a n d th e l a s t tru e m interm dow n th e
b lo c k of tru e m in te rm s
If th e s u b s e t o f tru e m in te rm s i s in te r s p e r s e d w ith f a l s e m in te rm s in
th e s e q u e n c e , th e n u s e th e m ap for. c y c lic p e rm u ta tio n a n u m b er of
tim e s so th a t a m axim um n u m b er o f tru e m in te rm s a re m a p p ed in to
th e lo w e r p o s itio n s of th e tr u n c a te d s e q u e n c e .
For e a c h of th e
re m a in in g tru e m in te rm s , tr a n s p o s it io n m a p s s u c h a s (nu ,my) c an be
u s e d , w h e re iru i s a f a l s e m interm s itu a te d w ith in th e tru n c a te d
sequence
.
{ m
. , . . . m l
2 ^ —b+1
/2n
a n d m. is a tru e m interm o u ts id e of
1
th is tr u n c a te d s e q u e n c e . If rm., m^ a re a d ja c e n t, th e n (rm ,rm ) c a n be
decom posed
^ ^
as
(m. , m. ) = (m, , m „ , . . . m
i j
^
) 2 +1
^ (m, ,Xnrz,)
(m , m , . . . m )i
, i
z
2n
If m . , m . a re n o t a d j a c e n t , th e n ( m. , m. ) c a n b e d e c o m p o s e d in to a
i
I
1 y
n u m b e r o f o th e r t r a n s p o s i t i o n s , e a c h o n e o f them b e in g e x p r e s s ib le
in form (A). F or i n s t a n c e , (m3 ,Hi5) c a n b e d e c o m p o s e d a s
On3 ,Im5) = (m3 ,m ^) (m ^ ,m 3) On3 ,m ^ ) . E a c h co m p o n en t of th e
rig h th a n d e x p r e s s io n c a n now b e e x p r e s s e d in form (A).
S te p 4 .
S et th e o u tp u t lo g ic d e v ic e ,to p ro d u c e f^ in a l l c a s e s
e x c e p t w h e n th e z e ro fu n c tio n h a s to b e p ro d u c e d ( c f . S te p I ) .
(A)
-32-
E xam ple 2 . 4 :
The fo llo w in g fu n c tio n g iv e n in tru th ta b le form
i s to b e r e a l i z e d .
X1
I
x„
2
x„
3
F
0 .
0
0
0
0
0
I
I
0 .
I
0
I
0
I
I
I
I
0
0
I
0
I
I
I
I
0
0
I
I
I
I
‘
.
0
The o rd e rin g g iv e n in T a b le 2 .3 i s u s e d h e re ,
M in term
I
F
.0
0
I
0
I
I
I
I
-33-
S te p I:
The n u m b e r o f tru e m in te rm s = 5 .
S te p 2:
0^ i s
S te p 3:
(m , m ) i s to be p r o d u c e d .
3
4
f)
0
(m3 , m4 ) = (m ,m , . . .m ) (m^,mg)(m^mg, . . . m ^ r
to b e u s e d a f te r a l l o th e r m a p s .
S te p 4: The o u tp u t lo g ic i s to b e s e t to p ro d u c e f ^ .
N ote th a t a t o t a l o f 13 m ap p in g o p e r a tio n s a re n e e d e d to p ro d u c e th e
f u n c tio n .
Such a h ig h num ber of m a p p in g .o p e ra tio n s a re n e c e s s a r y
b e c a u s e o n ly tw o m a p s a re a v a ila b le in th e m a ch in e to g e n e ra te a ll
p o s s ib le p e rm u ta tio n s .
(m
^ , m ) , . . , (m
r^1
r
S u p p o s e , th e m ap s (m^ ,m ^ ) , ( H ^ m 3) , . . . ,
m
2n- l ,
) a re a v a ila b le in h a rd w a re .
I t i s know n
2"
from G roup T heory t h a t w ith th is s e t of tr a n s p o s itio n m a p s , a l l
p o s s ib le p e rm u ta tio n s c a n b e g e n e r a te d .
T h e n , in th e a b o v e c a s e ,
n in e m ap p in g o p e ra tio n s to p ro d u c e Cm3 ,m ^) c a n b e r e p la c e d by a
s in g le ma p .
F u rth e r, th e
0 m ap s
c a n b e c o m p le te ly e lim in a te d if th e
o u tp u t lo g ic i s s u f f ic ie n tly fle x ib le s o t h a t th e (2n + l) f u n c tio n s ,
lis t e d in th e p ro o f of Theorem 2 , 2 . 6 a re g e n e ra te d b y i t .
Thus th e re i s a
w a y o f re d u c in g th e n u m b e r of m ap p in g o p e ra tio n s b y in c r e a s in g th e
■m ap p in g h a rd w a re a n d th e f le x ib ili ty of th e o u tp u t l o g i c .
-34-
N o te' th a t m a p p in g of th e in p u t is n e c e s s a r y b e c a u s e th e
.fu n c tio n to be r e a liz e d i s n o t a v a ila b le in th e o u tp u t l o g ic .
T h e re fo re ,
th e e x te n t to w h ic h th e o u tp u t lo g ic i s fle x ib le d e te rm in e s in how m any
o f th e c a s e s m ap p in g h a s to be e m p lo y e d , th e la t t e r n u m b er b e in g in
in v e r s e p ro p o rtio n to th e form er,.
S in c e th e f le x ib ility o f th e o u tp u t
lo g ic i s lim ite d , som e m ap s h a v e to be u s e d .
The la r g e r th e num ber
o f m ap s a v a il a b le in h a rd w a r e , th e l e s s e r th e n u m b er o f m ap p in g
o p e ra tio n s to be n e e d e d , in g e n e r a l, fo r th e r e a liz a tio n o f a s w itc h in g
fu n c tio n a n d , th e r e f o r e , f a s t e r i s th e w h o le o p e r a tio n .
T hus th e
n u m b er of m aps th a t s h o u ld be a v a ila b le in h a rd w a re m u s t be
d e te rm in e d by s p e e d re q u ire m e n ts a n d c o s t A tr a d e - o f f b e tw e e n
f le x ib ility of th e o u tp u t lo g ic an d a d d itio n a l m ap s is a l s o p o s s ib le
a n d th e .p r o p e r b a la n c e s h o u ld be a rriv e d a t on th e sa m e b a s i s .
C h a p te r 3
BULK TRANSFER WITH CELLULAR CASCADES
-3 6-
3 .1
T r a n s p o s itio n M a p s
It h a s b e e n m e n tio n e d in C h a p te r 2 th a t th e s e t o f tr a n s p o s i
tio n m ap s
{ (m^, n n + ^) | i = l , 2 , . . . , 2n - l
} a re s u f f ic ie n t to p ro d u c e
a r b itr a r y p e rm u ta tio n of th e s e t of m in te rm s .
S im ilar s e t o f s u f f ic ie n t
m ap s c a n be { (m ^ ,n n ) | i = 2 , 3 , . . . , 2 n ] . If th e m ap p in g d e v ice - of a
B. T. s y s te m i s a s s u m e d to be c a p a b le o f d ir e c tly im p le m e n tin g an y
tr a n s p o s itio n (nn ,m J , a q u e s tio n m ay b e a s k e d : W h at i s th e m aximum
n u m b er of tr a n s p o s it io n s n e c e s s a r y to r e a liz e a fu n c tio n , g iv e n th a t th e
o u tp u t lo g ic is o f knpw n f le x ib ili ty ?
p o s s ib le tr a n s p o s it io n s i s
(
2n
^
).
The to ta l'n u m b e r o f d iffe re n t
If th e o u tp u t lo g ic i s a s s u m e d to be
fle x ib le e n o u g h to p ro d u c e j u s t th e (2n + l) fu n c tio n s a s o b ta in e d in
T heorem 2 . 2 . 6 , th e n th e m axim um n u m b er o f tr a n s p o s it io n s th a t m ay
b e n e c e s s a r y to r e a liz e a n a r b itr a r y fu n c tio n i s 2n ~-t-.
O ne w a y in
w h ic h th e o u tp u t lo g ic c a n b e m ade fle x ib le i s by u s in g a M a itra
( 12)
c a s c a d e . In th e n e x t s e c tio n a m e a s u re of th e f le x ib ili ty o f M a itra
c a s c a d e in r e la tio n to b u lk tr a n s f e r s y s te m w ill b e o b ta in e d .
3 .2
O u tp u t L ogic w ith M a itra C a s c a d e
M a itr a ( ® ) h a s in tro d u c e d a ty p e o f o n e - d im e n s io n a l
g ra p h fo rm at fo r r e p r e s e n tin g s w itc h in g fu n c tio n s a n d w ith i t s h e lp , ■
c h a r a c te r iz e d th e c a s c a d e - r e a l i z a b l e f u n c tio n s .
T h e s e i d e a s w ill b e
u s e d to p ro v e a th e o re m b o u n d in g th e n u m b er o f tr a n s p o s itio n m aps
-37-
re q u ire d fo r r e a liz in g a lo g ic fu n c tio n in a b u lk tr a n s f e r s y s te m w ith a
M a itra c a s c a d e in th e o u tp u t l o g ic .
The o n e - d im e n s io n a l g rap h
fo rm at is sh o w n in F ig u re 3 . 2 . 1 fo r n = 3.
In c o n s tr u c tin g th e g ra p h ,
a t f i r s t a n o rd e rin g o f th e v a r ia b le s i s a s s u m e d (as in th e a b o v e
,
case:
x ^ ).
A h o riz o n ta l
lin e i s d iv id e d in to tw o e q u a l p a rts
in w h ic h th e l a s t v a r ia b le Xg is e q u a l to 0 o v e r th e r ig h t - h a l f p o rtio n
a n d i s e q u a l to I o v e r th e l e f t - h a l f p o rtio n .
The re g io n o v e r w h ic h .
Xg = 0 i s a g a in d iv id e d in to tw o p a r t s , o f w h ic h , th e r ig h t - h a l f
p o rtio n h a s th e n e x t v a r ia b le , Xg e q u a l to 0 a n d th e l e f t - h a l f p o rtio n ,
Xg e q u a l to I .
The v a lu e s o f Xg o v e r th e o rig in a l l e f t - h a l f p o rtio n ,
w h e re Xg = I , a re a m irro r im a g e o f th o s e in th e rig h t h a lf .
I n .th e
sam e m a n n e r, th e rig h tm o s t re g io n o v e r w h ic h Xg = 0 i s d iv id e d in to
tw o p a r t s , w ith th e rig h t a n d le f t p o rtio n s h a v in g x^ = 0 a n d x^ = I
x3
O
Il
CM
X
x2 = 1
x2 = 1
_«_______ i___I___ t_______ «___ ---- 1----------- 1------
0
F ig u re 3 . 2 . 1
I
1
0
0
I
X
CO
Il
O
= I
Il
O
The v a lu e s o f x ^ , in th e re g io n n e x t to th e l e f t , h a v in g x
X
CO
re s p e c tiv e ly .
•
I
0 ^
xI
O ne D im e n sio n a l G rap h F o rm at fo r T h ree V a ria b le F u n c tio n s
—3 8 —
x
= I , a re a m irror im a g e of th o s e in th e r ig h t.
T his p a tte r n i s ■
c o n tin u e d in t h e - l e f t - h a l f p o rtio n h a v in g Xo - I . E ach b la c k d o t on th e
h o r iz o n ta l lin e r e p r e s e n ts a d iff e r e n t c o m b in a tio n o f v a lu e s of th e
v a r ia b le s a n d t h e r e f o r e , s ta n d s fo r a m in te rm .
A fu n c tio n m ay be
p lo tte d on th is g ra p h w ith u p w ard v e r t i c a l lin e s from th e d o ts re p re s e n tin g
tru e m in te rm s a n d d o w n w ard v e r tic a l lin e s f a ls e m in te rm s .
A fu n c tio n i s c a lle d e v e n sy m m e tric if th e tru e te rm s in one
h a lf o f th e o n e - d im e n s io n a l g ra p h o c c u p y th e sam e m interm p o s itio n s
w ith r e s p e c t to th e c e n tre a s th e tru e te rm s on th e o th e r h a lf .
A fu n c tio n
i s odd sy m m e tric if th e tru e te rm s of o n e h a lf of th e g ra p h o c c u p y th e
sam e m interm p o s itio n s w ith r e s p e c t to th e c e n tre a s th e f a l s e te rm s
on th e o th e r h a lf .
A fu n c tio n is one s id e f l a t if th e tru e te rm s (fa ls e
te rm s) a re c o n fin e d w ith in o n e h a lf of th e g ra p h .
A M a itra c a s c a d e i s a o n e - d im e n s io n a l a rra y o f tw o - in p u t o n e o u tp u t c e l l s w h e re e a c h c e l l c a n p ro d u c e a n y lo g ic a l fu n c tio n of tw o
v a r ia b le s (F igure 3 . 2 . 2 ) .
If a fu n c tio n f(x n ' x n _ ] / • • -x I^ c a n
r e a liz e d by a M a itra c a s c a d e , th e n i t i s c a lle d M a it r a - r e a liz a b le (MR).
F ig u re 3 . 2 . 2 The M a itra C a s c a d e
-39-
An MR fu n c tio n is r e a liz e d w ith som e o rd e rin g of in p u ts (in F ig .
3 . 2 . 2 , th e o rd e rin g i s x , x
, . . . x j a n d a p p e a rs w ith so m e c h a r a c te r is tic
n n -1
I
p a tte r n w h e n p lo tte d on a o n e - d im e n s io n a l g ra p h w ith t h i s o rd e rin g of
v a ria b le s .
The fo llo w in g p ro p e rty h a s b e e n s tu d ie d by M a itra b u t w ill
b e d i s c u s s e d in a m o d ifie d form fo r s u b s e q u e n t r e f e r e n c e .
Theorem 3 . 2 . 1
L et f( x ^ , . . .x ^) b e a n MR fu n c tio n w ith th e o rd e rin g :
2V x n - I ' " ‘ "x I 1 ^ i t i s p lo tte d w ith th is o rd e rin g
t
th e n f m u s t be
e ith e r i.) odd sy m m e tric o r ii) e v e n sy m m e tric or iii) o n e s id e f l a t .
P ro o f: The th re e c a s e s m e n tio n e d in th e th e o re m e x h a u s t a l l p o s s ib le
s itu a tio n s fo r tw o v a r ia b le f u n c tio n s .
h a s o n ly tw o in p u ts .
Any c e l l in th e M a itra c a s c a d e
C o n s id e rin g th e l a s t c e l l of a c a s c a d e w h ic h
r e a l i z e s f , i t i s s e e n t h a t f c a n be e x p r e s s e d a s a fu n c tio n o f tw o
v a r i a b l e s - - x n a n d som e fu n c tio n f i ( x n _ i , . • . x ^ ) .
m ust be tr u e .
H e n c e th e p ro p e rty
Q.E.D.
R eferrin g to T heorem 3 . 2 . 1 , c o n s id e r tw o p o s s i b i l i t i e s . If
c a s e (i) o r c a s e (ii) i s tr u e , l e t f^ (xn _ ^ , . . .x ^ ) b e th e fu n c tio n in one
h a lf of th e o n e - d im e n s io n a l g ra p h .
If c a s e (iii) is tru e l e t f , (x
I n -1
b e th e fu n c tio n in th e n o n - f la t s id e (b o th s id e s f la t i s a. t r i v i a l c a s e ) .
S in c e f i s MR w ith o rd e rin g x n , x n _ ^ , . . .X ^ , h e n c e f^ m u s t b e M R.
. x„)
I
-40-
To s e e t h i s , n o te th a t f c a n b e e x p r e s s e d a s f = fij. * x Xl w h e re * i s one
o f th e .s ix te e n tw o - v a r ia b le fu n c tio n s ..
T h e re fo re , e ith e r f
of f
m ust ’
b e th e o u tp u t of th e l a s t b u t o n e c e ll in th e M a itra c a s c a d e re a liz in g , f .
I t i s know n t h a t c o m p le m e n ta ry o f a n MR fu n c tio n i s M R.
b e M R.
T h e re fo re , f^ m u s t s a t i s f y Theorem 3 . 2 . 1 .
fg c a n b e form ed from f ^ , fg from
T h u s , f^ m u s t
In a s im ila r m a n n e r,
-----f^ from f^_^ fo r k = 2 , 3 , . . .n - 3
a n d e a c h m u s t s a t i s f y T heorem 3 . 2 . 1 .
T h is r e s u lt g iv e s a t e s t p ro c e d u re
fo r MR fu n c tio n w ith a n a s s u m e d o r d e r in g .
T heroem 3 . 2 . 2
T here e x i s t s a M a itra - re a liz & b le fu n c tio n w ith
k m in te rm s tru e fo r e v e ry k s u c h th a t 0 g k.g 2n .
P ro o f: The p ro o f i s o b ta in e d by c o n s tr u c tin g a n MR fu n c tio n of a n y g iv e n
n u m b e r of tru e m in te rm s .
MR f u n c tio n .
A fu n c tio n w ith no m in term s tru e i s a tr iv ia l
C o n s id e r th e o n e - d im e n s io n a l g ra p h fo r n v a r ia b le s .
O ne s id e f l a t MR fu n c tio n s of d iff e r e n t s iz e c a n b e c o n s tr u c te d by ta k in g
o n e m in te rm , tw o m in te rm s , th re e m in te rm s , e t c . from o n e en d a s tru e
te r m s .
Q.E.D.
T h is th e o re m s h o w s th a t th e M a itra c a s c a d e h a s th e n e c e s s a r y
f le x ib ili ty a s s ta te d in th e proof o f T heorem 2 . 2 . 6 . T heorem 3 . 2 . 3 g iv e s a n
u p p e r b o und on th e n u m b e r o f n e c e s s a r y tr a n s p o s itio n s re q u ire d by
a M a itra c a s c a d e in o rd e r to r e a liz e a n a r b itr a r y f u n c tio n .
-41-
T heorem 3 . 2 . 3
To c o n v e rt a n y g iv e n fu n c tio n in to a n MR fu n c tio n ,
a t m o s t (2n - ^ - I ) tr a n s p o s it io n s a re n e c e s s a r y .
P ro o f: T h is p ro p e rty m ay be e a s i l y v e r if ie d fo r 3 - v a r ia b le f u n c tio n s by
u s in g M a it r a 's o n e - d im e n s io n a l g ra p h a n d c o n s id e rin g fu n c tio n s of u p
to 4 tru e m in te rm s (b e c a u s e c o m p le m e n ta ry of a n MR fu n c tio n i s a ls o
M R ). A cco rd in g to th e s ta te m e n t o f th e th e o re m ,, w e n e e d to u s e a t
m o st o n e tr a n s p o s itio n fo r c o n v e r s io n .
C o n s id e r th re e c a s e s fo r th e
3 - v a r ia b le g ra p h (F igure 3 . 2 . 1 ) :
i)
if)
Two tru e m in term s on e a c h s id e of th e o r ig in ,
One tru e m interm on o n e s i d e , Two tru e m in term s on th e
o th e r.
iii)
No tru e m interm on o n e s i d e .
C a s e (iii) is MR (one s id e f l a t ) .
C a s e (ii) c a n be c o n v e rte d to MR
fu n c tio n by in te rc h a n g in g th e s in g le tru e m interm on o n e s id e w ith a
f a l s e m interm on th e o th e r .
c a s e (i).
T here m ay b e th re e p o s s i b i l i t i e s for
The fu n c tio n m ay be odd sy m m e tric or e v e n sy m m e tric
a n d th e re fo re M R.
If n o n e of t h e s e , th e r e m u st b e o n e p a ir o f tru e
te rm s w h ic h a re m irro r sy m m e tric an d o n e p a ir w h ic h a re n o t.
C le a r ly ,
a s in g le tr a n s p o s itio n c a n m ake th e fu n c tio n e ith e r o dd or e v e n
sy m m e tric a n d th e re fo re M R.
The o th e r p o s s ib le c a s e s , c a n b e d e a lt
w ith in a m a n n e r s im ila r to o n e o f t h e s e th re e c a s e s .
-42-
L et u s now c o n s id e r n - v a r ia b le f u n c tio n s .
th e o re m i s tru e fo r (n - 1 ) - v a r ia b le f u n c tio n s .
A ssu m e th a t th e
We- s h a ll th e n n e e d
c o n s id e r o n ly fu n c tio n s- h a v in g m tru e min te rm s fo r 2n ~2 < m g 2n ■*■.
For th o s e f u n c tio n s h a v in g m - 2n “ ^ , o n e o f th e h a lv e s o f th e g rap h
w ill c o n ta in a t m o s t 2n ~^ tru e te rm s .
th e f a l s e te rm s on th e o th e r h a lf .
T h e s e c a n be in te rc h a n g e d w ith
W h en a l l of them a re in te r c h a n g e d ,
a t m o s t 2n “ 3 tr a n s p o s it io n s w ill h a v e b e e n u s e d .
The r e s u lta n t
fu n c tio n w ill be o n e s id e f la t a n d th e o th e r s id e — a fu n c tio n of
(n-1) v a r ia b le s w h ic h n e e d , a c c o rd in g to th e th e o re m , a t m o st
(2n ~3 _j.) tr a n s p o s itio n s fo r c o n v e r s io n .
In to ta l th e n , (2n “ ^ - I )
tr a n s p o s it io n s c a n do th e jo b .
N e x t w e c o n s id e r th e s p e c ia l c a s e w h e n e a c h h a lf c o n ta in s
2n ~^ tru e t e r m s .
O ne o f t h e s e w ill re q u ire a t m o st (2n - 3 - I )
tr a n s p o s it io n s fo r c o n v e r s io n in to MR f u n c tio n . A fter o n e h a lf h a s
b e e n c o n v e rte d in to MR f u n c tio n , th e re w ill b e on th e o th e r h a lf a t
J e a s t 2n ^ tru e te rm s w h ic h w ill b e a m irro r im ag e o f th e sam e
n u m b er of tru e te rm s on th e f i r s t h a lf o r a t l e a s t th e sa m e num ber
w h ic h w ill n o t b e a m irro r im a g e .
T h e re fo re , a t m o s t 2n _ ^ t r a n s p o s i­
tio n s w ill b e n e c e s s a r y to g e t odd o r e v e n sy m m e try .
T hus th e
t o t a l n u m b er of tr a n s p o s itio n s w ill b e a t th e m o st (2n ~^ - I ) .
-43-
N e x t c o n s id e r th e c a s e w h e re o n e h a lf (to b e' re fe rre d a s A) c o n ta in s
(21?-2 -K) tru e te rm s a n d th e o th e r h a lf (B) c o n ta in s (2n - 2 ' -L)- tru e te rm s ,"
w h e re K ^ L for Kz L e ith e r z e ro or p o s itiv e . - The c a s e o f n e g a ti v e L w ill
b e d i s c u s s e d a fte rw a rd s .
Let p .be. th e s m a lle s t in te g e r s u c h th a t 2P ^ (K+L)
T he o n e -d im e n s io n a l, g ra p h is d iv id e d in to 2n ~P b lo c k s of. 2 P te rm s . If "
w e m ark t h e s e a s 1 , 2 , 3 ,
.from o n e e n d , th e re w ill b e som e s - t h b lo c k
(to b e re fe rre d a s S) of 2P te rm s in .B , w h ic h w ill c o n ta in a t m o st, • ■
> - 2 -L
Z P -1 -
.Zn - P - 1 ;
tru e te rm s
L
?n-p-iy,
T he num ber of f a ls e te rm s in th is , b lo c k is a t l e a s t
2 P_
213- I
(|^ d e n o te s , th e b a s e v a lu e )
2n —p - I
If a l l of (2n ~2 -K) tru e te rm s o f A .are in te rc h a n g e d w ith th e f a ls e te rm s in
B ,.th e r e w ill b e (K + L) f a ls e te rm s in B . Let..this- in te rc h a n g e b e d o n e in
s u c h a w a y th a t th e 2P te rm s of S a re n o t d is tu rb e d b e fo re o th e r f a ls e te rm s
i.n B h a v e b e e n e x h a u s t e d . A fter th is h a s b e e n d o n e , th e a llo w a b le num ber
o f tr a n s p o s it io n s re m a in in g = 2n ~2 - I - ( 2 n -2 -K) = K - I .
The. in te rc h a n g e ,
m ay r e s u l t in tw o p o s s ib le s it u a t i o n s .
C ase I
But for th e s - t h b lo c k , B is fu ll of tru e te r m s .
In t h i s c a s e th e fu n c tio n , i n 8 -c a n b e tra n s fo rm e d in to a n MR fu n c tio n b y .
a t m o st (K-L) in te r c h a n g e s , a s K ^ 2P- 2 .
B y ,th is , th e o v e r - a l l fu n c tio n
w ill a ls o b e MR.
C a s e II
T h ere a re som e f a ls e te rm s -in B o u ts id e S .
-.44-
L et 2P~2 + f = N um ber of tru e te rm s in S, w h e re f < 2P- ^ .
(a)
If K ^ 2 P ~ 1 , th e n a l l tru e te rm s o f S m ay b e in te rc h a n g e d w ith f a l s e '
te rm s — a s u f f ic ie n t num ber o f t h e s e w ith a l l th e f a ls e te rm s in B b u t
o u ts id e .S a n d th e re m a in in g tru e te rm s w ith th e f a ls e te rm s w ith in S •'
in a m a n n er s u c h t h a t a o n e - s id e f l a t MR fu n c tio n in S r e s u l t s .
The
o v e r - a l l fu n c tio n is a l s o MR in t h i s c a s e .
(b)
L et K = 2P- 2 + c = L w h e re c < 2P- 2 . V alu es of K d iff e r e n t from L
w ill b e c o n s id e re d l a t e r .
(i)
C o n s id e r th e fo llo w in g s u b - c a s e s ."
c > f . As in ( a ) , a l l tru e te rm s in S c a n b e in te rc h a n g e d w ith .f a ls e
te rm s a n d a n MR fu n c tio n m ay b e o b t a i n e d .
(ii)
c < f . E x p re s s c a s
c = a j 2 P ~ ^ + ag 2 P~^ + ag2P~5
w h e re
a^ fo r i = 1 , 2 , 3 , . . . is a b in a ry v a lu e d v a r i a b l e , b e in g e ith e r 0 or I .
T hen K = 2P -2 + a^ 2P“ ^ +
2P“ 4 + . . . . . . .
C o n s id e r th e te rm s in 8 ;
T h ere w ill b e o n e h a lf o f th e s - t h b lo c k c o n ta in in g l e s s th a n 2P~2 tru e
te r m s .
W ith a t m o s t 2P~^ - I in te r c h a n g e s , th e s e a re m a p p ed (as m any a s
n e e d e d ) in to th e f a ls e te rm s in B o u ts id e S a n d th e re m a in in g in th e r e s t
o f th e s - t h b lo c k .
N e x t, c h o o s e th e s u b - b lo c k o f 2P- 2 te rm s in th e
o th e r h a lf of th e s - t h b lo c k c o n ta in in g a l e s s e r num ber o f tru e te r m s .th a n
th e o th e r s u b - b lo c k .
If e a c h s u b - b lo c k of 2P~2 te rm s in th is h a lf c o n ta in s
m ore th a n 2P""2 tru e t e r m s , th e n s in c e th e to ta l num ber o f tru e te rm s in S
is l e s s th a n 2 P ~ 1 , h e n c e in th e p re c e d in g s ta g e o f th e in te r c h a n g e , a
c o rre s p o n d in g ly l e s s e r num ber th a n 2P~2 -1 in te rc h a n g e s w e re u s e d .
45-
U se t h i s d iff e r e n c e o f in te rc h a n g e s to g e t a s u b -b lo c k , o f
2P~^ t e r m s ,
con­
ta in in g 2 P-S tru e te rm s or l e s s . ' If a^= I , th e n u s e a t m o s t 2P- ^ in te r ­
c h a n g e s to c le a r th e s u b - b lo c k of a l l tru e te r m s .
The. tru e te rm s c a n b e
m ap p ed in to th e f a ls e te rm s in B o u ts id e S , o r if th e re is n o n e , in to th e
f a l s e te rm s in S b u t o u ts id e th e h a lf a lr e a d y fre e d from tru e te r m s .
a
=. Oy do n o t c le a r th e s u b - b lo c k of tru e te rm s .
If
If a ^ - I , c h o o s e th a t '
h a lf of th e s u b - b lo c k of 2P 2 t e r m s , w h ic h c o n ta in s a l e s s e r num ber o f
tru e, te rm s an d p ro c e e d in th e m a n n e r s ta t e d b e f o r e . F in a lly , a n MR
fu n c tio n w ith one s id e f la t a t each , s ta g e o f its fo rm atio n is o b ta in e d .
If K is g r e a te r th a n L, th e n th e nu m b er of in te rc h a n g e s n e c e s s a r y to
tr a n s f e r tru e te rm s of A in to B.is.re d u c e d ... .L ater im th e p r o c e s s , ,th e----- —
n u m ber o f in te rc h a n g e s m ay b e in c r e a s e d b u t the- in c r e a s e w ill b e , a t
m o s t, t o th e sam e e x te n t a s th e r e d u c tio n e a r lie r .
T he c a s e of n e g a tiv e L m ay b e c o n sid e re d .n o w .2 ^ -2 -K a n d 2 n - 2 + l tru e te rm s r e s p e c t i v e l y , w ith
If A an d B c o n ta in
L , t h e n , in te r ­
c h a n g in g a s b e f o r e , an d th e n b rin g in g a l l f a ls e te rm s in to S , ' t he a llo w a b le
nu m ber of in te r c h a n g e s re m a in in g c a n b e sh o w n a s
= K -I -
(K-L). - i 2P - . (2P-1 + — ^
)}■
'
2n - P _ l ^
= 2P + L - I '
(2P~1
—
T h is num ber i s g r e a te r th a n 2P“ ? - 1 , n e e d e d to c o n v e rt th e fu n c tio n in S
in to M R.
Q.E-.D-.
T h ere e x i s t fu n c tio n s w h ic h n e e d a t l e a s t 2n ~^ - I tr a n s p o s it io n s to b e M R.
—4 6 -
An e x a m p le i s F = Sg(x^ ,Xg ,Xg , x ^ ) , w h e re Sg in d ic a te s th a t F is
tru e if a n d o n ly if tw o v a r ia b le s o u t o f fo u r a re t r u e . ' I t s h o w s th a t
th e bound on th e re q u ire d num ber of tr a n s p o s itio n s i s th e l e a s t u p p e r
b o u n d . It a l s o sh o w s th a t re s o rtin g to d iff e r e n t o rd e rin g o f v a r ia b le s
c a n n o t im p ro v e th e b o u n d , b e c a u s e th e fu n c tio n is s y m m e tric .
3 .3
D e te rm in a tio n o f N e c e s s a r y T r a n s p o s itio n s
The p ro o f o f Theorem 3 . 2 . 3 g iv e s a m eth o d of fin d in g th e
n e c e s s a r y m ap s to r e a liz e a n a r b itr a r y fu n c tio n w ith a b u lk tr a n s f e r
s y s te m h a v in g i t s o u tp u t lo g ic m ade u p of a fle x ib le M a itra c a s c a d e .
G iv e n a f u n c tio n , p lo t i t on a o n e - d im e n s io n a l g r a p h .
N e x t,
d e te rm in e th e tr a n s p o s it io n s n e c e s s a r y to c o n v e rt th e fu n c tio n in to
MR fu n c tio n in th e m a n n er d e s c r ib e d u n d e r T heorem 3 . 2 . 3 .
Now
s e t th e m ap p in g d e v ic e to p ro d u c e th e c o m p o s ite m ap m ad e u p of
th e d e te rm in e d tr a n s p o s it io n s a n d s e t th e o u tp u t lo g ic to p ro d u c e th e
MR fu n c tio n in to w h ic h th e g iv e n fu n c tio n h a s b e e n tra n s fo rm e d .
An e x a m p le o f c o n v e rtin g a n o n - M a itr a - r e a l iz a b le fu n c tio n in to
MR fu n c tio n i s w o rk e d o u t, g iv in g th e d if f e r e n t s t e p s .
-47-
E xam ple 3 .3
S te p I ,
F = SgCx^Xg/Xg'^)
P lo t th e fu n c tio n on a g ra p h (F igure 3 . 3 . 1 )
x.=0
F ig u re 3 . 3 . 1
P lo t of F =
(x^ ,x ^ ,x ^ ,x ^ ) in O n e -d im e n s io n a l G ra p h .
O n ly tru e te rm s a re sh o w n b y upw ard v e r tic a l l i n e . T h ere a re 3 su c h
te rm s on e a c h h a lf .
S te p 2 .
D e term in e w h ic h o f th e c a s e s in th e T heorem 3 . 2 . 3 a p p ly ,
i . e . , th e s p e c ia l c a s e o f 2n ^ tru e te rm s on e a c h s id e o r th e more
g en eral c a s e .
G e n e ra l c a s e ;
K= I, L = I.
True te rm s of a n y s id e c a n be
m a p p ed in to th e f a l s e te rm s of th e o th e r .
C h o o se th e rig h t h a lf tru e
te rm s fo r t h i s .
S te p 3 .
D e term in e p from 2 ^ = K + L.
W e h a y e 2^= 2 . T hus th e lo w e s t v a lu e of p is I . . .
—4 8 -
S tep 4 .
M ark th e b lo c k s of 2 te rm s s ta r tin g from th e le f t- m o s t term
to fin d a b lo c k c o n ta in in g m inim um n u m b er of tru e t e r m s .
T hird b lo c k from th e l e f t .
S te p 5 .
I t c o n ta in s tw o f a ls e te rm s .
In te rc h a n g e tru e te rm s of th e r ig h t - h a l f w ith th e f a ls e
te rm s of th e l e f t - h a l f , k e e p in g th e f a l s e te rm s of th e th ird b lo c k
u n c h a n g e d a s lo n g a s th e re a re o th e r f a l s e te rm s in th e l e f t - h a l f .
E x c e p t fo r th ird b lo c k , a l l o th e r te rm s a re now tru e in l e f t - h a l f .
S te p 6 .
In te rc h a n g e th e f a ls e te rm s of th e l e f t - h a l f e x c lu d in g
th e th ird b lo c k w ith tru e te rm s of th e th ird b l o c k .
S te p 7 .
C o n v e rt th e fu n c tio n in th e th ird b lo c k in to MR fu n c tio n by
th e re m a in in g i n t e r c h a n g e s .
S te p s 6 a n d 7 do n o t m ak e a n y m o d if ic a tio n s .
The r e s u l t a n t fu n c tio n i s .
sh o w n in F ig u re 3 . 3 . 2 .
F ig u re 3 . 3 . 2
The F u n c tio n F = S (x^
S u ita b le T r a n s p o s itio n s .
,x
,x ^) a f te r A p p lic a tio n of
—4 9 —
The re q u ire d n u m b er o f tr a n s p o s itio n s o b ta in e d in th is e x a m p le is
th e m inim um n u m b er p o s s i b l e .
cases.
But t h i s i s n o t n e c e s s a r i l y tru e in a l l
S o m etim es a sim p le in s p e c tio n m ay r e v e a l fe w e r tr a n s p o s itio n s
to .m a k e a fu n c tio n M a i t r a - r e a l i z a b l e . The re q u ire d n u m b er of
tr a n s p o s itio n s in a p a r tic u la r c a s e m ay b e a p p lie d from th e in p u t
a rra y to th e o u tp u t a rra y w ith th e h e lp o f a fle x ib le m a p p in g d e v ic e
c a p a b le o f im p le m e n tin g a n y tr a n s p o s itio n in a c c o rd a n c e w ith th e sc h e m e d e s c r ib e d in s e c tio n 2 . 1 of C h a p te r 2 .
An a lte r n a tiv e i s to
p la c e a n a rra y o f m ap p in g e le m e n ts b e tw e e n in p u t a n d o u tp u t a rra y ,
e a c h e le m e n t b e in g c a p a b le of im p le m e n tin g a n y tr a n s p o s it io n .
a fle x ib le m ap p in g e le m e n t is show n in F ig u re 3 . 3 . 3 .
Such
The p a ir of
c a s c a d e s w ith f le x ib le lo g ic c e l l s , sh o w n in th e c e n te r of th e
F ig u re , a re u s e d to d e t e c t th e o c c u r re n c e o f th e p a ir of m in term s
b e tw e e n w h ic h a n in te rc h a n g e is n e c e s s a r y .
T h is i s d o n e by
s e ttin g th e c e ll fu n c tio n s to p ro d u c e c o m p le m e n ta tio n in p ro p e r
v a r ia b le s a n d g a tin g th e o u tp u ts of th e c e l l s th ro u g h a n AND b lo c k
w h ic h p ro d u c e s a tru e o u tp u t o n ly if th e d e s ir e d c o m b in a tio n of in p u ts
to th e c a s c a d e i s p r e s e n t . ■ The o u tp u t o f th e OR g a te i s tru e w h en
a n y o n e of th e p a ir of m in term s i s p r e s e n t;
i t i s f a ls e o th e r w is e .
A
tru e o u tp u t from th e OR g a te i s u s e d to c o m p le m e n t p ro p e r v a r ia b le s in
■th e to p m o s t c a s c a d e , w ith th e c e l l s th e r e h a v in g b e e n s e t e a r lie r to
p ro d u c e s u ita b le fu n c tio n s o f th e ir in p u ts .
-50-
fle x ib le
c e ll
F ig u re 3 . 3 . 3
F le x ib le M ap p in g E lem en t fo r T r a n s p o s itio n M ap s
(3- v a r ia b le )
W ith th e ty p e of m ap p in g e le m e n t j u s t d i s c u s s e d , i t i s sim p le to
p ro d u c e m a n y - to - o n e m ap s of th e s e t o f m in te rm s .
F ir s t, d e fin e a
fu n c tio n w ith th e g ro u p o f m in te rm s , a l l of w h ic h a re to b e m apped
o n to o n e m in te rm .
N e x t, r e a liz e th e fu n c tio n e m p lo y in g a n y s u ita b le
n e tw o rk a n d u s e th e o u tp u t from th i s n e tw o rk in th e sam e m an n er a s
th e o u tp u t o f th e OR g a te in F ig u re 3 . 3 . 3 .
M a n y - to -o n e m ap p in g o f
-51-
th e s e t o f m in te m s c a n b e l e s s c o m p le x th a n p e rm u ta tio n of th e s e t
a n d m ay recm ire l e s s e r n u m b er of m ap p in g o p e ra tio n s in s u ita b le c a s e s
E xam ple 3 .3 i s a n in s ta n c e o f s u c h c a s e .
U sin g o n ly p e rm u ta tio n
m a p s , i t w a s sh o w n t h a t th re e in te r c h a n g e s w e re n e c e s s a r y to c o n v e rt
th e g iv e n fu n c tio n in to a n MR f u n c tio n .
W h en m a n y - to - o n e m ap s a re
a llo w e d , i t i s p o s s ib le to a u g m e n t th e g iv e n fu n c tio n b y in c lu d in g
th e m i n t e r m s -----x ^ x ^ x ^ x .
fu n c tio n i s o b ta in e d .
a u g m e n te d f u n c tio n .
an d x^XgX^x^
—
so t h a t a n MR
The o u tp u t lo g ic i s s e t to r e a liz e th is
To p ro d u c e th e d e s ir e d lo g ic , h o w e v e r, m apping
d e v ic e c a n b e s e t e a r lie r to m ap t h e s e tw o te rm s to so m e f a l s e te rm ,
s u c h as-x-jXgXgX^ of th e fu n c tio n r e a liz e d by th e o u tp u t lo g ic .
3 .4
Bulk T ra n s fe r in C a s c a d e s
C o n s id e r th e M a itra c a s c a d e sh o w n in F ig u re 3 . 4 . 1 . a .
O ne m ay re g a rd th e s e t o f d ir e c t in p u ts { x ^ X g , . . .x ^} a lo n g w ith
th e s e t o f c e ll o u tp u ts { a ^ , a g , . . . a ^ _ ^ }
a s a n ew s e t o f v a r ia b le s
w h ic h c a n be tr e a te d in th e sam e m a n n e r a s th e v a r ia b le s of th e in p u t
a rra y o f a b u lk tr a n s f e r s y s te m d i s c u s s e d e a r lie r .
S im ila rly , th e
o u tp u t a rra y m ay b e a n o th e r a rra y c o n s is tin g o f th e d ir e c t in p u ts
{ y ^ y g , • • . ym} to a n u m b er of c e l l s e i t h e r in th e sa m e c a s c a d e o r in
a d if f e r e n t c a s c a d e a lo n g w ith th e o u tp u ts ( b ^ f b g , . .
c e lls .
The m a p p in g d e v ic e m ay c o n s i s t o f a m e c h a n ism fo r
° f th o s e
-52-
(a)
F ig u re 3 . 4 . 1
(a) In p u t C a s c a d e
(b)
(b)
O u tp u t C a s c a d e
r e p o s itio n in g of th e v a r ia b le s s o th a t o n ly lo g ic - f r e e m ap s a s
d is c u s s e d p re v io u s ly c a n be im p le m e n te d .
The s e t [ y i ' Y g ' • • i ^m)
m ap p ed in to th e o u tp u t a rra y is a s u b s e t of th e s e t o f v a r ia b le s
{ X j , . . .Xfi, a , , . . . a n _ j} a n d i s tra n s fo rm e d in th e c a s c a d e to p ro d u ce
a n ew s e t of d e p e n d e n t v a r ia b le s [ b j , . . . , b m_ j }
. The m ap p in g b a c k
from th e o u tp u t a rra y in to th e in p u t a rra y n e e d n o t b e id e n tity m a p p in g .
For m ap p in g from o n e a rra y in to a n o th e r , a p ro p e r s u b s e t o f th e s e t
of in d e p e n d e n t an d d e p e n d e n t v a r ia b le s m ay ta k e p a r t.
A p ro p e r
s u b s e t is n e c e s s a r y b e c a u s e th e n u m b er of d ir e c t in p u ts in a c a s c a d e
i s a lw a y s l e s s th a n th e sum to ta l of th e in d e p e n d e n t a n d d e p e n d e n t
v a ria b le s .
The m ap p in g i s a s s u m e d to b e o n e - to - o n e in th e s e n s e
th a t e a c h m em ber of th e s u b s e t is m ap p ed to a d iff e r e n t (d ire c t) c e ll
in p u t.
d e v ic e .
Any o n e of th e c a s c a d e s m ay s e rv e a s th e o u tp u t lo g ic
The c a s c a d e s m ay be lo g ic a lly fle x ib le an d c a n b e r e s e t
a f te r e a c h m ap p in g o r m ay b e lo g ic a lly f ix e d .
The f in a l o u tp u t fu n c tio n
i s o b ta in e d a s o n e m em ber o f a n a rra y a t th e end of a s u ita b le n u m b er of
m a p p in g s .
It i s c le a r ly p o s s ib le to h a v e s e v e r a l o u tp u t f u n c tio n s a p p e a r
-53-
in th e d iff e r e n t m em b ers of th e f in a l a r r a y .
F u rth e r, o n e m ay h a v e
m ore th a n o n e c a s c a d e in o n e a r r a y .
L et u s c o n s id e r a sim p le a n d s p e c if ic c a s e .
c o n s i s t o f th e v a r ia b le s
{ x^x
a rra y o f th e v a r ia b le s { y . , . . .y
I
z IB
L et th e in p u t a rra y
, . . . x n , a ^ , . . .a ^ _ ^ ] a n d th e o u tp u t
, b , . . .b
I
Ju
,} .
I
L et th e m ap p in g
b e lo g ic - f r e e a n d th e o u tp u t lo g ic c o n s i s t o f th e in p u t a n d o u tp u t
cascad es.
L et t h e s e c a s c a d e s be lo g ic a lly f le x ib le .
O ne m ay show t h a t s u c h a s y s te m h a s lo g ic a l u n iv e r s a li ty if th e
in p u t a n d o u tp u t c a s c a d e s c o n ta in a t l e a s t one e x tra d ir e c t c e ll in p u t
in a d d itio n to th e t o t a l n u m b er of o r ig in a l in p u ts { x ^ X g , . . .x n 3
To s e e t h i s , c o n s id e r a n y fu n c tio n o f n v a r i a b l e s .
•
A term of th e
fu n c tio n m ay c o n ta in , a t m o st n l i t e r a l s a n d c a n b e r e a liz e d in a
c a s c a d e of (n-1) c e l l s , e a c h r e a liz in g e ith e r a n AND(OR) o r a
NIM P(IM P) f u n c tio n .
L et th e term r e a liz e d b e r e p r e s e n te d b y a .
A fter r e a liz in g th e term in th e in p u t c a s c a d e , a m a p p in g m ay be
p erfo rm ed s u c h t h a t th e v a r ia b le s { x ^ /X g , . . .x ^ 3 a re m ap p ed
id e n t i c a l l y in to th e o u tp u t c a s c a d e a rra y w h ile th e v a r ia b le a ^ is
m a p p ed to th e e x tra c e l l in p u t y n+1 p la c e d a t th e e n d o f th e o u tp u t
cascade.
By s e ttin g th e p ro p e r c e l l f u n c tio n s .in th e o u tp u t c a s c a d e ,
a s e c o n d term o f th e fu n c tio n m ay b e r e a liz e d an d c o m b in e d w ith th e
-54-
f i r s t term by h a v in g th e l a s t c e ll in t h a t c a s c a d e perform a n OR(AND)
f u n c tio n .
M ap p in g b a c k n o w , o n e m ay r e a liz e .a th ird term a n d in ,th is
m a n n e r th e p ro c e d u re m ay b e c o n tin u e d u n til a ll th e te rm s h a v e b e e n
re a liz e d .
T his ty p e o f B. T. s y s te m h a s a n e q u iv a le n t in th e T w o -ra il C a s c a d e
(
of S hort
16 )
. W h ile a n e x tra a r te r ia l c o n n e c tio n i s u s e d in th e
T w o -ra il C a s c a d e to p ro p a g a te th e p a rtia lly -f o rm e d f u n c tio n , an
e x tra c e ll in p u t is u s e d h e re to r e p r e s e n t th e p a rtia lly -f o rm e d fu n c tio n
a n d co m b in e i t to th e n e w ly -fo rm e d te rm ., As p o s s ib le in th e T w o -ra il
C a s c a d e a l s o , one m ay r e a liz e a s u ita b le c o lle c tio n o f te rm s in
o n e c a s c a d e a t a tim e , s o th a t th e n u m b e r o f tim e s m a p p in g h a s to ■
b e u s e d c a n b e re d u c e d to a n e x te n t d e p e n d in g on how th e g iv e n fu n c tio n
m ay b e d e c o m p o s e d in to c a s c a d e - r e a l i z a b l e f u n c tio n s .
No u s e h a s b e e n m ade in th e a b o v e a rg u m e n t of th e in te rm e d ia te
c e l l f u n c tio n s a^ (o r b^) fo r i< n .
I t i s p o s s ib le to h a v e e x tra c e ll
in p u ts w h e re n e c e s s a r y to r e p r e s e n t a n d u s e them in fo rm in g a p a rt
or th e w h o le of th e fu n c tio n in a m ore e f f ic ie n t m a n n e r.
In th e
fo llo w in g , a n e x a m p le o f r e a liz in g a g iv e n fu n c tio n u s in g b u lk
tr a n s f e r in c a s c a d e s is w o rk e d o u t.
-55-
E xam ple 3 .4
F =
(x^ + X3) + (x2 + x
+ x ) X5 + X2 X3 X^
T h is fu n c tio n m ay b e r e a liz e d in a c c o r d a n c e w ith th e p ro c e d u re
s u g g e s te d e a r l i e r by r e a liz in g one term of th e fu n c tio n in a c a s c a d e ,
m ap p in g th e o u tp u t to th e o th e r c a s c a d e a n d c o m b in in g w ith th e n e x t
term r e a liz e d in th a t c a s c a d e , m app in g b a c k a n d so o n .
F ig u re 3 . 4 . 2
(a)
(b)
F ig u re 3 . 4 . 2
(a)
In p u t C a s c a d e
s h o w s th e tw o c a s c a d e s .
(b)
O u tp u t C a s c a d e
The c e ll in p u ts { y ^ , y 2 , . . . y ^ a r e th e sam e
a s th e inputs[x-j, , x ^ , . . .x ^ )
. The c e ll in p u t y^ - a g a n d Xy = b g . The
c e ll f u n c tio n s in th e in p u t c a s c a d e a re in itia lly
3 ; =X^Xg
-56-
To r e a liz e th e s e c o n d te rm , m ap th e in p u t a rra y in to o u tp u t a rra y a s
s p e c if ie d .
The c e ll f u n c tio n s in th e o u tp u t c a s c a d e a re i n i t i a l l y
bJ = Y l = X 1
b2 = b I y S = x l x 3
bS = b4 = b3 = b2
b 6 = b S + y 6 = x I x 3 + X1X2
To r e a liz e th e n e x t te rm , c h a n g e a p p r o p r ia te ly th e c e ll fu n c tio n s in
th e in p u t c a s c a d e a n d m ap th e o u tp u t a rra y b a c k a s s p e c if ie d a b o v e .
The c e l l fu n c tio n s in th e in p u t c a s c a d e a re now
a3 = a 2 = a l
a 4 = X5a 3 = X5X2
a S = S4
a 6 = x 7 + a 5 = x i x 3 + x I x 2 + x Sx 2
In t h i s m a n n e r, to r e a liz e th e f u n c tio n , fiv e m a p p in g s a re n e c e s s a r y .
To i l l u s t r a t e t h a t s u ita b le u s e of in te rm e d ia te fu n c tio n s c a n re d u c e
th e n u m b e r o f m a p p in g s to a c o n s id e r a b le e x te n t , a n o th e r r e a liz a tio n
w ill be g iv e n b e lo w .
The in p u t a n d o u tp u t c a s c a d e s a re sh o w n in
F ig u re 3 . 4 . 3 . a a n d 3 . 4 . 3 . b .
-57-
F ig u re 3 . 4 . 3
A rray C o n fig u ra tio n on M ap p in g fo r th e R e a liz a tio n of
F = Xj (x 2 + Xg) + (Xg + Xg + X^) Xg + XgXgXg'
(a) In p u t C a s c a d e A rray
(b) O u tp u t C a s c a d e Array
( a f te r a m apping) (c) In p u t C a s c a d e A rray (a fte r tw o
m a p p in g s ) .
Let th e c e ll fu n c tio n s in th e in p u t c a s c a d e b e s e t to p ro d u c e th e
fo llo w in g f u n c tio n s (F igure 3 . 4 . 3 . a):
-58-
Sl=Xg
^3 = = 2 + = 3 + = 4
8 / =r X c ( x
'5 ' "2
S5
+ Xo + X , )
=6
N e x t, u s e 'a s u ita b le m ap to h a v e (F ig u re 3 . 4 . 3 . b)
^l==I
y3==z
? 4 = =3
Yg = 34
T h e n , s e t th e c e ll fu n c tio n s to h a v e (F igure 3 . 4 . 3 . b)
b I = y I y 2 = x I ( x2
b 2 = IrS = x 2
b3 = y3y4 = X2X3
b4 = y3y4y5 = X2X3X6
b5 = b4 + y 6 =
x 2x 3x 6
+ x 5 (x2 + x 3 + x4>
L et x! b e th e v a r ia b le m a p p ed b a c k fo r th e f i r s t tim e to th e p re v io u s
i
in p u t p o s itio n x^.
U se s u ita b le m ap (F igure 3 . 4 . 3 . C ) s o th a t
-59x'
= b
I
4
l
= x
l
( x9 + x )
z
S
= b J = X 2 X 3 X 6 + X 5 (X g + X g + X ^ )
O th e r v a r ia b le s a re no lo n g e r n e c e s s a r y .
N ow , s e t th e c e ll fu n c tio n
s u c h th a t
a , = x . + x . = X 1 (X 2 + X 3 ) + X 2 X 3 X 6 + X 5 ( X 2 + X 3 + X 4 )
- F
It i s p o s s ib le to r e a liz e th e a b o v e fu n c tio n in s e v e r a l o th e r w a y s
An e f f ic ie n t r e a liz a tio n i s la r g e ly d e p e n d e n t on a s u ita b le d e c o m p o s i­
tio n o f th e f u n c tio n .
C h a p te r 4
PARALLEL BULK TRANSFER SYSTEM
-61-
4.1
P a ra lle l T ra n s fe rs
It h a s b e e n sh o w n in C h a p te r 2 th a t a rb itra ry lo g ic c a n be
r e a liz e d u s in g b u lk tr a n s f e r te c h n iq u e .
To do t h i s , c o m p le x m aps of
th e in p u t a rra y a re s o m e tim e s re q u ire d a n d c o m p o se d o u t o f a lim ite d
nu m ber of b a s i c m ap s a v a ila b le in th e m ap p in g d e v ic e .
As i t i s a
le n g th y a n d tim e -c o n s u m in g p r o c e s s , i t i s n e c e s s a r y to c o n s id e r a
s e r i o - p a r a l l e l a p p ro a c h to th e p ro b le m .
O ne m ay d iv id e th e d a ta on th e
in p u t a rra y in to s m a lle r s u b - a r r a y s a n d th e n , on th e s e s u b - a r r a y s
b u lk tr a n s f e r te c h n iq u e c a n be a p p lie d in p a r a lle l .
The g e n e r a l
n a tu re of th e s y s te m in th is sc h e m e c a n be v is u a liz e d a s a m u lti­
le v e l B. T. s y s te m w ith a t r e e - l i k e s tru c tu re (F igure 4 . 1 ) .
M ap p in g a n d L ogic
O u tp u t
F ig u re 4 . I
M u lti- le v e l Bulk T ra n s fe r S y stem
-62-
To in v e s t ig a te th e c h a r a c t e r i s t i c s o f th is ty p e o f p a r a lle l
B. T. s y s t e m . w e f a c e th e fo llo w in g p ro b le m s:
a n in p u t a rra y o f v a r i a b l e s .
S u p p o se w e a re g iv e n
W e p a r titio n th e a rra y in to a s e t of
d is jo in t o r p a r tia lly jo in t s u b - a r r a y s a n d r e a liz e a rb itra ry fu n c tio n s
on th e v a r ia b le s of e a c h of th e s u b - a r r a y s to g e t a s m a lle r a rra y of
o u tp u ts w h ic h w e s u b je c t to th e sam e tre a tm e n t a s b e fo re an d th e n
r e p e a t th e p r o c e s s u n til a s in g le fu n c tio n i s r e a l i z e d .
fu n c tio n s c a n be p ro d u c e d in t h i s m a n n e r?
w e s e t u p th e s y s te m to r e a liz e i t ?
W h a t k in d of
G iv e n a fu n c tio n , how do
If a l l fu n c tio n s a re n o t r e a liz a b le ,
how do w e t e s t fo r th e r e a l i z a b i l i t y of a fu n c tio n ?
T h e se q u e s tio n s
g e n e r a lly p e r ta in to th e s u b je c t of d e c o m p o s itio n of s w itc h in g
fu n c tio n s .
The s u b je c t of d e c o m p o s itio n o f s w itc h in g fu n c tio n s h a s b e e n
s tu d ie d in d e t a i l b y A sh e n h u rs t
( 9)
(4 )
^ . a n d C u rtis
.
H o w e v e r, th e
ty p e o f d e c o m p o s itio n to be c o n s id e re d h e r e , th o u g h r e s t r i c t e d ,
p r e s e n ts a fo rm id a b le problem b e c a u s e o f s h e e r s iz e w h e n th e s e
m e th o d s a re s o u g h t to b e a p p lie d .
T h e s e a re c h a rt m e th o d s fo r th e
p u rp o s e o f g e n e r a l d e c o m p o s itio n .
The problem a s s o c i a t e d w ith
d e c o m p o s itio n o f a la rg e a rra y in to jo in t o r d is jo in t s u b - a r r a y s of
know n s iz e an d g e o m e try a n d s tu d y o f th e r e a liz a b le fu n c tio n s s e e m s
im p o s s ib le to h a n d le in te rm s o f d e c o m p o s itio n c h a r ts a n d n e e d s som e
C
-63-
s o r t o f a lg e b r a ic a p p r o a c h . Roth a n d K a r p ^ ) h a v e u s e d th is k in d of
.
a p p ro a c h in d e a lin g w ith th e pro b lem o f m in im iz a tio n o f B o o lean g r a p h s .
The fo llo w in g m a tte r w ill be d e v e lo p e d a lo n g th e l a t t e r l i n e .
A b u lk tr a n s f e r s y s te m m ay b e c o m p ared w ith r e s p e c t to lo g ic a l
u n iv e r s a li ty w ith th e r e c e n tly p ro p o s e d U n iv e rs a l L o g ic a l M o d u le ( ^ )
I
o r ULM' .
The term "ULM " w ill be u s e d in th e fo llo w in g fo r a b u lk
tr a n s f e r s y s te m o r a n y o th e r d e v ic e h a v in g lo g ic a l u n iv e r s a li ty , b e c a u s e
th e t h e o r e tic a l d e v e lo p m e n t i s in d e p e n d e n t o f how th e d e v ic e i s
im p le m e n te d .
C o n s id e r th e in p u t a rra y of th e B. T. s y s te m in F ig u re 2 . 1 . 1 .
L et u s a s s u m e th a t th e a rra y o f c e l l s is d iv id e d in to s d is jo in t s u b s e ts
w h e re e a c h s u b s e t d o e s n o t n e c e s s a r i l y c o n ta in th e sam e n u m b er of
c e lls .
A lso a s s u m e th a t.th e r e a re a s e t o f U L M 's e a c h o f w h ic h c an
r e a liz e a r b itr a r y lo g ic a l fu n c tio n s on th e v a r ia b le s of e a c h s u b s e t a n d ,
f u r th e r , th e o u tp u ts of th e s e U L M 's a re s im ila rly (as in th e o rig in a larra y ) fe d a s in p u t to n e x t h ig h e r le v e l U L M ' s a n d .th e p r o c e s s is
re p e a te d u n til a s in g le fu n c tio n i s r e a l i z e d .
(N ote th a t th e in p u ts to
a n ULM a t a n y le v e l com e from a d is jo in t s e t of o u tp u ts of th e p re v io u s
le v e l a n d t h e s e s e t s n e e d n o t be of th e sam e s i z e ) .
C le a r ly if th e
lo g ic a l fu n c tio n s p erfo rm ed by e a c h ULM is k n o w n , th e u ltim a te
A cco rd in g to c o n v e n tio n a l te rm in o lo g y , a n n - v a r ia b le ULM h a s , in g e n e r a l,
m in p u ts w h e re m § n . S in c e w e a re n o t c o n c e rn e d w ith th e s tru c tu re of
th e U LM , w e s h a ll m e a n , by n - in p u t U L M , a ULM of n v a r i a b l e s .
- 64-
fu n c tio n can. be d e te rm in e d in te rm s o f th e o rig in a l v a r ia b le s by tra c in g
b a c k from th e o u tp u t to th e in p u t.
But th e r e v e rs e p ro b le m — g iv e n a
f u n c tio n , to d e c id e if i t i s r e a liz a b le by s u c h a s c h e m e , a n d w h en
r e a l i z a b l e , to fin d th e fu n c tio n s to b e p erfo rm ed b y e a c h ULM — is
n o t so s im p le .
To know th e r e a liz a b le fu n c tio n s w e m u s t fin d a
c r ite r io n fo r r e a l i z a b i l i t y .
R e a liz a b ility t e s t s fo r a m u ltile v e l s y s te m ,
w h e re e a c h le v e l c o n ta in s a n u m b er o f m u lti- in p u t U LM 1s w ill th e re fo re
b e d e v e lo p e d h e r e . The a lg o rith m to b e p r e s e n te d h a s so m e a d v a n ta g e s ,
b e c a u s e o f th e f a c t th a t i t is a p p lic a b le to th e s u m -o f - p r im e - im p lic a n ts
form o f a fu n c tio n so t h a t th e v a s t n u m b er of m in term s of th e fu n c tio n n e e d
n o t be c o n s id e r e d .
The s u m -o f - p r im e - im p lic a n ts form i s a s s u m e d -to .
b e o b ta in e d from a n o rig in a l d is ju n c tiv e - f o r m e x p r e s s io n b y som e
s ta n d a rd p ro c e d u re s s u c h a s th e m eth o d o f c o n s e n s u s ^
. F u rth e r,
th e a lg o rith m c o m b in e s th e a d v a n ta g e s of a lg e b r a ic m eth o d w ith th o s e
of ta b u la r te c h n i q u e . F in a lly , th e c o m p le x ity of th e a lg o rith m d o e s
n o t in c r e a s e w ith th e s iz e o f s u b s e ts in to w h ic h th e a rra y is o rig in a lly
p a r titio n e d .
H o w e v e r, c o m p u ta tio n s in c r e a s e w ith th e n u m b er of
in p u ts o f a U L M .
The c a s e w h e re th e p a rtitio n in g o f th e a rra y is s u c h th a t th e .
r e s u l t a n t s u b - a r r a y s a re n o t d is jo in t w ill b e c o n s id e re d in a la te r
s e c tio n .
As m ay be g u e s s e d , th is c a s e i s m ore c o m p le x , p a r tic u la r ly
-65-
b e c a u s e o f th e f a c t th a t th e re a re m ore th a n o n e w a y of r e a liz in g
c e r ta in fu n c tio n s a t a n y le v e l a n d n o t a l l <_>f th e s e m ay u ltim a te ly
le a d to a r e a liz a tio n .
4 .2
D is jo in t D e c o m p o s itio n
In th is s e c tio n w e s h a ll s ta r t by c o n s id e rin g sim p le c a s e s
of d is jo in t d e c o m p o s itio n an d g ra d u a lly b u ild u p m ore c o m p le x c a s e .
A g e n e r a l a lg o rith m fo r t e s t in g r e a l i z a b i l i t y w ill be g iv e n a t th e e n d .
P----------- -Aj —
~— —-|
F ig u re " 4. 2. 1
y---------- — ----------- 4
T w o - le v e l N etw ork w ith T w o -in p u t ULM a t th e L a st L ev el
-66-
C o n s id e r th e fo llo w in g s im p le c a s e :
An a rra y A i s p a rtitio n e d
in to d is jo in t s u b s e ts A 1 a n d A (F igure 4 . 2 . 1 ) .
t
2
fu n c tio n on A, a n d U
1 . 2
U
I
i s a n y a rb itra ry
on A a n d FL i s th e f in a l o u tp u t f u n c tio n .
2
3
Let
F b e a g iv e n fu n c tio n e x p r e s s e d a s a sum o f prim e i m p l i c a n t s . If F
c a n be r e a liz e d by th e n e tw o rk of F ig u re 4 . 2 . 1 , th e n F c a n be
e x p re sse d a s:
F=U 3 (U 1 , U 2 )=U 3{ U (Xj 7X2 , . . . ,XfcK U 2 (xk + 1 , . . . , x n )}
As th e d e c o m p o s itio n i s d i s j u n c t i v e , th e o c c u rre n c e of th e v a r ia b le s
X j , X2 , . . . , x k in F m u st be a c c o u n te d fo r b y th e fu n c tio n 'U j a n d th e
o c c u rre n c e of th e v a r ia b le s x k + J , x k + 2 , . . . ,x ^ b y th e fu n c tio n U3 T h e re fo re w e c a n m ake a d e c o m p o s itio n ta b le (T able 4 . 2 . 1 ) w ith tw o
c o lu m n s h e a d e d by Aj a n d A g.
T a b le 4 . 2 . 1
D e c o m p o s itio n T ab le
E ach prim e im p lic a n t (PI) of F o c c u p ie s a d is t i n c t row in t h i s ta b le .
T h o se l i t e r a l s of th e PI t h a t a re form ed o f th e v a r ia b le s o f Aj a re p la c e d
-67-
■u n d e r
a n d th o s e form ed o f th e v a r ia b le s o f Ag a re p la c e d u n d e r Ag,
b o th p a r ts o c c u p y in g th e sam e ro w .
For n o ta tio n , a s u b s c r ip t I w ill
b e u s e d fo r a p ro d u c t term u n d e r A
such a s
e tc .
S im ila rly ,
a s u b s c r ip t 2 w ill be u s e d fo r a p ro d u c t term u n d e r Ag, s u c h a s P g ,
Qg e t c .
In g e n e r a l, th e r e f o r e , P i's of F w ill b e r e p r e s e n te d a s
P 1 Pn , P 1Q 0 , Q 1 Q 0 e t c . w h e re P1 , P1 , Q 1 a p p e a r u n d e r A1 an d
I
Z
I , ^
JL
Z
i
l
l
I
P2 ' ^ 2 ' ^ 2 a Pp e a r u n ^ e r A g.
.We s h a ll a s s u m e U 2 to be a n o n - d e g e n e r a te fu n c tio n o f
an d
U g , a s o th e r w is e , F m u s t be in d e p e n d e n t o f th e v a r ia b le s of one or
b o th of t h e s e f u n c t i o n s . W e s h a ll c o n s id e r h e re a lin e a r a rra y of
s u b - a r r a y s , b u t th e re is no l o s s o f g e ? re ra lity b e c a u s e , w h a te v e r be
th e d im e n s io n of th e o rig in a l a r r a y , s in c e th e d e c o m p o s itio n is
d i s j u n c t i v e , th e s u b - a r r a y s c a n a lw a y s b e a rra n g e d in a lin e a r fo rm .
F ir s t , a few th e o re m s w ill b e p ro v ed on w h ic h th e a lg o rith m w ill be
based.
T he th e o re m s a re tru e for a n y r e a liz a b le fu n c tio n F .
T heorem 4 . 2 . 1
L et P^P 2 b e a prim e im p lic a n t of F .
of e ith e r U 1 o r U 1 a n d P 0 i s a PI o f e ith e r U
T hen P^ is a PI
or U0 .
Proof: L et P^Pg b e .a PI o f F . W e s h a ll f i r s t prove t h a t P^ i s e ith e r in
U^ o r in U^ a n d Pg i s e ith e r Ug or in U g .
-6 8 The fo llo w in g a re th e p o s s ib le in c lu s io n s itu a tio n s fo r
A.
For P^:
i)
P1 = U 1
«)
P1= U 1
iii)
■
P K U , P ^ U a n d th e r e e x i s t m in te n n s M 'a n d M 9
i
I
I
I
I
Z
c o n ta in e d in P^ s u c h t h a t
> B.
an d
.
F or P g:
1)
P2 = U 2
2)
pZ ^ 2
3)
Pg
P g ^ Hg
a n d th e r e e x i s t m in te rm s Mg and
c o n ta in e d in P^ s u c h th a t MgG U g ,
M ^ e Ug .
C o n tra ry to o u r a s s e r t i o n , a s s u m e th e fo llo w in g i s tru e :
C ase I
iii) i s tru e fo r P^ a n d 3) is tru e fo r P g . A ss ig n v a lu e s to
th e in p u ts (x ^ , . . . ,x ^ ) s u c h t h a t th e fo llo w in g tru th ta b le i s o b ta in e d .
In p u t A ss ig n m e n t
U!
U2
F
'
r— I
M^ = I
I
0
I
M
= I
0
I
I
Mg = I , M^ = I
0
0
I
co
r-H
M = I ,
M
Il
S
I
r—i
I
Il
I
= I
,
F i s s e e n to be a c o n s ta n t fu n c tio n w h ic h c o n tr a d ic ts th e h y p o th e s is
t h a t F i s n o n - d e g e n e r a te .
-69-
C a s e II
i) i s tru e fo r P
I
a n d 3) i s tru e fo r P .
2*
A ss ig n v a lu e s to th e in p u ts s u c h t h a t P^ = I a n d h e n c e U
S in c e Pg = I w h e n e v e r
w henever U 1 = I .
1
= I or
H ence F = U
fu n c tio n o f U^ an d U ^ .
U = P +
1 , 1
I
= I.
= I , th is m e a n s th a t F i s tru e
+ S(U , U ) w h e re S(U 1 , U 9) is som e
I
2
i
z
.
But P^ £
U , th e re fo re
g (x , , . . . , x )
I
k
fo r som e fu n c tio n g ( x 1 , . . . , x )
1
k
F = P 1 + P t x 1 , , . . , X k) + S(U f U 2)
T h is im p lie s t h a t P
I
i s a n im p lic a n t of F . T h is c o n tr a d ic ts th e
a s s u m p tio n th a t P^Pg is a PI of F .
C a s e III
ii) is tru e fo r P 1 a n d 3) i s tru e fo r P g .
A rguing a s in c a s e II, w e c a n sh o w th a t F = Pj^+hCx ,...,X k ) +
R ( U p U g ) w h ic h g iv e s a c o n tr a d ic t io n .
C a s e IV a n d V, a s s ta t e d b e lo w , a re a n a lo g o u s , s in c e w e s h a ll a rriv e
a t th e c o n tr a d ic tio n th a t Pg is a n im p lic a n t of F .
C a s e IV
iii) i s tru e fo r P 1 a n d I) i s true, fo r P g .
C ase V
iii) i s tru e fo r P 1 a n d 2) i s tru e fo r P g .
The re m a in in g fo u r c a s e s conform to th e th e o re m .
-70-
an d
. an d
-P 1
P! ^ u 1
an d
an d
P1
B2 '
U2 -
P G
2
P2 =
B2 c! I
CO
C a s e IX
I
Ul
C a s e VIII
s U
oT 1
C a s e VTI
pi
Ul
C a s e Vl
The th e o re m w ill b e p ro v e d if w e c a n show th a t in t h e s e l a s t
fo u r c a s e s
i s a PI of U^ or
an d P^ i s a PI of U 2 o r
. In c a s e VI,
F = U 1U2 H-GtU1 , U2)
s in c e P^ P^ i s a PI of F , a n d G (U ^ , Ug) i s som e fu n c tio n in U 1 a n d U g .
S u p p o se P 1 i s n o t a PI of U 1 .
Thus th e r e e x i s t s a PI of U 1 , s a y P *
s u c h t h a t P1C P1*." But w h e n P1* = I 1 Pg = I w e h a v e U 1Ug = I ,
t h a t i s F = I . T hus P^P^ c a n n o t b e a PI of F . A c o n tr a d ic tio n .
H e n ce
P m u s t b e a PI of U . In a s im ila r w a y , w e c a n show t h a t P 9 is a
1
1
.
z
/
PI o f U . In a n a n a lo g o u s f a s h io n w e c a n p ro v e th a t in c a s e VII, P.
'2
I
i s a PI o f U 1 , Pg i s a PI of U ^ , in c a s e V III, P 1 i s a PI o f U^ a n d P^
i s a PI o f Ug a n d in c a s e IX, P^ i s a PI o f U 1 an d Pg is a PI o f U g .
H e n ce th e th e o r e m .
T heorem 4 . 2 . 2
’
Q.E.D.
If P, P„ an d Q . Q 0 a re tw o P i's of F a n d if P 0 =Q„
J-Z
^
2
th e n QjG U ^ (U 1) if PjC U ^(U 1) a n d fu r th e r , P 1 a n d Q 1 a re P i's of
U ^ (U ^ ).
C o n v e r s e ly , if P1 , Q 1 ^ U ^(U 1) , th e n Q ^ P g - F if P 1 P ^
F.
-71-
F u r th e r, i f P
I
and Q
I
a re P i's of U (U 1) a n d P P . i s a PI o f F , then.
I - *I ,Z .
Q P 0 i s a PI o f P .
M
P ro o f: F ir s t P art:
L et P^P 2 a n d Q Q 2 b e tw o P i's of F s u c h t h a t
F u rth e r, W L O G , l e t P ^ s
a n d Pgs U g . If p o s s ib le , le t Q^c U 1 .
C o n s id e r th e in p u t c o n d itio n s s u c h th a t P^Pg i s t r u e .
T h en ,
F = U 3 ( Ur U 2) = ! .
E x p re s s e d in te rm s of U^ , U g , a n d Ug ,
w h en U^ = I , Ug = I , th e n U 3 = I...........(i)
C h an g e th e in p u t c o n d itio n s a s n e c e s s a r y to m ake Qg t r u e .
Pg = Q , w e h a v e now Q ^Q g tru e a n d c o n s e q u e n tly F = I .
S in ce
E x p re s s in g
in te rm s o f U g , Ug , a n d U 3 ,
w h e n Ug = O, Ug = I , th e n U 3 = I . . . (ii)
From (i) a n d ( i i ) , w e c a n w rite F = Ug + T w h e re T i s som e
e x p r e s s io n in v o lv in g , in g e n e r a l, U^ a n d U g - A lso , s in c e
? 2 c U g, Ug = Pg + S w h e re S i s som e sum of p ro d u c t te rm s o f th e
o r ig in a l v a r i a b l e s . T h e re fo re ,
F = P2 + S + T.
I t fo llo w s t h a t P 1 P 0 i s n o t a PI of F .
L z
T h e re fo re if P U ^ , th e n Q gs U^ .
P g ^ Ug th e n QgE Ug .
C o n tra d ic tio n .
S im ila rly i t c a n b e sh o w n th a t if
T hat Pg a n d Qg a re P i's of Ug(Ug) fo llo w s •
-72-
from T heorem 4 . 2 . 1 .
Proof of th e c o n v e r s e :
W L O G , Pg c Ug .
tru e .
L et P j , Q g
a n d P^P^c F . A ssu m e
C o n s id e r th e in p u t c o n d itio n s s u c h t h a t P^Pg is
T h e re fo re ,
F = T J 3 (u P u 2) = ! .
E x p re s s in g in te rm s o f TJ1 , TJg an d TJ3 , th e r e f o r e , w e m ay w r ite ,
F = TJ1 TJ2 + T
w h e re T i s a g a in som e e x p r e s s io n in v o lv in g TJ1 a n d TJ2 .
Now c h a n g e th e in p u t c o n d itio n s a s n e c e s s a r y to m ake Q
t r u e . The
tru e c o n d itio n o f Pg n e e d n o t b e c h a n g e d fo r th is p u r p o s e . As
= I , Ug = I , h e n c e from a b o v e , F = I .
T h erefo re Q 1PgG F if.
PlPg= F.
Let P 1 a n d
b e P i's of TJ1 a n d P 1Pg a PI of F . T h e re fo re ,
Q 1Pg G F . If Q 1 P 2 i s n o t a PI, le t Q 1 lfcPg* b e a PI of F w h e re
Q 1= Q 1*
and
Pg = Pg *
s u c h t h a t a t l e a s t o n e of th e im p lic a tio n r e la tio n s i s p ro p e r.
L et Q(=
Q 1* .
W e th e n h a v e Q 1 Pgc ^ l * P2 c F • ^ b e n ,. u s in g
f i r s t p a r t, Q 1*= U 1 ; T h e re fo re Q 1 c a n n o t b e a PI of U 1 .
C o n tra d ic tio n
-73-
S o Q 1 = Q i *.
L et P^c P^*.
T hen,
P 1 P2 C Pl P2 * - F (sinC e Q l P 2 * £ F ) *
So P 1 P^ c a n n o t be a PI o f F .
H e n ce P^ = P * .
C o n tr a d ic tio n .
T h a t i s , w ith th e g iv e n c o n d itio n s , Q 1Pg is a PI o f F .
Q.E.D.
.C o ro lla ry I .- The s e t of P i's of U 1 a n d U 1 o b ta in e d u n d e r A1 c o m p le te ly
d e fin e U 1 a n d U 1 .
•
I
I
c o m p le te ly d e fin e U
S im ila rly th e s e t o f P i 's o b ta in e d u n d e r A9
'6
Z
and U .
it
P ro o f: If th e o rig in a l e x p r e s s io n fo r F i s a c o m p le te sum of P i 's th e n th e
c o ro lla ry fo llo w s from th e c o n v e rs e o f Theorem 4 . 2 . 2 .
W hen U^ is n o t
u n a te , in s a y , U 1 or U 1 , th e n a l l P i's of U 1 an d U 1 w ill a p p e a r u n d e r A1 ..
If Ug i s u n a te , in , s a y U1 , th e n th e P i's of U 1 c a n b e o b ta in e d from U 1
b y c o m p le m e n ta tio n .
Let F b e n o t a c o m p le te sum o f P i ' s .
Let
M 1C
I
U 1 b e a m interm
I
t h a t i s n o t in c lu d e d in th e P i's of U 1 o c c u rrin g u n d e r A1 .
P 1 P2
a PI of F s u c h th a t P ^ U 1 .
Let
T h e n , b y T heorem 4 . 2 . 2 ,
M 1PgG F 1 H e n c e , M 1Pg m u st be in c lu d e d in som e P i's o f F .
a s s u m p tio n Pg i s n o t a PI o f F .
M P 9c ‘ Q P .
By
H e n c e 3 Q ^ u n d e r A1 s u c h th a t
It fo llo w s t h a t M c Q
I
I
By Theorem 4 . 2 . 1 , Q
I
I
-74-
i s a prim e im p lic a n t of U .
b u t n o t in c lu d e d in a PI o f U
' I
i t c a n be show n t h a t
a n d Ug a n d
—
T h u s , th e a s s u m p tio n
■£
,
a p p e a rin g u n d e r A i s f a l s e .
S im ila rly ,
is c o m p le te ly d e fin e d b y th e te rm s u n d e r A^
a re c o m p le te ly d e fin e d b y th e te rm s u n d er. A ^.
Q .E . D.
T heorem 4 . 2 . 3
L et U ^ (x ^ , . . . , x^) a n d . U^ (x^^ ^ , . . . ,x ^ ) b e tw o
B oolean fu n c tio n s d e fin e d on d is jo in t .s u b s e ts of v a r i a b l e s . If
a n d Ug a re e x p r e s s e d a s a sum o f prim e im p lic a n ts lik e
U^ =
+ Q^. + . . . , a n d U^ = P^ +
+ . . . , th e n e a c h r e s u lta n t
p ro d u c t term w h en
F ( x .......... x ) = U 1 U 2 = (P 1 + Q 1 + . . . ) ( P 2 + Q 2 + . . . )
i s w r itte n a s a sum of p ro d u c ts i s a p rim e im p lic a n t of F .
P ro o ffW ith th e c o n d itio n s g iv e n in th e a b o v e th e o re m , l e t P, P„ a n d
■
i L
Q 1Q 2 b e tw o d i s t i n c t p ro d u c t te rm s o f F .
From th e d e f in itio n of
prim e im p lic a n t i t fo llo w s th a t am ong th e p ro d u c t te rm s of F , if a
r e la tio n s h ip i s fo u n d b e tw e e n a p a ir o f te rm s s u c h th a t (i) o n e term is
o f th e form Rx 1 an d th e o th e r term o f th e form Sx 1 w h e re R a n d S a re
p ro d u c ts of l i t e r a l s n o t in v o lv in g X1 a n d e ith e r R - S or S £ R o r,
(ii) o n e term is su b su m e d by th e o th e r , th e n an d th e n o n ly , a l l te rm s
o f F do n o t r e p r e s e n t prim e im p lic a n ts .
-75-
As P^Pg a n d
Q
2 c a n be a n y tw o g e n e r a l te rm s o f F , w e n e e d
to sh o w t h a t c a s e (i) a n d (Ii) c a n n o t o c c u r b e tw e e n P^Pg a n d Q ^Q 2 *
Let u s s u p p o s e , c a s e (i) i s tr u e .
W LOG, le t
a n d P P = x . P *P a n d Q Q = x .Q *Q
1 2
1
Z
1 2
1 I Z
c [ x , ..
•
w ith P = x .P . * an d
I
11
Q^ = X jQ * w h e re P^* a n d Q ^* a re th e p ro d u c ts of l i t e r a l s re m a in in g
a f te r d e le tio n o f x^ a n d X jfrom P^ a n d Q^ r e s p e c tiv e ly .
W LO G , th a t P 1 ^P 2Q Q 1 * Q2 .
N
ow
,
( P ^ Q 1 ) an d
d e fin e d on d is jo in t s e t s o f v a r i a b l e s .
A lso a s s u m e ,
( P 3 ,Q 2]. a re
T h e re fo re , w e m u st h a v e ,
P1*= Q *
and
P2= Q 2
It fo llo w s th a t
i s n o t a sum of prim e im p lic a n ts b e c a u s e c a s e (i)
i s s a t i s f i e d b e tw e e n P^ a n d Q 1 . T h is is c o n tra d ic to ry to th e a s s u m e d
c o n d itio n s .
T h e re fo re c a s e (i) c a n n o t b e tru e a b o u t P^P 2 an d Q 1 Q 2 .
To p ro v e th a t c a s e (ii) c a n n o t p o s s ib ly o c c u r, i t is . s u f f ic ie n t to
sh o w th a t th e re e x i s t s a t l e a s t o n e m interm in P 1 P2 w h ic h d o e s n o t
o c c u r in Q 1Q 2 a n d v ic e v e r s a .
S in c e P 1Pg a n d Q 1Qg a re d i s t i n c t , e ith e r P 1
b o th .
W L O G , a s s u m e P 1 / Q . As P
1
n o t su b su m e d by th e o th e r .
1
T h e re fo re
I
and Q
I
Q 1 o r Pg
^Q
or
a re P i's o f U 1 , o n e i s
I
-763
P 1Xi
I
O 1 and
Q 1X ^ P 1
w h e re X , X, a re p ro d u c ts of th e lite ra ls .-o f som e v a r ia b le s i i r
I
] .
{ X^
• •,./ Xj^ } • It fo llo w s th e re fo re th a t
p A
0 Iq 2
p2 ^
and
Q 1XjQ 2 A p I p 2 T h e re fo re c a s e (ii) a ls o c a n n o t o c c u r .
H e n ce th e th e o r e m ,
Q.E.D.
The fo llo w in g th e o re m c a n be p ro v e d v e ry sim p ly fo r th e tw o -v a ria b le •
case.
H o w e v e r, fo r e a s e in g e n e r a liz a tio n , i t is p ro v e d in so m ew h a t
d e ta il.
T heorem 4 . 2 . 4
U
V
i s u n a te in U (U ) o r U (U9)' if a n d o n ly if th e
I 4
I ^
sum of th e te rm s u n d e r A 1 (Ag) i s n o t e q u a l to I .
i
P ro o f: F ir s t P a rt: W E O G , l e t Ug b e u n a te in U 1 a n d l e t S P 1 = sum o f I*S
th e te rm s u n d e r A1 = I , w h e re
I
i
r e p r e s e n ts som e g e n e r a l term u n d e r
A1 . W e s h a ll sh o w a c o n tra d ic tio n in t h i s a s s u m p tio n .
U 1 = I f c i , . . . ,X1,) / l
S p‘ = I = U 1 H- U1 .
W e have
o r O (a s F i s n o n -d e g e n e ra te ) an d
-77-
The f i r s t c o n d itio n im p lie s th e p r e s e n c e o f som e p ro d u c t o f th e l i t e r a l s
of th e v a r ia b le s o f A1 in a t l e a s t o n e PI o f F .
The s e c o n d c o n d itio n
im p lie s t h a t th e r e m u st b e a t l e a s t a p a ir o f t h e s e p ro d u c ts so th a t
th e sum m ay b e I a n d fu rth e r th a t th e p ro d u c ts m u st b e lo n g to
c o m p le m e n ta ry fu n c tio n s a s U 1 7^ I o r 0.
I t fo llo w s th a t th e r e e x is t s
a t l e a s t a p a ir of P i's of F , P 1 P 0 an d Q Q s u c h t h a t P s U
I ^
I 4
1 1
Q 1S u
.
an d '
W LO G , l e t P2 S U g . At f i r s t , c o n s id e r Q 2 S U2 . W hen
P 1 P 2 i s tr u e , th e n U 1 = I , U 2 = I a n d Ug = I .
W h en Q 1 Q 2 i s tr u e ,
th e n U 1 = Oi U 2 = I , an d U 3 = I . T h e re fo re ,
^3 = ^ 2 + %
w h e re T i s som e e x p r e s s io n in v o lv in g U 1 a n d U 2 in g e n e r a l.
U2 - Q 2 + ^ 2
S
But
(from e a r lie r a ss u m p tio n )
w h e n S i s som e e x p r e s s io n in v o lv in g th e o rig in a l v a r ia b le s
{ X j ^ 1 , . . . , x n } . T h e re fo re ,
U 3 = Q 2 + P2 + S + T
I t fo llo w s t h a t P 1 P 2 a n d Q 1 Q 2 a re n o t P i's of F .
N e x t, c o n s id e r Q 2 S U 2 ..
C o n tra d ic tio n .
A rguing in a s im ila r m a n n e r a s b e f o r e ,
w e c a n show
U g = U 1 U 2 -P U 1F 2 + R
w h e re R i s som e fu n c tio n of U 1 a n d U 2 . F u rth e r, u s in g th e sam e
a rg u m e n ts a s in T heorem . 4 . 2 . 2 , w e c a n sh o w U 1 U 2 a n d U 1 U 2 a re
-78-
P I 's o f U .
T hat i s ,
is n o t u n a te in U ^ .
C o n tr a d ic tio n ,
T h e re fo re , th e c o n c lu s io n i s th a t if Ug i s u n a te in U p th e n th e
sum o f th e 't e r m s u n d e r
c a n n o t b e e q u a l to I .
Proof of c o n v e r s e : L et £ Pi ^ I a n d a s s u m e U q i s n o t u n a te in U or
_
i
3
1
U p W e s h a ll sh o w th is a s s u m p tio n to b e c o n tr a d ic to r y . A ccording
to a s s u m p tio n , b o th U^ a n d U^
o c c u r in th e P i's o f U^ . W L O G ,
l e t P^P^ be a PI of F s u c h th a t P^c U^ a n d P ^s U g .
a n o th e r PI o f F s u c h th a t Q s
1
u
1
.
L et Q gQ g be
T h a t Q 1 c U 1 s h o u ld b e p re s e n t
1
1
fo llo w s from th e a s s u m p tio n Ug i s n o t u n a te in Ug o r U^ .
F o r, if
£ Pg = U , th e n u s in g T heorem 4 . 2 . 2 , w e c o u ld w rite
U3 = U 1 * T
w h e re * i s o n e m em ber of th e s e t (• • , + } a n d T i s a n e x p r e s s io n fre e of
U g . W e c o u ld th u s sh o w Ug to be u n a te in U g , c o n tra ry to o rig in a l
a s s u m p tio n .
By th e C o ro lla ry to T heorem 4 . 2 . 2 , th e P i's n e c e s s a r y to d e fin e
Ug an d Ug m u st be p r e s e n t u n d e r A .
The sum of t h e s e P i's w ill be
e q u a l to (U 1 + U 1) or I .
T his c o n tr a d ic ts th e g iv e n c o n d itio n
.I
I
I
i
2 P, 7^ I . H e n c e U i s u n a te if £ P^ ^ I Jl
Q.E.D.
i 1
3
i
-79-
T heorem 4 . 2 . 5
and
If a fu n c tio n c o n ta in s tw o P i's of th e form
(or Q ) , th e n i t i s n o t r e a l i z a b l e .
B roofhS uppose th e r e e x i s t tw o P i's lik e P^P^ an d
W L O G , th a t P e
. If p o s s i b l e , le t
a ls o s
t r u e , U = I a n d ir r e s p e c tiv e o f w h a t U 2 i s , F = I .
in F . A ssu m e ,
. W h en
is
T h e re fo re ,
F = U^ + T w h e re T is som e e x p r e s s io n in v o lv in g U^ a n d Ug in g e n e r a l ;
But
U1 =P
1
+ Q 1 + S (by a ss u m p tio n )
I - I
w h e re S i s som e e x p r e s s io n in v o lv in g th e o rig in a l v a r i a b l e s .
F = Pj + Q
T h at i s , P-^Pg
Q ^ - U^ .
+ S + T
n o t a PI of F .
H e n c e , P^ a n d
T h erefo re
C o n tr a d ic tio n .
m u st b e lo n g to c o m p le m e n ta ry f u n c tio n s .
A ls o , W L O G , le t Pg c
\J .
So le t ■
W h en Pj,Pg is t r u e , w e h a v e
U^ = I , Ug = I , a n d Ug = I .............................................(I)
W h en Q j i s tru e so th a t U j = O a n d in p u t c o n d itio n s a re s u c h th a t '
Ug = I , th e n w e h a v e , s in c e Q j is a PI o f F
' .
U j = ° , U 2 = ! , a n d U 3 = ! .............................. ....
T h e re fo re , F = Ug
= U2 + R
. (2 )
U2 ^
. . . . . . .
. from
(].) an d (2)
w h e re R i s som e e x p r e s s io n in v o lv in g U j a n d Ug in g e n e r a l.
But
-80-
w h e re Q i s s o m e .e x p r e s s io n in v o lv in g th e o rig in a l v a r ia b le s
t x k + l' ‘ • ' xn I '
T h e re fo re ,
P = P2 + Q + R .
T hat i s ,
i s n o t a PI o f F .
C o n tra d ic tio n .
H e n c e , th e fu n c tio n c a n n o t be r e a l i z a b l e .
Q.E.D.
T h e s e th e o re m s e s t a b l i s h th e b a s i s of a p ro c e d u re fo r te s tin g
r e a l i z a b i l i t y of a fu n c tio n w ith d i s jo in tly d e c o m p o s e d s t r u c t u r e .
T heorem 4 . 2 . 1 s t a t e s th a t if w e d iv id e th e P i's of F in
th e m a n n e r of T a b le 4 . 2 . 1 , w e g e t th e P i's of th e c o n s ti tu e n t fu n c tio n s
U 1 an d U 1 fo r k = 1 , 2 .
k ■
k
By th e c o ro lla ry to T heorem 4 . 2 . 2 ,
i t h a s b e e n show n th a t th e P i's of Uj, (U^) o b ta in e d in t h i s m an n er
c o m p le te ly d e fin e Uv (U ) .
K k
T heorem s 4 . 2 . 2 a n d 4 . 2 . 4 a l s o
c le a r ly in d ic a te how to d is tin g u is h a Pl of U ^ from a Pl of U ^ .
H o w e v e r, if a l l P i's o f F a re n o t p r e s e n t in th e -fu n ctio n s p e c if ic a tio n ,
i t m ay n e e d a l i t t l e m ore e ffo rt to d e te rm in e w h e th e r a PI u n d e r , s a y
A j , b e lo n g s to U j o r U j .
The f i r s t PI m ay be a s s ig n e d a r b itr a r ily
to U j o r U j w h ile th e o th e r s a re to b e c o m p ared w ith th e p re v io u s
o n e s to s'ee if th e c o n d itio n s o f T heorem 4 . 2 . 2 a re d ir e c tly a p p lic a b le .
O th e r w is e , u s in g c o m p a ris o n , if a PI Q j i s s e e n to h a v e a m interm
-81-
com m on w ith a n o th e r PI, a lr e a d y a s s ig n e d t o , s a y
(by th e p ro o f o f T heorem 4 . 2 . 1 ) .
As c o n v e n ie n t, th is c o m p a riso n m ay
a l s o be d o n e w ith th e s e c o n d p a rt Pg o f ^a PI P ^ P g .
Q ^Q g b e tw o P i 's of F .
, th e n -
L et P^Pg an d
T h e n , if i t is fo u n d th a t Pg h a s a m lnterm
com m on w ith Q g , th e n a s s i g n
to
if P ^
.
T he- p ro o f of th is
s ta te m e n t fo llo w s from sim p le a rg u m e n ts s im ila r to .o n es g iv e n
b e fo re — to sh o w t h a t , if in th is c a s e ,
A.
, th e n Qg an d Pg
m u st be P i's o f F , c o n tra ry to o rig in a l a s s u m p tio n .
The th e o re m s a re e a s i l y s e e n to a p p ly in c a s e s w h e re th e o rig in a l
a rra y i s d iv id e d in to m ore th a n tw o s u b - a r r a y s .
The g e n e r a liz e d
th e o re m s a re b e in g s ta t e d w ith o u t p ro o f h e r e . The p ro o fs w ill be
fo u n d in A ppendix B.
L et A b e 'a s e t o f b in a ry v a r ia b le s
in to s d i s j o i n t s u b s e ts A ^ , Ag , . . . , Ag .
{ x ^ , Xg , . . . , x n } , p a rtitio n e d
L et
, U g , . . . , U g b e a r b itr a r y
s w itc h in g fu n c tio n s on th e v a r ia b le s o f A ^ , Ag, . . . , Ag r e s p e c tiv e ly a n d
U b e a n y s w itc h in g fu n c tio n on U]_ ,U g , . . . , U g .
L et F (x j ,x ^ , . . . ,x ^ )
b e a fu n c tio n of n v a r i a b l e s ,, r e a liz a b le a s U .
T heorem 4 . 2 . 1 '
L et
Pq P^ . . . P^..... Pjc b e a PI o f F .
PI o f e ith e r U^ o r U^, fo r i = a , b , . . i , . . k .
T hen P^ i s a
-82-
T heorem 4 . 2 . 2 '
L et P X arid Q Y b e tw o P i's o f F w h e re X an d
i
i
Y r e p r e s e n t p ro d u c ts of l i t e r a l s of th e v a r ia b le s o f a n y s u b s e ts
o th e r th a n A. . If X = Y, a n d P. E u . ( U. ) , th e n Q. q U. (U.) a n d
I
i l l
i l l
b o th .P^, Q i a re P i's o f Ui (Ui ) .
a n d Pi X c P .
T hen QiX c F .
C o n v e r s e ly , le t Pi , Qi c U^ (U J
F u rth e r, if P^, Qi a re P i's o f Uj^ U J
a n d P1X i s a PI o f F , th e n Q.X i s a PI of F .
I.
I
C o ro lla ry I'
The s e t o f P i’s of U i a n d Ui O b tain ed from F w ith
th e u s e o f Theorem 4 . 2 . 1 ' c o m p le te ly d e fin e U i an d Ui fo r a l l i .
T heorem 4 . 2 . 3 '
Let U (X1 , . . . x k) , U 2 (xk + 1 . . . x m) , . . . U s (xp / . . .Xn )
b e s B o o lean fu n c tio n s d e fin e d on d is jo in t s u b s e ts of v a r i a b l e s .
If
U 1 , U g , . . .U ^ a re e x p r e s s e d a s a sum o f prim e im p lic a n ts , su c h a s
U 1 = (P, + Qi + R 1 + . . . ) , th e n e a c h r e s u l t a n t p ro d u c t term w h en
F f r 1 .......... 2V
= U 1 U 2 " - - U s,
i s e x p r e s s e d a s a sum of p ro d u c ts in a prim e im p lic a n t o f F .
T heorem 4 . 2 . 4 '
U i s u n a te in U 1 o r U i fo r a n y i, if a n d o n ly if th e
sum yj P? o f th e te rm s o f th e v a r ia b le s o f A. o b ta in e d from F is n o t
J
e q u a l to I .
T heorem 4 . 2 . 5 '
If a fu n c tio n c o n ta in s tw o P i's o f th e form PiX
a n d Qi fo r a n y i , w h e re X i s fre e of th e l i t e r a l s of th e v a r ia b le s of Ai ,
-83-
th e n th e fu n c tio n i s n o t r e a liz a b le a s U .
T y p e s o f D e c o m p o s a b le F u n c tio n s
I
C o n s id e r F ig u re 4 . 2 . 1 .
A g iv e n fu n c tio n m ay in c lu d e th e
fo llo w in g tw o ty p e s o f p ro d u c ts sh o w n in T ab le 4 . 2 . 2 .
T a b le 4 . 2 . 2
T y p es o f Prim e Im p lic a n ts in T w o -in p u t ULM c a s e
Type I - P 1 , P 2
T ype II - Q Q 2
A cco rd in g to T heorem 4 . 2 . 5 , a r e a liz a b le fu n c tio n m ay c o n ta in
(a) o n ly ty p e I o r (b) o n ly ty p e II p r o d u c ts . For c a s e (a), a n etw o rk
w ith O R -g a te a t th e f in a l le v e l i s n e c e s s a r y .
The f i r s t - l e v e l lo g ic
fu n c tio n s U 1 a n d U 2 a re d e te rm in e d b y th e s e t of p ro d u c ts p r e s e n t
u n d e r A^ a n d A g.
S in c e a ULM i s a s s u m e d w ith in p u ts fe d b y th e
v a r ia b le s of e a c h s e t , h e n c e t h e s e fu n c tio n s c a n a lw a y s b e r e a liz e d .
C a s e (b) i s s lig h tly m ore c o m p le x .
H ere e a c h p ro d u c t
t e s t e d to s e e if i t b e lo n g s to U^ o r U^ .
m u st be
The sam e i s tr u e fo r Q g .
W h ile th is is b e in g d o n e , a tru th ta b le of th e v a r ia b le s U^ a n d U 2 m ay
b e f ille d u p to d e te rm in e th e fu n c tio n a t th e fin a l l e v e l . In th e f i r s t
l e v e l , th e fu n c tio n s U ^ , Ug c a n b e r e a liz e d o r th e ir c o m p le m e n ta rie s
w ith s u ita b le m o d ific a tio n a t th e s e c o n d l e v e l .
W e s h a l l w o rk o u t a n
—8 4 e x a m p le b e lo w fo r i l l u s t r a t i o n .
E xam ple 4 . 2 . 1 :
F = X 1 X3 X3 X4 + X 1X3 + X 1X4 + XgX3 + X3 X4 an d 'A
A2 = t X 3 ' X4 1
S te p I :
■
+ =2=4
A1 = C x 1 ,Xg 3
*1
T h is i s a n
D e c o m p o s itio n T a b le fo r F = X 1XgXgX^ + X1Xg + X 1X^
+%
Ag = [ x 3 , X4 ]
X3
■
K1
x4
x„
X
3
x„
2
x .
4
X1Xg
X3 X^
2
.
e x a m p le of c a s e (b ),
S te p 2 .
,
.
M ake a ta b le (show n in T a b le 4 . 2 . 3 ) .
T a b le 4 . 2 . 3
= [x ^ X g ]
A n a ly z e .
A p plying T heorem 4 . 2 . 4 , U 3 is n o n - u n a te .
.
L et X1- U 1 . A pplying T heorem 4 . 2 . 2 ,
X g - XJ1 . A pplying T heorem 4 . 2 . 4 ,
x 1x 2 £ ^ l
.
-
—85 —
S im ila rly Xg £ Ug
" 4 ^ 2
S te p 3 :
T a b le 4 . 2 . 4
M a k e a tru th ta b le (T able 4 . 2 . 4 ) .
A T ruth T a b le fo r Ug in te rm s of In te rm e d ia te L ev el
F u n c tio n s U , U^ in E x am p le 4 . 2 . 1
.
S te p 4 :
U
U
u I
u2
u3
0
0
I
0
I
0
I
0
0
I
I
I
S y n th e s iz e . ■
3
= U (§> U
2
I
I
= X 1 + X0
2
I
U 2 = X3 + x 4
I t m ay be p o in te d o u t th a t th e a n a l y s i s i s d o n e in a g e n e r a l m an n er
a n d c o n s ti tu te s a s u f f ic ie n t t e s t fo r r e a liz a b ility ..
i
■
-86-
The c a s e of la rg e r (more in p u ts ) U LM 1s in th e in te rm e d ia te
le v e l s a n d th e c a s e o f m u ltile v e l n e tv /o rk s m ay now b e c o n s id e r e d .
F ir s t, le t u s ta k e a tw o - le v e l, th r e e - in p u t ULM (in te rm e d ia te le v e l)
n e tw o rk (F igure 4 . 2 . 2 ) .
F o llo w in g p re v io u s d i s c u s s i o n , c o n s id e r
th e ty p e of p ro d u c ts th a t m ay be p r e s e n t in a s u m -o f-P I form of
e x p r e s s io n fo r th e f u n c tio n .
T h e s e a re lis t e d in T ab le 4 . 2 . 5 .
I---------------------------------- A------------------------------ —|
F ig u re 4 . 2 . 2
T w o -le v e l N etw ork w ith T h re e -in p u t ULM a t th e
L a s t L ev el
-87-
T a b le 4 . 2 . 5
T y p es o f Prim e Im p lic a n ts in T h r e e -in p u t ULM C a s e
A
M em b er
B
C
P0
3
Type I
P1
■I
P
Type II
QnQn
I 2
R2n*?3 •
Type III
—
m em b ers o f ty p e II I.
m em bers o f ty p e I , ^
m em b ers of ty p e II
T here a re s e v e n w a y s in w h ic h th e s e
ty p e s m ay c o m b in e a n d o c c u r in a f u n c tio n .
n o t r e p r e s e n t r e a liz a b le f u n c tio n s .
ty p e I P i 's .
S 1S
I
T 1T2T3
It i s n o te d th a t th e re a re
and
2
All th e s e c o m b in a tio n s do
C o n s id e r a fu n c tio n F c o n ta in in g
If P^ i s in F , th e n h ig h e r ty p e (II an d III) P i's in v o lv in g
p ro d u c t te rm s o f th e form
c a n n o t b e in F (by Theorem 4 . 2 . 5 ' ) .
T h e re fo re , i f P^ o c c u r s a s a PI in F , th e n o n ly
can occur a s a
PI o f F . If tw o m em b ers o f ty p e I a re in F , th e n ty p e II a n d ty p e III
cannot o c cu r.
re s tric tio n .
All m em b ers o f ty p e I c a n a l s o o c c u r w ith th e sam e
W h en ty p e I P i's a re a b s e n t , o n ly ty p e II P i's o r ty p e III
P i's o r u n d e r s p e c i a l c o n d itio n s , ty p e II P i's w ith ty p e III P i's c a n b e
p r e s e n t in F .
I t s h o u ld b e c le a r th a t p e r m is s ib le c o m b in a tio n of ty p e s m ay n o t
g iv e a r e a liz a b le f u n c tio n , an d fo r te s t i n g r e a l i z a b i l i t y . T heorem 4 . 2 . 2 '
—8 8 -
m u st u ltim a te ly b e u s e d .
S till la r g e r ULM n e tw o rk m ay b e a n a ly z e d
in a s im ila r m a n n e r.
A la rg e r.n u m b e r of le v e l s d o e s n o t in v o lv e m ore c o m p le x ity
th a n is a s s o c i a t e d w ith co m p u tin g a n in c r e a s e d nu m b er o f in te rm e d ia te
f u n c tio n s d u e to a n in c r e a s e d n u m b er o f l e v e l s . ■ F or c o n v e n ie n c e in
d e a lin g w ith m u ltile v e l n e tw o r k s , a m o d ific a tio n in n o ta tio n is
in tro d u c e d h e r e . W e s h a ll l e t U li r e p r e s e n t a g e n e r a l ULM in th e
IJ
n e tw o r k . The f i r s t s u b s c r ip t i in U ^ re f e r s to th e i - t h le v e l in th e
n e tw o rk , w h ile th e s e c o n d s u b s c r ip t j to th e j - t h p o s itio n in th a t,
le v e l.
I t h a s b e e n s e e n t h a t to d e te rm in e w h a t th e f u n c tio n U ^
sh o u ld b e , th e fu n c tio n p ro d u c e d a t th e (i+j) - t h le v e l m u s t be k n o w n .
T hus th e p ro c e d u re s h o u ld be to b e g in a t th e f in a l le v e l a n d w ork
b a c k w a rd a f te r co m p u tin g th e fu n c tio n s of th e p re c e d in g l e v e l .
A
g e n e r a l a lg o rith m b a s e d on th e p re v io u s d i s c u s s io n s m ay now b e g iv e n .
G e n e ra l A lgorithm
An a s s u m p tio n w h ic h u n d e r lie s th e d is c u s s io n h e re i s th a t, e x c e p t
fo r th e fu n c tio n s w h ic h e a c h ULM .m u st perform in o rd e r to r e a liz e a
f u n c tio n , o th e r p a ra m e te rs lik e th e n u m b e r of s u b - a r r a y s , th e n um ber
o f l e v e l s a n d th e n u m b er o f in p u ts to a n ULM a re a l l p re d e te r m in e d .
T hough th e a lg o rith m i s a p p lic a b le fo r a n a s s u m e d n e tw o rk s tr u c tu r e ,
—89 —
i t s s im p li c ity , perm i t s te s t i n g a la r g e n u m b er o f v a r i e t i e s w ith a s m a ll
am ount of e ffo rt.
L et A1 , A ,
I
2
A i s d iv id e d .
,A
be th e s u b - a r r a y s in to w h ic h th e o rig in a l, a rra y
s
L et F b e a f u n c tio n , e x p r e s s e d a s a sum o f prim e
im p lic a n ts fo r w h ic h a r e a l i z a b i l i t y t e s t i s to b e p e rfo rm e d .
S te p I : A rrange th e P i's in a m a trix w ith c o lu m n s h e a d e d by
A1 , A . . . . ,A a n d e a c h row r e p r e s e n tin g o n e PI.
i
2
s
The e n try in th e
( i , j ) - t h p o s itio n of th e m a trix i s a p ro d u c t of l i t e r a l s of a s e t of
v a r ia b le s s u c h th a t th e p ro d u c t o f l i t e r a l s i s a p a rt o f th e PI re p re s e n te d
b y th e i - t h row an d th e v a r ia b le s to w h ic h th e l i t e r a l s b e lo n g a re in th e
s u b - a r r a y A j. The P i's a re e n te r e d in g ro u p s a c c o rd in g to a s c e n d in g
o rd e r o f ty p e n u m b e rs .
S te p 2:
C o n s id e r th e ULM a t th e l a s t l e v e l .
C o rre s p o n d in g to .
e a c h o f i t s in p u ts , g ro u p th e s u b - a r r a y s (by tra c in g b a c k from th e
o u tp u t to th e in p u t a rra y ) w h ic h p ro d u c e th is in p u t.
C a ll th is group
of s u b -a rra y s a s I - s e t .
• S te p 3 : E xam ine th e te rm s o f a n I - s e t and d e te rm in e if th e o u tp u t
fu n c tio n i s u n a te in th e v a r ia b le r e p r e s e n te d b y th e I - s e t .
(T est
p ro c e d u re b e c o m e s s im p le r if i t is fo u n d to b e u n a t e ) . If th e I - s e t is
—9 0~
to o la rg e a n d d e te rm in a tio n o f u n a te n e s s i s to o in v o lv e d , th e n p ro c e e d
w ith o u t d e te rm in in g u n a t e n e s s .
A s s ig n th e f i r s t term o f a n I - s e t to a fu n c tio n .
C o m p are th e
s u c c e e d in g te rm s w ith th e p re v io u s te rm s a s d i s c u s s e d b e fo re an d
a s s i g n them to th e fu n c tio n o r to i t s c o m p le m e n ta ry in a c c o rd a n c e
w ith T heorem 4 . 2 . 2 '.
If th e o u tp u t fu n c tio n is u n a te in th e in p u t
v a r ia b le r e p r e s e n te d by th e I - s e t , th e n a l l th e te rm s of th e I - s e t m u s t
to g e th e r b e lo n g to th e sam e fu n c tio n (p re c e d in g l e v e l ) .. If a term
c a n n o t be a s s ig n e d f i r s t , k e e p i t in a w a itin g l i s t fo r t h a t I - s e t and
co m p are a f t e r a s u f f ic ie n t n u m b er of te rm s h a v e b e e n a s s i g n e d .
(If
th e re a p p e a r s no in d ic a tio n fo r a s s ig n in g i t to th e fu n c tio n o r th e
c o m p le m e n ta ry , a s d i s c u s s e d p r e v io u s ly , th e n , th e fu n c tio n is n o t
r e a liz a b le )
M ak e a c h e c k t h a t th e fu n c tio n a n d i t s c o m p le m e n ta ry a re
d e te rm in e d c o m p le te ly s u c h th a t e a c h term s a t i s f i e s th e c o n d itio n s of
T heorem 4 . 2 . 2 '.
In o rd e r to c h e c k , i t i s n e c e s s a r y to s e e , a lo n g w ith,
th e c o n d itio n s o f T heorem 4 . 2 . 2 ', t h a t e a c h PI of a fu n c tio n h a s a t
l e a s t o n e l i t e r a l c o n flic tin g w ith e v e r y PI o f i t s c o m p le m e n ta ry ,
w h e n p r e s e n t, a n d th a t th e sum o f th e P i's of th e s e tw o fu n c tio n s
e q u a ls I .
-91-
S te p 4 : W rite th e fu n c tio n s p erfo rm ed by th e p re c e d in g le v e l U L M 's
a s su m s o f F I 's o b ta in e d from s te p 3 .
(E ith er a fu n c tio n or i ts
c o m p le m e n ta ry c a n b e r e a liz e d by a ULM .)
D e te rm in e th e fu n c tio n p e rfo rm ed b y th e l a s t le v e l ULM in te rm s
of th e o u tp u ts o f th e p re c e d in g le v e l U L M 's by w ritin g a s e t o f tru e
c o n d itio n s in te rm s of t h e s e o u tp u ts .
S trik e o u t th e te rm s of th e
c o m p le m e n ta ry fu n c tio n s (p re c e d in g le v e l) from th e I - s e t s .
S te p 5: T ake th e f u n c tio n s in th e I - s e t s a n d , c o n s id e rin g e a c h
p re c e d in g le v e l ULM a s th e l a s t U LM , fo llo w s te p s 2 - 4 ,
a n d c o n tin u e
u n til fu n c tio n s p erfo rm ed by a l l U L M 's a re d e te rm in e d .
E xam ple 4 . 2 . 2
X 1X g X ^ X y X g
F “ XX gX^X yX g
+ XgXgX^XyXg
XgXgX^XyXg
+ X 1X g X ^ X g X 1 QX 11
+ X 1X 3 X 4 X 9 X 10 X 11 + X 2 X 3 X 4 X 9 X 10 X 11
+ X ^ ^ g X id K ^
+
X 3X 4 X g X g X y X g
+ X 3X 4X 3 X g X y X g
+ X 3X 4 X 5 X g X 9X 10 X 11
+ X 3 X 4X 5 X g X 9 X 10 X 11
+
X 1X g X g X y X g
+ X 1X g X g X y X g
+ X j X g X g X g X qX
+ X 1X g X g X 9 X 10 X 11
The n e tw o rk w e s h a ll t e s t fo r r e a liz in g t h i s fu n c tio n i s sh o w n in F ig u re 4.2.3 .
—9 2 -
x 9 x I Ox I l
F ig u re 4 . 2 . 3
S te p I :
The S tru c tu re of th e N etw o rk U sed fo r T e s t- r e a liz a tio n
o f th e F u n c tio n of E x am p le 4 . 2 . 2
a
T ab le i s m a d e .
S te p 2: The s u b a r ra y s c o rre s p o n d in g to e a c h 1 - s e t a re g ro u p e d .
The r e s u l t s of t h e s e s te p s a re sh o w n in T ab le 4 . 2 . 6 .
—9 3 T a b le 4 . 2 . 6
A D e c o m p o s itio n T ab le fo r th e F u n c tio n F in E xam ple 4 .2 .'2 .
A=
XX.
x x
x„x
x„x„
7 .8
XnX.
x„x
7 8
■
x x
XqX
9 10 11
XqX
X rX
XnX
XrX
X 1X,
X 1X
X X
9 10 11 ■
-94-
S te p 3 : Ug i s u n a te in Ugg 'a n ^ U^ ^ • (It m ay a p p e a r d if f ic u lt to
c h e c k b y a d d itio n ;
b u t i t c a n b e c o n c lu d e d .th a t Ug i s u n a te in U g i
b e c a u s e Ug i s a fu n c tio n of tw o v a r ia b le s a n d is u n a te .in U g g .)
Let x x X ze U
a n d x x £U
1 3 4
21
/ o Z Z
; th e n , U
o^
U 0 U 9 9 . A c h e c k is
/.i zz
m ade to s e e th is re la tio n is tru e fo r a l l P P s of U g 1 a n d U g g .
S te p 4: U g 1
+
= X 1X g X 4
X 1X 3 X 4
+ XgX3 X4 +
X 3 X 3 X 4 + X 1X 3 X 5
+ X i X 2X 6 + X 3 X 4X 5SE6 . + X 3X 4 X 5X 6
%
= X 7X8
* x Sx IOx I l
S te p 5: To d e te rm in e U g 1 in te rm s o f U 1 1 , U 1 3 , U 13 a n d U 33 in
te rm s of U 14 a n d U 1 5 .
U 31 i s n o t u n a te in U 31* , U 13* , an d U 1 3 * (w here b y U* b o th U and
U a re im p lie d ) .
L e t-X1^ U 1 1 . T h e n x 3 ^ U 1 1 . F u rth e r x^x^p U 11 a s
X 3X3 X3 X4^ U3 1 - T h e re fo re , U 11 = X 3 + X3
L e t X g X ^ e U 3 3 . T h e n x g X ^ e U 3 3 . F u rth e r XgX2J e U 13
a s X 1XgX4F U 3 1 . A lso
X3X4
e U 33 . T h erefo re U 33
L et Xg £ U jg . T hen X gE U 13 a n d XgXgE U 13 .
T h e re fo re
U 13 = X5 + Xg.
=
X3S
X4 .
-95From T a b le 4 . 2 . 6 , w e h a v e
if U 11 = I , U 12 = I ,
th e n U 21 = I
,
if U 11 = 0 , U ^3 = I ,
th e n U 21.= I a n d
if U 12 = 0 , U 13 = 0 ,
th e n U 21 = I .
T h e re fo re
U 21 = U 11U 12 + ^ 1 2 U 13 + U l l u I 3
A c h e c k i s m ad e a t th i s s ta g e to s e e if th is re la tio n i s tru e fo r a l l th e
P i's of U
U
22
/ U 12 a n d U 1 3 . N ex t d e te rm in e U ^ in te rm s O fU 14 an d U3 5
i s u n a te in U * an d U 1 * . L e t x x
14
15 ■
78
XgX1QX1 1 G U 1S-
U 22 ~ U 14 ^ U 15
^
U
and
14
T hen
.
%14 = =7=8
U 1 5 = x Sx IOx I l
F ig u re 4 . 2 . 4 s h o w s th e fu n c tio n s of th e d iff e r e n t U L Mt s in th e
n e tw o r k .
A s im p le e x a m p le of a n o n - f e a liz a b le fu n c tio n w ill b e w orked
o u t b e lo w .
E xam ple 4 . 2 . 3
'I
F = x l x 2X6 + X1X4
a n d A1 — ^
, x2
X3J
+ X2X5 + X3X6
, Ag — £ x ^ , X3 , Xg J
-96-
U U
11 12
+ U
11
U
12
I
13
F ig u re 4 . 2 . 4
S p e c ific a tio n of th e N etw o rk to R e a liz e th e F u n c tio n
o f E xam ple 4 . 2 . 2
S te p I : As b e f o r e , a ta b le i s m ad e (T able 4 . 2 . 7 )
-9 7-
T a b le 4 . 2 . 7
A D e c o m p o s itio n T ab le fo r F = XjXgXg + x i x 4+ X2X5 + X3X6
Aj =
C x i , X2 , X3 }
X1
X4
x2
x5
CO
X
X6
Xi X2
X6
S te p 2: A n a ly s is :
V
x6 1
L et x ^ s
T hen
But X1X2C x i E
A2 = [ V
U^asXgX^^F.
a n d X1X2 C X2C U 1 .
T h is is ''im p o s s i b le .
H e n c e th e
fu n c tio n c a n n o t be r e a l i z a b l e .
A p o in t th a t m ay be n o tic e d in th is p a r tic u la r e x a m p le is th atth e p ro d u c t te rm s o c c u rrin g u n d e r A1 a re n o t a l l P i's ( ’ . * x ^ x g c X1) .
T hus th e fu n c tio n d o e s n o t s a t i s f y the. c o n d itio n s of T heorem 4 . 2 . 1 .
4.3
N o n - d is jo in t D e c o m p o s itio n '
The d e c o m p o s itio n of a n a rra y into, n o n - d is jo in t s u b - a r r a y s
c a n b e e x p e c te d to p e rm it a w id e r c l a s s of lo g ic a l fu n c tio n s to be
r e a l i z e d , b e c a u s e , if th e id e a i s s tr e tc h e d to th e e x tre m e , th e r e s u lt
m ay b e a t r i v ia l d e c o m p o s itio n o f a n a r r a y in to a n u m b e r o f -s u b -a rra y s .
-9 8—
e a c h s u b - a r r a y b e in g e q u a l to th e o rig in a l a rra y so th a t th e re w ill be no
p roblem in r e a liz in g a r b itr a r y fu n c tio n s on th e a rra y by r e a liz in g i t on
th e s u b - a r r a y s .
In t h i s s e c tio n , so m e s im p le c a s e s o f n o n - d is jo in t
d e c o m p o s itio n w ill be d i s c u s s e d .
Let Aj , A g , . . .A g be a lin e a r a rra n g e m e n t o f s u b - a r r a y s s u c h th a t
e a c h a d ja c e n t p a ir of s u b - a r r a y s - - Ai an d Ai+ i - - h a v e so m e v a r ia b le s
com m on b e tw e e n them (F igure 4 . 3 . 1 ) .
A^ an d At h a v e e a c h o n ly one
n e ig h b o rin g s u b - a r r a y Ag a n d Ag ^ r e s p e c tiv e ly , w h ile a l l o th e r s u b ­
a rra y s h a v e tw o n e ig h b o r s .
Let
, Dg , . . . Di , . . . D s b e th e s u b s e ts
of A ^ A g ____A .____ Ag r e s p e c tiv e ly s u c h th a t FX c o n ta in s a l l th o s e
v a r ia b le s of Ai th a t b e lo n g o n ly to A^ a n d to no o th e r s u b - a r r a y s .
Let
C j , C ^ , . . . C g j b e e a c h a s e t of v a r ia b le s s u c h th a t C i c o n ta in s a ll
th o s e v a r ia b le s th a t a re com m on to A^ a n d Ai+ i for i~ l , 2 , . . . , ( s - 1 ) .
F ig u re 4 . 3 . 1
L in e a rly A rranged N o n - d is jo in t S u b -a rra y s
-99-
The c a s e to be c o n s id e re d i s w h e n A is c o m p o se d o f tw o s u b a r r a y s — A^. a n d A^ (figure 4 . 3 . 2 ) .
The p o s s ib le ty p e s o f P i's th a t m ay
be p r e s e n t in a g iv e n fu n c tio n a re l i s t e d u n d e r T ab le 4 . 3 . 1 .
F ig u re 4 . 3 . 2
T ab le 4 . 3 . 1
D e c o m p o s itio n in to Two S u b -a rra y s
T y p e s of Prime Im p lic a n ts in Sim ple N o n - d is ju n c tiv e C a s e
M em b ers
T y p es
I
PV P2
II
O 1x J , X1O
III
rI r
IV
2
s I h s2
V
h
As b e f o r e , P ^ ,
, e t c . — le t t e r s (e x c e p tin g X, Y, Z) w ith
s u b s c r ip t I — a re u s e d to r e p r e s e n t p ro d u c ts o f th e l i t e r a l s of th e
v a r ia b le s of D ^;
s im ila r ly , P ^ , Q ,
e t c . fo r th e v a r ia b le s of
-100X , Y , Z fo r th e v a r ia b le s o f C .
I
I
I
I
The c o m b in a tio n of d iff e r e n t ty p e s of P i's o f a fu n c tio n c a n be
s tu d ie d fo r r e a liz a b ilit y in a c c o r d a n c e w ith th e p re v io u s d is c u s s io n
a n d a s y n th e s is p ro c e d u re c a n a l s o b e w o rk e d o u t in a s im ila r f a s h io n .
M o st o f th e th e o re m s a p p ly w ith s lig h t m o d ific a tio n .
w ith o u t p ro o f.
The p ro o fs a re g iv e n in A ppendix B.
T h e s e a re s ta te d
The X 's
a n d Y 's b e lo w m ay be v a c u o u s .
Theorem 4 . 3 . 1
~
;
-
L et P X Pn be a PI o f F . Then P .X , i s e ith e r a PI
I l z
I I
of U 1 or U 1 a n d X 1 P 0 i s e ith e r a PI of U n or U 0 .
I
I
I Z
/
6
T heorem 4 . 3 , 2
■
If
anc^
a re tw o P i's o f F an d if
X1P2 = Y1Q 2 , th e n Q 1Y1 s U ^(U 1) if P1X1= U 1 ( F j ) a n d fu rth e r P Xj
a n d Q 1Y1 a re P i's of U ^ U ^ .
th e n Q 1^ 1P g - P if P j X j P2S F .
'
C o n v e r s e ly , if P ^ , Q 1X j S U j ( Uj ),
F u rth e r, if P j X j an d Q j X j a re P i's of
U 1 ( U ) ' a n d P 1X1P0 i s a PI of F , th e n Q 1X P is a PI o f F ,
I
I
I i z
1 1 2
As th e s u b - a r r a y s a re n o t d i s j o i n t , so m e te rm s in a g iv e n
fu n c tio n m ay se e m e lig ib le fo r in c lu s io n in m ore th a n o n e c o n s titu e n t
fu n c tio n s U j , U 2 , e t c .
to r e a l i z a t i o n .
But n o t a l l th e a lte r n a t iv e s m ay u ltim a te ly le a d
E xam ple 4 . 3 . 2 i s a n illu s tr a tio n of t h i s c ase'.. The
p h ra s e 'th e term o c c u rrin g u n d e r Aj (A^)1 w ill m ean h e re th a t if
-101-
P1X P
Ag .
i s a PI of F , th e n P 1X
o c c u rs u n d e r A an d X1P9 o c c u rs u n d e r
T h e n , c o ro lla ry to T h. 4 . 2 . 2 a p p lie s in n o n - d is jo in t c a s e .
B ut, th e o re m 4 . 2 . 3 i s n o t a p p l i c a b l e .
4 . 2 . 5 a p p ly a s s u c h .
H o w e v e r, th e o re m s 4 . 2 . 4 a n d
The th e o re m s c a n a l s o be g e n e r a liz e d in th e
sam e m a n n er a s in th e d is jo in t c a s e (sh o w n in A ppendix B) .
For a fu n c tio n to b e d e c o m p o s a b le in th e m a n n er of F ig u re 4 . 3 . 2 ,
o n e o f s e v e r a l c a s e s m ay o c c u r (w ith r e f . to T ab le 4 . 3 . 1 ) . ,
C ase I .
The fu n c tio n c o n s i s t s of o n ly ty p e I or ty p e II o r ty p e V
P i's or P i's o f a n y c o m b in a tio n o f th e s e t y p e s .
■
T h is c a s e i s s im ila r
to ty p e I of T a b le 4 . 2 . 2 a n d th e s y n th e s is p ro c e d u re s a m e .
Type I .
PI m u s t n o t o c c u r w ith a n y o th e r ty p e (by T h . 4 . 2 . 5 ) .
C ase 2.
The fu n c tio n c o n s i s t s of ty p e II a n d ty p e III P i's '.
Let R]G U
and
Q 1X1= U 1
and
Q lX i= U 1
and
V
U2
x I= U 2
T h e n , e ith e r
or
X s UI
2
A ls o , T heorem 4 .3 .2 m u st b e s a t i s f i e d fo r e n s u rin g r e a l i z a b i l i t y .
A fu n c tio n w ith o th e r c o m b in a tio n s o f T ypes II , I I I , IV an d V
P i's c a n b e s im ila rly a n a ly z e d .
I t i s o b v io u s th a t a s th e tw o s u b ­
a rra y s a re m ade to sh are, som e v a r i a b l e s , a g r e a te r v a r ie ty o f fu n c tio n s
-102b e c o m e s r e a l i z a b l e . N e tw o rk s u s in g th r e e - in p u t U L M 's in th e
in te rm e d ia te le v e l s m ay a ls o b e s im ila r ly s tu d ie d .
The l i s t of
d iff e r e n t p o s s ib le ty p e s of P i 's in th is c a s e b e c o m e s q u ite la r g e .
Two e x a m p le s to i l l u s t r a t e th e p re v io u s d is c u s s io n s a re g iv e n b e lo w .
E xam ple 4 . 3 . 1
P = X iX gX j
+
+ X iX gX g
X iX gX j
+ XgXg
+ X iX gX gX g
+ XgXg
+ XgXj
+ X iX gX g
+ X iX gX gX g
+ X jX gX g
The n e tw o rk to t e s t for th e r e a liz a tio n o f F i s show n in F ig u re
4.3.3.
X1
I
fig u re 4 . 3 . 3
X
2
X„
3
X,
4
*3
*4
*5
*6
A N etw o rk to R e a liz e F in E xam ple 4 . 3 . 1
-103-
S te p I :
T a b ie 4 . 3 . 2
A ta b le i s m ad e (T able 4 . 3 . 2 ) .
A D e c o m p o s itio n T a b le fo r th e F u n c tio n F in E xam ple 4 . 3 . 1
5
6
x x
XrX
X 1 X,
XrX
S te p 2:
A n a ly s is .. The u n a te n e s s d e te rm in a tio n i s o m itte d h e re
I f it . x P 2= u i
th e n
X1X3E U 1 a n d X4E U 1
and
X5X g c
u2
—1 0 4 -
F u rth e r,
x 3c
as
U2
as
X jX 2 C ^
T h e re fo re ,
X5X gcU 2 .
x I x Z= V 1 ■
T h e re fo re ,
*
as
x 4= 5 Z
S 5 C tJ 2
as
X3 =
as
TT1
S te p 3: W e h a v e , U 1 =
F i s n o n - d e g e n e r a te
Xl X2S s^ F
. F is n o n - d e g e n e r a te .
+ Xj X2 + X4
U2
= X3 + x ^ x
A tru th ta b le of U 3 in te rm s of U 1 a n d Ug i s sh o w n (T ab le 4 . 3 . 3 ) .
T a b le 4 . 3 . 3
A T ruth T a b le o f U 3 in te rm s o f In te rm e d ia te L ev el
F u n c tio n s U 1 an d U 0 in E x am p le 4 . 3 . 1
I
/
ui
uZ
I
I
I
0
0
I
I
0
0
0
I
tY
.
.
0
From th e t a b l e , i t is s e e n th a t U 3 = U 1U 2 + U l Ug .
E xam ple 4 . 3 . 2
• F =
X 1X 4
+■ x 2x 4 +
X3 X4
+ x5 +
X 6X 7 + X 8 X 9
The n e tw o rk to b e t e s t e d fo r r e a liz in g th e fu n c tio n i s sh o w n in F ig u re 4 . 3 , 4 . .
-105-
In th e fig u r e ,
i s som e fu n c tio n f ( x j . . . x^) and
fu n c tio n g ( x ^ . . .x ^ ) .
U^-j o r to
V22
In g e n e r a l, th e PI
is a n o th e r
of F m ay b e lo n g e ith e r to
or to b o t h . The p u rp o s e o f th is e x a m p le i s to show th a t
Xrj c a n n o t b e lo n g to
an d th e re b y d e m o n s tra te th e e x is t e n c e of th e
p roblem o f d e te rm in in g th e p ro p e r s e t o f te rm s for th e c o n s titu e n t
fu n c tio n s in a p re c e d in g le v e l s u c h th a t a g iv e n fu n c tio n m ay be
u ltim a te ly r e a l i z e d , p ro v id e d i t is r e a liz a b le by th e g iv e n n etw o rk
s tru c tu re .
In th e f ir s t s te p a ta b le i s m ade (T ab le 4 . 3 . 4 ) .
-106-
T a b le 4 . 3 . 4
A D e c o m p o s itio n T able fo r F + XgXy
+ XgX^ + XgX^ + X5
+ XgXg
D3=Fxgl
O
Il
x"
C 1=Tx3 ] O2= I x 4 I G2= ( x 5l
CO
X
I-•
U
Il
X
I---- ------------------- -----12 "--------------------------
Dj=TXg,Xg]
/
X1
X4
X2
X4
x3
X4
x5
x6
X7
X8 = 9
C o n s id e rin g th e fu n c tio n U 3 in te rm s o f the v a r ia b le s U 31 an d
U3 2 , i t is s e e n th a t U 3 c o n s i s t s o f o n ly ty p e I an d ty p e V P i's and
th e re fo re a n OR g a te a t th e fin a l le v e l is a l l th a t is n e c e s s a r y .
M o re o v e r, U 3 i s u n a te in U ^ an d U 32 s o th a t w e m ay w rite
U 2 1 = X 1X 4
+ ' X 2X4
+ X3 X4 + x 5
' U 22 = x 5 + X6X7 + X8X9
-107-
As i t i s n o t know n w h e re
b o th th e f u n c tio n s .
s h o u ld g o , it is IrviiAuded in
But c o n s id e r Ug^ . . I t i s a sum o f P i's o f ty p e s I ,
I I , a n d I I I . ■So i t i s n o t r e a liz a b le b y th e n e tw o rk .
L et U g i = x i x 4 + X2X4 + X3X4*-
a mar^ner s im ila r to th e
p re v io u s e x a m p le , i t c a n be fo u n d t h a t U 21 =
Un
= Xi + X 2 + X 3
U 32
= X 4 - + X 3X4
• U^g a n d
= X4
A lso U 00 = U
+ U1
22
13
J
U
U
13
14
= X
5
+ X-X1
6
= X 0X0 .
8 9 ■
T hus i t is s e e n th a t if x^ Is n o t in c lu d e d in U g i ' th e n th e fu n c tio n
i s r e a liz a b le b y th e g iv e n n e tw o r k .
M e n tio n m ay b e m ade of th e f a c t
t h a t in U ^g a n im p lic a n t (not PI) x ^ x ^ i s o b ta in e d .
T h is i s u n lik e th e
s itu a tio n in, d is jo in t d e c o m p o s itio n .
T he a lg o rith m fo r t e s t - s y n t h e s i s o f n o n - d is jo in tly a rra n g e d
n e tw o rk s tr u c tu r e i s n o t s e p a r a te ly g iv e n a s a l l th e s te p s a re sam e a s in
th e a lg o rith m in d is jo in t c a s e .
The p o s s ib le a m b ig u ity in d e te rm in in g
th e c o n s ti tu e n t f u n c tio n s o f a p re c e d in g l e v e l , a s i l l u s t r a t e d , m ay b e
-108-
re s o lv e d b y f i r s t a s s u m in g th e te rm s c o n c e rn e d to be p r e s e n t in a l l th e
c o n s titu e n t fu n c tio n s fo r w h ic h th e y a re e lig ib le a n d c o n tin u in g th e
a n a ly s is u n til e ith e r th e g iv e n fu n c tio n i s fo u n d to b e r e a liz a b le or
som e c o n s ti tu e n t fu n c tio n s a re fou n d to b e u n re a liz a b le ..
In th e
l a t t e r c a s e , th e re m o v a l of s u ita b le te rm s m ay b e a tte m p te d on a
c u t-a n d -try b a s is .
C h a p te r 5
PARALLEL BULK TRANSFER'SYSTEM W ITH
FLEXIBLE INPUT DOMAIN
-110-
5 .1
The Problem o f V a ria b le G ro u p in g
The a lg o rith m fo r th e t e s t - s y s t h e s i s o f f u n c tio n s r e a liz a b le w ith
p a r a lle l b u lk tr a n s f e r s y s te m w h ic h h a s b e e n p re s e n te d in th e p re v io u s
C h a p te r i s d e p e n d e n t on a p re d e te rm in e d g ro u p in g of v a r i a b l e s .
Such
a s itu a tio n c o rre s p o n d s to a p a r a lle l B.-T. s y s te m w h e re in d iv id u a l
B. T. u n its p r o c e s s a fix e d s e t of v a r ia b le s on th e in p u t a r r a y .
A lte rn a te ly ,
i t i s p o s s ib le to th in k of th e in p u t d o m ain of a B.T.. u n it a s a fle x ib le
s e t o f v a r i a b l e s , w h ic h m ay be a n y w h e re on th e in p u t a r r a y .
C o n s e q u e n tly ,
th e q u e s tio n of a p ro p e r g ro u p in g of v a r ia b le s a r i s e s in th e te s t in g of
a g iv e n fu n c tio n for r e a l i z a b i l i t y .
Of c o u r s e , i t i s p o s s ib le to a s s u m e
o n e p a r tic u la r g ro u p in g a t a tim e a n d a p p ly th e p re v io u s a lg o rith m
u n til o n e w o rk s or a l l p o s s ib le g ro u p in g s f a i l . ' But it i s a la b o rio u sp ro c e ss .
The p u rp o s e o f th is C h a p te r i s to s tu d y som e a s p e c t s of th is
p ro b lem a n d d e v e lo p a n a lte r n a t e p ro c e d u re fo r d e te rm in in g th e g ro u p in g
o f d iff e r e n t v a r ia b le s th a t m ay le a d to th e r e a liz a tio n o f th e fu n c tio n
th ro u g h d e c o m p o s itio n .
The d e c o m p o s itio n o f a fu n c tio n in to s u b fu n c tio n s u s u a lly
c o n s i s t s in fa c to rin g o u t o f s u ita b le te rm s a n d c o lle c tio n o f o th e r te rm s
in to p ro p e r g r o u p s .
It i s s im p le r to d e te rm in e w h ic h te rm s to f a c to r
o u t in th e c a s e o f d is ju n c tiv e d e c o m p o s itio n th a n w h e n n o n - d is ju n c tiv e
-111-
d e c o m p o s itio n is a l s o a llo w e d .
It is fu rth e r sim p lifie d , if a r e s tr ic tio n is
p u t on th e l o g ic - r e a liz in g e le m e n ts w h ic h h a v e so lo n g b e e n a s s u m e d to b e
U L M 's s o t h a t o n ly u n a te fu n c tio n s o f th e in p u ts a re p r o d u c e d . A part from
th e f a c t th a t th e u n d e rly in g m a th e m a tic a l s tru c tu re of u n a te fu n c tio n s c a n
b e e x p lo ite d , th is r e s tr ic tio n m ay c o n s id e r a b ly re d u c e th e c o m p le x ity of
lo g ic a l m o d u le .
G iv e n a s w itc h in g fu n c tio n o f a n y n u m b er o f v a r i a b l e s , th e re a lw a y s
e x i s t s a la rg e e n o u g h ULM th a t c a n r e a liz e i t .
H o w e v e r, b e c a u s e of
e n g in e e rin g d if f ic u l tie s i t is n o t p r a c tic a b le to p ro d u c e ULM of la rg e
n u m b er o f v a r i a b l e s .
T hree or f o u r - v a r ia b le ULM m ay b e ta k e n a s s t a n d a r d .
T h u s , w ith a g iv e n fu n c tio n , th e a tte m p t sh o u ld be to d e c o m p o s e i t in to
s u b - f u n c tio n s of s m a ll num ber o f v a r i a b l e s .
In .th is C h a p te r, w e s h a ll
m ake a s tu d y of so m e s p e c if ic c a s e s o f d is ju n c tiv e a n d n o n - d is ju n c tiv e
d e c o m p o s itio n w ith th e h e lp o f th e p ro p e rtie s of r e a liz a b le fu n c tio n s
s tu d ie d in th e p re v io u s C h a p te r.
The c a s e s in c lu d e u n a te - lo g ic e le m e n t
n e tw o rk a n d lim ite d - in p u t ULM n e tw o rk .
5 .2
U n a te L ogic N etw ork
M u k h o p a d h y a y ^ ^ h a s s tu d ie d p ro p e rtie s of u n a te c a s c a d e s
a n d s u g g e s te d m in im iz a tio n a lg o rith m s fo r th e m . The u n a te c a s c a d e s
a re s im ila r in s tru c tu re to th e M a itra c a s c a d e — th e d is tin c tio n b e in g
th a t th e c e l l s in th e c a s c a d e a re r e s t r i c t e d to perform o n ly u n a te
fr u n c tio h s of th e i r tw o i n p u t s . In t h i s s e c t i o n , th e m ore g e n e r a l tre e
-112n e tw o rk of u n a te lo g ic e le m e n ts h a v in g m ore th a n tw o in p u ts w ill be
s tu d ie d .
E ach lo g ic e le m e n t is a s s u m e d to b e c a p a b le of p ro d u c in g a n y
u n a te fu n c tio n o f i t s i n p u ts .
It w ill be c a lle d a U n ate M o d u le ( U M) .
C o n s id e r th e tr e e n e tw o rk sh o w n in F ig u re 5 . 2 . 1 .
Uij for p ro p e r i , j w ith 15 i § n an d 15 jS m
a s b e fo re .
The fu n c tio n
h a s th e sa m e m ean in g
W ith th e r e s tr ic tio n now th a t e v e ry fu n c tio n U ij i s u n a te ,
th e fo llo w in g r e s u lt i s o b ta in e d .
U
11
F ig u re 5 . 2 . 1
T heorem 5 . 2 . 1
A T ree N etw ork
If th e o u tp u t fu n c tio n p ro d u c e d by e v e ry m odule in th e
g e n e r a l d is ju n c tiv e n e tw o rk i s u n a te , th e n th e f in a l o u tp u t fu n c tio n F
i s u n a te in th e o rig in a l v a r ia b le s
( x-^ ,X 2 , . . .Xfi } .
-1 1 3 P ro o f: E x p ress. F in te rm s of th e o u tp u ts o f th e U M 's o f th e l a s t bu t
o n e s ta g e :
F
U
n
( U*'
, IT*
n-1, I
n-1,2
. ..u*n - 1 , m.)
w h e re U*
m e a n s e ith e r U , . o r U n . b u t n o t b o th .- E x p re s s
n -1 ,i
n-1,i
n-1,i
U*
in te rm s o f th e p re v io u s le v e l v a r ia b le s fo r a l l i .
n-l,i
th is m a n n er b a c k up to th e f i r s t l e v e l .
P ro ce e d in
S in c e th e n e tw o rk i s d is ju n c tiv e
a t a l l l e v e l s , th e o c c u r re n c e of a v a r ia b le i s a c c o u n te d fo r by o n ly
o n e UM a t a n y l e v e l . As th e fu n c tio n s p ro d u c e d a re u n a t e , th e
fu n c tio n F c a n n o t c o n ta in a v a r ia b le in b o th prim ed a n d u n p rim ed form
a t an y s ta g e .
T hus F i s u n a te in th e o rig in a l v a r ia b le s .
Q .E . D .
As a r e s u l t of th is p r o p e r ty , o n ly u n a te fu n c tio n s re q u ire to
b e t e s t e d fo r r e a l i z a b i l i t y w ith a u n a te tr e e n e tw o rk . F o r n o n -u n a te
f u n c tio n s , m ore th a n o n e tr e e m u st be u s e d fo r r e a l i z a t i o n . ■
G e n e ra lly th e r e .a r e tw o ty p e s o f te rm s in a fu n c tio n s p e c ifie d
a s a sum o f prim e im p lic a n ts :— (i) te rm s in v o lv in g o n ly th o s e v a r ia b le s
th a t do n o t o c c u r in m ore th a n one te rm ;
(ii) te rm s in v o lv in g v a r ia b le s
th a t o c c u r in m ore th a n o n e te r m . As a n e x a m p le , le t
-114-
F = Xj +
+ X3X4 + x 5 x gx 7 + X5 XgX8 *
H e re th e f i r s t th r e e te rm s
b e lo n g to ty p e (i) a n d th e l a s t tw o te rm s to ty p e ( i i ) . To g e t a n e s tim a te
o f how m an y U M ' s m ay b e n e c e s s a r y to r e a l i z e a g iv e n f u n c tio n , l e t u s
r e s t r i c t c o n s id e r a tio n to fu n c tio n s w h ic h c o n ta in o n ly te rm s o f ty p e ( i ) .
G iv e n a s p e c if ic s i z e k o f a n U M , th e n u m b e r o f U M 's n e c e s s a r y to
r e a liz e a p rim e im p lic a n t t h a t i s a p ro d u c t o f m l i t e r a l s in th e m a n n er
sh o w n in F ig u re 5 . 2 . 2 , i s g iv e n by
Nj = I +
w h e re
m -k
k -1
f" r e p r e s e n ts th e c e ilin g v a l u e .
If e a c h PI o f th e f u n c tio n .is
c o n s id e r e d s e p a r a t e l y , th e n th e t o t a l n u m b e r .of U M ' s re q u ire d to r e a liz e
i P i's s e p a r a te ly i s
N = I +
(IH1-W
k -1
+ i +
<m2-W
k -1
+ ...+
(m j-k )
k -1
w h e re m ,m , . . .m a re th e n u m b er o f l i t e r a l s in th e f i r s t , s e c o n d , . .
I
2
i
i - t h PI r e s p e c t i v e l y .
F u r th e r , to r e a l i z e th e fu n c tio n c o n s is tin g o f ■
(i-k )
U M ' s to p ro d u c e a
k -1
T h u s , a t m o s t, (N +M) U M ' s a re n e c e s s a r y to r e a liz e
i p rim e i m p l i c a n t s , i t w ill n e e d M = I +
s in g le o u tp u t.
a n y g iv e n f u n c tio n w ith k - v a r ia b le U M 1s .
S y n th e s is o f D is ju n c tiv e U n a te N e tw o rk o f k - in p u t E le m e n ts
G iv e n a f u n c tio n F , a p ro c e d u re fo r t e s t - s y n t h e s i s w ith a u n a te
n e tw o rk o f k - v a r ia b le lo g ic e le m e n ts w i l l b e c o n s id e re d h e r e .
Of th e
t
-115-
F ig u re 5 . 2 . 2
A T ree w ith k - I n p u t L ogic E le m e n ts
tw o ty p e s o f prim e im p lic a n ts of a g iv e n fu n c tio n d i s c u s s e d e a r lie r
in th is s e c tio n , ty p e (i) P i's a rc a lr e a d y in d is ju n c tiv e ly d is s o c ia te d
form a s g iv e n ;
i t i s n e c e s s a r y to r e a liz e th e s e te rm s by g ro u p in g in
c o n s is te n c e w ith th e s iz e of th e lo g ic e l e m e n t s .
As a n e x a m p le , le t F = a b + c d e f + g .
A u n a te tre e of 3 -
in p u t e le m e n ts r e a liz in g F i s g iv e n in F ig u re 5 . 2 . 3 .
For th e ty p e (ii)
P i 's , d e c o m p o s a b le f e a tu r e s , if a n y , s h o u ld be in v e s tig a te d an d b ro u g h t
o u t.
firs t.
F or c o n v e n ie n c e , w e d e fin e c e r ta in te rm s an d p ro v e a p ro p e rty
-116-
a b
g
c
d
e
f
U
U
U
= ab + g
2
3
= cde
= U 1 + fU
1
2
F ig u re 5 . 2 . 3 A N etw ork to R e a liz e F = a b + c d e f +
D e fin itio n 5 . 2 . 1
g
A fu n c tio n w ill be c a lle d k - u n a te d e c o m p o s a b le
if i t c a n be r e a liz e d in a tre e by u s in g k - in p u t or s m a lle r u n a te lo g ic
e le m e n ts in a d is ju n c tiv e m a n n e r.
D e fin itio n 5 . 2 . 2
The le n g th of a PI of a u n a te fu n c tio n i s th e num ber
of l i t e r a l s in th e P I.
The le n g th d iff e r e n c e o f tw o P i's o f a u n a te
fu n c tio n is th e d iff e r e n c e in th e num b er o f l i t e r a l s c o n ta in e d in the
tw o P i ' s .
The m axim um le n g th o f a PI fo r a n n - v a r ia b le u n a te fu n c tio n
c a n be n . T h is o c c u rs w h en th e re i s o n ly o n e PI in th e fu n c tio n an d i t
c o n s i s t s of th e l i t e r a l s of a l l th e v a r ia b le s o f th e f u n c t i o n s .
The m axim um
p o s s ib le le n g th d iff e r e n c e of an n - v a r ia b le u n a te fu n c tio n i s ( n - 2 ) .
T his
-117-
o c c u r s w h e n th e re a re tw o ■P I1s in th e fu n c tio n w ith one PI c o n s is tin g of
.
\
o n ly one lite r a i a n d th e o th e r c o n s is tin g o f th e re m a in in g (n -1 ) l i t e r a l s .
D e fin itio n 5 . 2 , 3
If F(x , x , . . .x ) c a n b e e x p r e s s e d a s a c o m p o sitio n
I /
n
in v o lv in g Uv ( x . , . . .x .) a s a n a rg u m e n t s u c h th a t { x . , . . .x , } i s a s u b s e t
K
i J
i
j
o f I X1 , . . .x } , th e n U1 ( x ,. . .x .) i s a s u b - f u n c tio n o f F .
1
n
K
i j
Two fu n c tio n s
a re d i s j o i n t if th e y a re d e fin e d on d is jo in t s e t s of v a r i a b l e s .
Lemma 5 . 2 . 1
L et F be k - u n a te d e c o m p o s a b le .
L et xA be a PI
o f F , w h o s e l i t e r a l s o c c u r in a t l e a s t tw o d is jo in t s u b fu n c tio n s of F z
w h e re A r e p r e s e n ts a p ro d u c t of l i t e r a l s .
L et
be a s u b fu n c tio n of a t
m o s t k v a r ia b le s of F c o n ta in in g th e prim e im p lic a n t xB w h e re B - A..
If
h a s a n y o th e r PI, th e n th e re i s a t l e a s t a n o th e r PI o f F , w h ic h w ill
c o n ta in th e l i t e r a l s of A o r a s u b s e t of t h e s e lite r a ls s u c h t h a t a t m o st
(k-2) of th em a re a b s e n t .
Proof: W ith th e g iv e n c o n d itio n , le t B b e v a c u o u s in xB .
In th is c a s e ,
x i s a . PI o f U ^ . By T heorem 4 . 2 . 2 ' , th e f a c to r A m u st b e a s s o c i a t e d
w ith e v e ry PI of
.
Thus F w ill c o n ta in o th e r P i's h a v in g th e fa c to r A.
In th e o th e r e x tre m e , B c a n b e a p ro d u c t of (k-2) l i t e r a l s .
c a s e , th e r e e x i s t s o n e o th e r PI o f
In th a t .
c o n s is tin g of a s in g le lite r a l
-1 1 8 o n ly .
By T heorem 4 . 2 . 2 ' , t h i s PI o f
m u s t b e a s s o c i a t e d w ith a f a c to r
c o n ta in in g a l l th e l i t e r a l s o f A b u t th o s e o f B.
T hus F w ill c o n ta in
a n o th e r PI h a v in g a f a c to r w h ic h w ill c o n ta in a l l b u t (k -2 ) l i t e r a l s o f A.
W h en th e s i z e o f B l i e s b e tw e e n t h e s e tw o e x tr e m e s , th e com m on f a c to r
c o rre s p o n d in g ly i s o f in te r m e d ia te s i z e .
Q.E.D.
The im p o rta n c e o f th is th e o re m i s in c o n n e c tio n w ith d e te rm in in g
a s u b - f u n c tio n o f a g iv e n fu n c tio n , c o n ta in in g a p a r tic u la r v a r ia b le .
To d o t h i s , i t i s n o t n e c e s s a r y to c o n s id e r a l l te rm s o f th e g iv e n fu n c tio n
b u t o n ly th o s e t h a t h a v e a f a c to r o f p ro p e r le n g th co m m o n .
T h is
p ro p e rty a lo n g w ith th e f a c t t h a t th e fu n c tio n m u st b e u n a te m a k es
t e s t - s y n t h e s i s of u n a te n e tw o rk s s im p le r th a n th e g e n e r a l d is ju n c tiv e
n e tw o rk s y n t h e s i s .
In th e n e x t s e c t i o n , a n a lg o rith m fo r g e n e r a l
d is ju n c tiv e n e tw o rk s y n th e s is w ill b e g iv e n , in d ic a tin g th e s p e c ia l
a s p e c t s fo r th e u n a te n e tw o rk a t p ro p e r p l a c e s .
5 .3
An A lgorithm fo r G e n e ra l D is ju n c tiv e N e tw o rk S y n th e s is
The tr e e n e tw o rk g iv e n in. F ig u re 5 . 2 . 1 m ay b e th o u g h t of
a s a g e n e r a l d is ju n c tiv e , n e tw o rk of u n iv e r s a l lo g ic a l m o d u le s . In s tu d y in g
a s y n th e s is p ro c e d u re fo r i t , w e d e fin e a t f i r s t a term a p p lic a b le h e r e .
-119-
D e fin itio n 5 . 3 . 1
A fu n c tio n i s c a ll e d k -d e c o m p o s a b le i f i t c a n be
r e a liz e d in a tr e e by u s in g k - v a r ia b le U L M 's in a d is ju n c tiv e m a n n e r.
A ssu m e t h a t a fu n c tio n i s s p e c if ie d a s a c o m p le te sum of prim e
im p lic a n ts .
T h is w ill b e a lw a y s th e c a s e if th e fu n c tio n i s u n a te .
To t e s t i t fo r k - d e c o m p o s a b ility , th e fo llo w in g s te p s a re d e v e lo p e d .
S te p I .
T e s t if th e fu n c tio n i s u n a t e .
If u n a te th e n u s e s im p lif ic a tio n
a s m e n tio n e d a lo n g w ith th e fo llo w in g s t e p s .
S te p 2 .
I n s p e c t a n d e n te r in a s u b - f u n c tio n a l l s u c h te rm s a s
c o n ta in in g v a r ia b le s t h a t o c c u r in m ore th a n o n e te rm .
O th e r te rm s
o f F c o n s i s t o f o n ly th o s e v a r ia b le s t h a t o c c u r o n ly o n c e .
T h e s e a re
a lr e a d y in d is ju n c tiv e l y d i s s o c i a t e d form a n d m ay b e r e a liz e d in
c o n s i s t e n c e w ith th e s iz e o f th e lo g ic d e v i c e . From th e s u b - f u n c tio n
o b ta in e d , a b o v e , f a c to r o u t a n y com m on l i t e r a l s a n d g e t th e r e s id u a l
f u n c tio n .
S te p 3 .
C o n s id e r th e s e t o f v a r ia b le s in th e s u b - f u n c tio n s e p a r a te d
(o r th e r e s i d u a l f u n c tio n a s th e c a s e m ay b e ) .
T ake a v a r ia b le a n d
l i s t i t w ith a l l th e o th e r v a r ia b le s w h ic h a s s o c i a t e w ith i t in d iff e r e n t
te rm s .
T ake e a c h o f th e l a t t e r v a r ia b le s a n d a d d to th e l i s t th e n ew
v a r ia b le s t h a t a s s o c i a t e w ith them a n d c o n tin u e in t h i s m a n n e r.
R efer
-1 2 0 to t h i s l i s t a s th e l i s t o f " c o n n e c te d " v a r i a b l e s .
If a t a n y tim e i t i s
fo u n d t h a t th e l i s t d o e s n o t grow b e y o n d k v a r i a b l e s , th e n th e re e x i s t s
a d is ju n c tiv e s u b - f u n c tio n o f a t m o s t k v a r i a b l e s , w h o s e te rm s c a n
b e s e p a r a te d o u t.
R e p la c e t h i s s u b - f u n c tio n b y a s in g le v a r ia b le
U . . T ak e a n e w v a r ia b le n o t in c lu d e d in th e p re v io u s l i s t a n d p ro c e e d
in th e s a m e m a n n e r u n til a l l th e v a r ia b le s a re e x h a u s te d . . G e t th e
r e s i d u a l f u n c tio n a f t e r s e p a r a tin g a l l th e s u b - f u n c tio n s . '
F o r e x a m p le , l e t F = a b + b e + c a + d e + d f . T e s tin g F fo r
3 - d e c o m p o s a b ility , i t i s s e e n t h a t th e f i r s t term a b in v o lv e s v a r ia b le s
a a n d b w h ic h a re a s s o c i a t e d w ith th e te rm s c a a n d a b r e s p e c t i v e l y .
T h e s e th r e e v a r ia b le s a re n o t a s s o c i a t e d w ith a n y o th e r n e w v a r i a b l e s .
T h e re fo re , a s u b - f u n c tio n U j = a b + b e + c a c a n b e fo r m e d .
S te p 4 .
In s te p 3 , if th e r e i s a l i s t c o n ta in in g m ore th a n k
v a r ia b le s ta k e a te rm o f F c o n ta in in g so m e o f t h e s e v a r i a b l e s .
b e s u c h a PI a n d x a l i t e r a l o c c u rrin g in P j .
L et P j
To d e te rm in e w h ic h s u b ­
f u n c tio n c o n ta in s x , th e f i r s t s te p i s to b re a k P j in to tw o p a r ts :
P j = x I H , w h e r e .H i s so m e p ro d u c t o f l i t e r a l s n o t in v o lv in g x .
C o m p are H w ith o th e r P i's to fin d th e com m on l i t e r a l s in e a c h c a s e . ’
A PI h a v in g la r g e r n u m b e r of com m on l i t e r a l s m ay b e c o n s id e r e d f i r s t .
L et Pj b e a PI o f th e g iv e n fu n c tio n w h ic h c a n b e w r itte n a s Pj - G | I
-121-
w h e re G an d I a re p ro d u c ts of l i t e r a l s w ith I com m on to b o th
H.
an d
If th e fu n c tio n is to be t e s t e d fo r k - u n a t e d e c o m p o s a b ility , Pi
s h o u ld b e c h o s e n s u c h th a t i t c o n ta in s a f a c to r h a v in g a l l b u t a t m o st
(k-2) l i t e r a l s of H .
N ow , H c a n b e w ritte n a s H = IJ w h e re J i s e ith e r
v a c u o u s o r a p ro d u c t of som e l i t e r a l s .
It i s th e n p o s s ib le to e x p re s s
(P] + Pi ) a s P^ + P^ = x I J + G I = (x J + G) I .
If x j a n d G c a n be
g ro u p e d u n d e r th e sam e s u b - f u n c tio n , th e n fo r a l l te rm s in v o lv in g
x j , th e re m u s t be c o rre s p o n d in g te rm s in v o lv in g G , s a tis f y in g
T heorem 4 . 2 . 2 '.
F u rth e r, i t im p lie s th a t a n im p lic a n t of th e r e s id u a l
fu n c tio n (xj + G ), w h ic h i s n o t a prim e im p lic a n t, c a n n o t b e p r e s e n t.
If th e s e c o n d itio n s a re n o t s a t i s f i e d , th e n G c a n n o t b e a term
o f th e s u b - f u n c tio n c o n ta in in g x j .
th e n .
A n ew term of F m u st b e c o n s id e re d
If th e re i s no term s a tis f y in g th e c o n d itio n s , th e r e m ay b e tw o
p o s s i b i l i t i e s — (i) th e s u b - f u n c tio n c o n ta in in g x h a s o n ly o n e PI or
(ii) th e r e i s no s u b - f u n c tio n of p ro p e r s iz e in c lu d in g th e v a r ia b le x.
C a s e (i) m ay be t e s t e d b y fa c to rin g o u t th e com m on p ro d u c t from a l l
th e te rm s c o n ta in in g x a n d d e fin in g a fu n c tio n Uj fo r th e com m on p ro d u c t
T he o c c u rre n c e o f th e v a r ia b le s of Uj m u s t b e r e p la c e a b le w ith U j or
Uj w ith s u ita b le a rra n g e m e n t of te r m s .
th e s e c o n d p o s s i b i l i t y m ay b e a c c e p t e d .
If t h i s is n o t p o s s i b l e , th e n
If th e re is o n ly o n e term in F .
I
-
12 2 -
c o n ta in in g x ( x ) , th e n a te rm c o n ta in in g x(x) m ay b e ta k e n in s te a d of th e
te rm c o n ta in in g x ( x ) , a n d th e s te p 4 m u s t b e r e s t a r t e d .
If th e r e is one
te rm c o n ta in in g x a n d o n e term c o n ta in in g x , th e n th e te rm s c o n ta in in g
x a n d x s h o u ld form a 2 - v a r ia b le s u b - f u n c tio n fo r d e c o m p o s a b ility .
The c a s e w h e n th e r e i s o n ly o n e te rm c o n ta in in g x(x) a n d n o term
c o n ta in in g x(x) i s u n r e a liz a b le e x c e p t in . c a s e th a t term i s th e o n ly
term in th e r e s id u a l f u n c tio n a t th a t s t a g e .
The c a s e (ii) d o e s n o t
m e an t h a t F i s n o t d is ju n c tiv e l y d e c o m p o s a b le .
It. m ay in d ic a te th a t
th e in p u t x o c c u r s in a h ig h e r le v e l o f th e t r e e .
The n e x t s te p sh o u ld
th e n b e to c h o o s e a n o th e r v a r i a b l e , s a y y , a n d go th ro u g h th e a b o v e
p ro c e d u re .
If c o n d itio n s a s m e n tio n e d e a r l i e r a re s a t i s f i e d fo r G , th e n
G c a n b e g ro u p e d w ith x j in a p a r tia lly - f o r m e d s u b - f u n c tio n .
In th is
c a s e , m ove to s te p 5 .
S te p 5 .
L et Pj be a PI w h ic h h a s so m e l i t e r a l s com m on w ith I
s u c h t h a t i t i s p o s s ib le to w rite I = ML a n d Pj = N L .
ta k e o u t L a n d w rite
In th is c a s e ,
P^ + P^ + P^ = L(xJM + GM + N ).
C om pare
GM w ith N to e n s u r e , a s b e f o r e , t h a t b o th c a n b e in c lu d e d in th e sam e
s u b -fu n c tio n .
x JM .
I t i s n o t n e c e s s a r y to c o m p a re N w ith th e o th e r te rm
I f i t i s p o s s ib le to in c lu d e N 7 a u g m e n t th e p a rtia lly -f o rm e d
-123-
fu n c tio n a c c o r d in g ly .
a s in S te p 5 .
T ake a n ew term an d fo llo w th e sam e p ro c e d u re
If i t i s n o t p o s s ib le to in c lu d e N , ta k e a n o th e r term of
F a n d fo llo w S te p 5 .
W h en th e re a re n o m ore te rm s o f F to b e c o m p a re d ,
l e t T b e th e com m on f a c to r of th e s e t. o f te rm s w h ic h a re s u ita b le
fo r in c lu s io n in th e f u n c tio n .
R e p la c e th e r e s id u a l fu n c tio n o b ta in e d
b y f a c to rin g o u t T by a s in g le v a r ia b le Ujc - It m u st be p o s s ib le to
r e p la c e a l l o c c u r r e n c e s o f th e v a r ia b le s o f
by
or
. If n o t,
■th e fu n c tio n is n o t d is ju n c tiv e l y d e c o m p o s a b le .
W h en th e s u b - f u n c tio n i s fo rm e d , a c h e c k m u s t.b e m ad e to
s e e if th e n u m b er of v a r ia b le s h a s e x c e e d e d k .
W h en i t e x c e e d s k ,
th e s u b - f u n c tio n m u st b e a g a in d e c o m p o s a b le in o rd e r to b e r e a liz e d
w ith k in p u t s .
I t is p o s s ib le to c h e c k th is a t e v e ry s ta g e of
a u g m e n ta tio n of th e p a rtia lly -f o rm e d s u b - f u n c tio n an d s to p th e tr i a l
\
w h e n th e n u m b er of v a r ia b le s e x c e e d s k a n d s ta r t w ith a n ew v a r i a b l e .
B ranch to s te p 2 a t th e e n d of S te p 5 .
The p ro c e d u re c a n be
■ term inated w h e n th e g iv e n fu n c tio n c a n b e e x p r e s s e d a s a f u n c tio n of
a n u m b er o f s u b - f u n c tio n s w h ic h m ay a g a in b e fu n c tio n s o f s t i l l s m a lle r
s u b - f u n c tio n s a n d s o o n , in a m a n n er s u c h t h a t to r e a liz e a n y s u b ­
fu n c tio n s no m ore th a n k d i s jo in t in p u ts a re n e c e s s a r y .
-124-
Some e x a m p le s w ill be w o rk e d o u t b e lo w .
E x am ple 5 . 3 . 1
F = X^XgXg + x ^ x g x g + xgXgXg + x ^ x g x g + x^XgXg
+ X g X g X g + XgX^ + X 3 X 4
W e s h a ll t e s t F fo r 3 - u n a te d e c o m p o s a b ility .
S te p I .
The fu n c tio n i s u n a te .
S te p 2 .
E a c h term h a s a t l e a s t o n e v a r ia b le th a t o c c u r s in
a n o th e r te rm .
S te p 3 .
The l i s t o f c o n n e c te d v a r ia b le s c o n ta in s a l l v a r ia b le s
o f th e f u n c tio n .
S te p 4 .
The term
C h o o se th e term XgX^.
X3X4
L et XgX, = Xg
c o n ta in s H; s o , c o m b in in g , XgX/ +
H w h e re H = x ^ .
X3X4
= (xg +
X3)
X4
A c h e c k r e v e a ls th a t X3 a n d Xg a re in te r c h a n g e a b le in th e f u n c tio n .
S te p 5 .
T here i s no th ird term c o n ta in in g X4 .
Let
T hen F c a n be e x p r e s s e d a s
F = (xg+Xg)x^Xg + (xg+Xg)X]Xg + (xg+x3 )X5Xg + (xg+ xg)x 4
= U ifX iX g
+ X ^ X g + X g X g + X^)
= Xg + X3
.
-125-
S tep 2...
A fter s e p a r a tin g th e com m on f a c to r U j , c o n s id e r th e
r e s i d u a l fu n c tio n R = X1X5 + x Xg + X5Xg + x „ .
The term x 4 o c c u rs
s in g ly .
S e p a ra tin g x ^ , w e h a v e Ug = XjXg + x^Xg + XgXg..
S te p 3 .
The fu n c tio n Ug c o n s i s t s of 3 v a r ia b le s .
T h erefo re F
c a n b e d e c o m p o s e d a s F = U ^(U ^ + x ) w here- U^ = Xg + Xg an d
U 2 = x I x 5 + x 1x 6 + X5X6-
E x am ple 5 . 3 . 2
-
F = X 1 X3 X4 X5 + X3 X3 X4 X5 + X 1X3 X4 X5 + X3 X4 X5 .
W e s h a ll t e s t F fo r 2 - d e c o m p o s a b ility .
S te p I .
The fu n c tio n is n o t u n a t e .
S te p 2 .
F = X5 Cx3XgX4 + X3 XgX4 + X 3XgX4 + XgX4)
= XgR
w h e n R i s th e r e s id u a l fu n c tio n .
S te p 3 .
The l i s t o f c o n n e c te d v a r ia b le s in c lu d e s a l l th e v a r ia b le s
o f th e r e s id u a l f u n c tio n .
Step 4 .
L e t x 3XgX4 = X 4 I x 3X3 = X4 1FI
P ic k in g th e te rm XgXgX1J a s i t c o n ta in s th e li t e r a l Xg a n d c o m b in in g ,
X 3X3 X4 + X3 X3 X4 = X3 Cx3 X4 ' + X 3X4 ) . T he c o n d itio n fo r g ro u p in g
XgX^ w ith X 3X4 in th e sa m e s u b - f u n c tio n i s s a t i s f i e d .
-126-
S te p 5 .
T here i s no o th e r term w ith x-
L e t x 3 Cx2X4 + X1X4 ) = X 3X4 Cx2 + X1)
= X3 X4 U 1 ' w h e re U 1 = x "2 + X 1
T hen R =
X 3 X 4 Cx 1
+ 3< ) + X4 (
x^Xg
+ X3 )
= ^ 3 x 4 D l + U (iTl + x Sf
x Sx 4 u I + X 4U 1 + X4X3
S te p 2,
No m o d if ic a tio n ,
S te p 3 .
No m o d if ic a tio n .
S te p 4 .
Let X3 X4 U 1= X3 H w h e re H = X4 U 1
No o th e r term of R c o n ta in s a n y l i t e r a l o f H .
R e je c t th is term a n d ta k e X3X4 i n s t e a d .
XzjU1 ,
we have,
X 3X 4
L et X3X4 = X3 G .
C om bining
+ X4 U 1 = X4 (x 3 + U 1) . It is fo u n d th a t TT1
c a n b e in c lu d e d w ith xq .
S te p 5 .
T here is no o th e r term c o n ta in in g x 4 .
T hen R = X 3 X4 U 1 + X4 CU1 + X3 )
= IL2X4 + X4 U2
Let U 2 = X3 + U 1
-1.2 7 Th us F i s 2 - d e c o m p o s a b l e a s
w ith
R = Ug A
/
^ 2 = x 3 + ^ l anc^
u I = Xi + x 2 .
5 .4
S y n th e s is of N o n - d is ju n c tiv e N etw o rk
Any a r b itr a r y lo g ic fu n c tio n c a n b e r e a liz e d by u s in g a n a d e q u a te
n u m b er of k - in p u t U L M 's fo r a n y RB 2 , w ith a s u ita b le n o n .-d is jo in t
n e tw o rk s t r u c t u r e . T h e re fo re i t i s im p o rta n t to d e v e lo p a lg o rith m w h ic h
a tte m p ts to re d u c e th e c o s t of r e a liz a tio n by m in im izin g th e num ber
of re q u ire d lo g ic e l e m e n t s :
By a m o d ific a tio n o f th e p ro c e d u re
s u g g e s te d fo r d is ju n c tiv e n e tw o rk s in s e c tio n 5 .3 it is p o s s ib le to
s u g g e s t a p ro c e d u re fo r r e a liz in g a fu n c tio n w ith a n o n -U is ju n c tiv e
n e tw o r k . An a d v a n ta g e in fo llo w in g t h i s m eth o d o f s y n th e s is is th a t
th e p a r tia lly d is ju n c tiv e fe a tu re of a fu n c tio n c a n b e u t i l i z e d to
re d u c e th e n u m b er o f lo g ic a l e le m e n ts in th e r e a liz a tio n o f th e fu n c tio n .
The m o d ifie d a lg o rith m is g iv e n b e lo w w ith th e d if f e r e n c e s o n ly
m e n tio n e d .
S te p I .
Sam e a s in S e c tio n 5 . 3 .
S te p 2 .
Sam e a s in S e c tio n 5 . 3 .
-1 2 8 S te p 3..
Sam e a s in S e c tio n 5 . 3 .
S te p 4 .
T he a u g m e n ta tio n o f a s u b - f u n c tio n b y in c lu d in g a n ew
term i s c a r r ie d o u t a c c o rd in g to th e c o n d itio n s m e n tio n e d in S e c tio n 5 . 3 .
W h en no s e c o n d te rm i s to b e fo u n d fo r i n c lu s io n , th e fo llo w in g ru le
is a d o p te d .
L et P b e a p ro d u c t o f l i t e r a l s p r e s e n t in so m e term of th e g iv e n fu n c tio n .
L et Q b e a n o th e r s u c h p ro d u c t o f l i t e r a l s . F o r so m e te rm s
{ PM^ I i = 1 , 2 , . . , M = a p ro d u c t o f l i t e r a l s } in F , in w h ic h P is
p r e s e n t , l e t th e r e b e c o rre s p o n d in g te rm s { QM i | i = 1 , 2 , . . . }
Q is p re s e n t.
in w h ic h
L et th e r e b e o th e r te rm s o f F in v o lv in g P a n d Q in w h ic h
th e a b o v e i s n o t tr u e .
In s u c h a c a s e , d e fin e a s u b - f u n c tio n
Uj = P + Q a n d m ove to S te p 5 .
I t m ay b e m e n tio n e d t h a t in n o n ­
d is ju n c tiv e c a s e , i t i s p o s s ib le to h a v e so m e term o f F c o n ta in in g
a s a p a r tia l p ro d u c t a n im p lic a n t (n o t PI) o f U j .
S te p 5 .
The s u b - s t e p s a n d c o n d itio n s a s g iv e n in s e c tio n 5 .3
a r e a p p lie d .
A fter a s u b - f u n c tio n Uj h a s b e e n fo rm e d , th e o c c u r re n c e s
o f th e v a r i a b l e s o f Uj in F a re r e p la c e d w ith Uj o r Uj w h e re v e r p o s s i b l e .
The n e x t s te p i s to go b a c k to S te p 2 a n d r e p e a t th e p ro c e d u re
u n til th e f u n c tio n c a n b e d e c o m p o s e d in to s u b - f u n c tio n s o f d e s ir a b le
s iz e .
An e x a m p le i s w o rk e d o u t b e lo w .
-129-
E xam ple 5 . 4 . 1
F = X1X2 + XgX4 Xg + X3 X4 X 6 + X5 XgX8 + X 5X7Xg
.. W e s h a ll r e a liz e F w ith 3 - in p u t o r s m a lle r H E M 's .
S tep I .
The fu n c tio n is u n a t e .
S te p 2 .
S e p a ra te x^Xg a n d g e t th e r e s id u a l fu n c tio n (R ).
S te p 3 .
The l i s t o f c o n n e c te d v a r ia b le s in v o lv e s all. v a r ia b le s
o f th e r e s id u a l fu n c tio n .
S te p 4 .
L e t x 3 X4Xg =
X3 F I w h e r e H = X 4 Xg.
C o m b in in g Xgx 4 Xg, w e h a v e
X g X 4 X g + X g X 4 X g = X 4 [ X g X g + X g X g ]'
S te p 5 .
T here is no o th e r term w ith x 4 .
N ow , l e t X4 Cx3 Xg -1- X3 X 6 ) = X4 X3 ( X g + X g ) = X4 X3 U 1
w h e re U 1 = Xg + X g.
Then w e h a v e ,
R = X4 X3 U 1 + X5 X 6X8 + X 6X7X8.
T hough U 1 in v o lv e s th e v a r ia b le s { Xgz x g } , th e o c c u r re n c e of
XgXg in som e te rm s c a n n o t b e r e p la c e d b y U 1 or U j .
S te p 2 . .
The term X4 X3 U 1 is s e p a r a te d from th e r e s t of th e fu n c tio n
S te p 3 .
No m o d if ic a tio n .
-130-
S te p 4 .
T hen
Let X5XgXg = Xg I XgXg
= XgH
X5 XgXg + X 5X7 X5 = X 5X g fx 5 + X7)
S te p 5 .
f u n c tio n .
No o th e r term w ith Xg or Xg is p r e s e n t in th e r e s id u a l
S o , l e t X5Xgfx 5 + X7) = X 5XgU2
w h e re
= X5 + x
T hen F = X4 X3 U 1 + X 5X5 U 2 + X 1X2
S te p 2 .
LetU3 =X^x5U1
,
U 4 = X 5 X5 U 2
and
U5 = x l x 2
T hen F = U 3 + U 4 + U 5
The r e a liz a tio n i s sh o w n in F ig u re 5 . 4 . 1 .
x„ x
F ig u re 5 . 4 . 1
A N etw ork to R e a liz e F =
+ X 5 X 6X 8
+ X 6 X 7X 8
X 1X 2
+
X 3X 4 X g
+
X 5X4X 5
-131-
The a b o v e p ro c e d u re o f s y n th e s is o f n o n - d is ju n c tiv e n e tw o rk s is
g iv e n h e re a s a n e x te n s io n o f th e t e s t - s y n t h e s i s m eth o d fo r d is ju n c tiv e
n e tw o rk s g iv e n b e fo re . A te c h n iq u e fo r fin d in g th e m inim um n u m b er of
k - in p u t U L M 's to r e a liz e a g iv e n fu n c tio n m ay re q u ire e x a m in a tio n of
a la rg e n u m b er of p o s s i b i l i t i e s .
Roth a n d K a r p ^ ) h a v e d e a l t w ith
s im ila r p ro b le m s u n d e r c e r ta in c o n s tr a in ts . •
C h a p te r 6
CONCLUSIONS
-133-
6.1
Sum m ary
•
A s tu d y o f th e c e ll u la r b u lk tr a n s f e r s y s te m from th e v ie w ­
p o in t o f i t s lo g ic a l c a p a b i l i t i e s h a s b e e n p r e s e n te d in th is t h e s i s .
The
m o d e l a d o p te d fo r th e B. T. s y s te m c o n s i s t s of a n in p u t a r r a y , a m a p p in g 1
d e v ic e , a n o u tp u t a rra y a n d a n o u tp u t lo g ic .
The in flu e n c e of v a rio u s
f a c to r s lik e f le x ib ili ty of th e m ap p in g d e v ic e , f le x ib ility in th e o u tp u t
lo g ic , p a r a lle lis m of o p e ra tio n h a s b e e n in v e s t ig a te d .
It h a s b e e n sh o w n in C h a p te r 2 t h a t th e s y s te m c a n b e m ade
lo g ic a lly u n iv e r s a l w ith a p ro p e r c o m b in a tio n of o u tp u t lo g ic a n d m a p s .
F u rth e r, in r e a liz in g a r b itr a r y lo g ic , th e r e e x i s t s a tr a d e - o f f am ong th e
n u m b e r o f m ap p in g o p e r a t i o n s , n u m b er of in d e p e n d e n t m ap s an d am o u n t
o f f le x ib ili ty in th e o u tp u t lo g ic s u c h th a t in c r e a s in g one of th e s e
q u a n titie s te n d s to d e c r e a s e th e o t h e r s . A sim p le d e s ig n fo r a B. T.
s y s te m h a s b e e n g iv e n fo r i l l u s t r a t i o n .
In C h a p te r 3 , th e e f f e c t of in tro d u c in g f le x ib ility in th e o u tp u t
lo g ic by th e u s e o f f le x ib le - lo g ic c a s c a d e s h a s b e e n c o n s id e r e d .
C e rta in b o u n d s on th e n u m b er of re q u ire d m ap p in g o p e r a tio n s ( tr a n s p o s i­
tio n ty p e ) h a v e b e e n o b t a in e d . A m o d ific a tio n of th e b u lk tr a n s f e r s y s te m
i s a l s o d i s c u s s e d in w h ic h th e in p u t a n d o u tp u t a rra y s c o n s i s t o f th e
p rim ary o r m ap p ed in p u ts to g e th e r w ith a s e t of fu n c tio n s d e riv e d from
-1 3 4 th e in p u ts b y som e b u i l t - i n l o g i c .
I C h a p te r 4 d e a ls w ith th e to p ic o f b u lk tr a n s f e r in p a r a lle l .
■Assuming th e in p u t a rra y to b e d iv id e d in to m any s u b - a r r a y s , a n in v e s tig a
tio n h a s b e e n m ade a b o u t th e ty p e o f f u n c tio n s th a t c a n b e r e a liz e d b y a .
n u m b er of B. T. u n its o p e ra tin g in p a r a l l e l .
Some p ro p e rtie s of
r e a liz a b le fu n c tio n s h a v e b e e n p ro v e d a n d a n a lg o rith m h a s b e e n
p r e s e n te d fo r t e s t in g r e a l i z a b i l i t y o f fu n c tio n s on th e a s s u m p tio n th a t
th e in p u t d o m ain of e a c h of th e b u lk tr a n s f e r u n its w o rk in g in p a r a lle l
i s f ix e d .
In C h a p te r 5, th e c a s e o f fle x ib le in p u t d o m ain h a s b e e n
c o n s id e re d a n d a n a lg o rith m fo r th e t e s t - s y n t h e s i s of lo g ic fu n c tio n s
on th is b a s i s h a s b e e n d e v e lo p e d .
A part from a b e tte r u n d e rs ta n d in g o f th e c a p a b iliti e s of a B. T.
s y s te m , th e a b o v e r e s e a r c h h a s th ro w n lig h t on th e fo llo w in g t o p i c s . '
i)
T ra n sfo rm a tio n o f a n a rb itra ry g iv e n fu n c tio n in to M a itra -
r e a liz a b le f u n c t i o n s .
The r e s u l t g iv e n in s e c tio n 3 .2 s h o w s th a t th e re i s a n e x p o n e n tia l
d e p e n d e n c e of th e m axim um num ber o f tr a n s p o s itio n s n e c e s s a r y to c o n v e rt
'a g iv e n fu n c tio n in to M a it r a - r e a liz a b le fu n c tio n on th e n u m b e r of
v a ria b le s .
The sa m e d e p e n d e n c e i s k now n to e x i s t fo r th e m aximum
n u m b er o f c e l l s n e c e s s a r y in a c u tp o in t o r s im ila r - ty p e a r r a y s to r e a liz e
-1 3 5 a n a rb itra ry f u n c tio n .
ii)
T e s t - s y n th e s is of lo g ic f u n c tio n s r e a liz a b le in d is ju n c tiv e
a n d sim p le n o n - d is ju n c tiv e m u lti- le v e l n e tw o rk .
The a p p lic a b ili ty o f th e s y n th e s is a lg o rith m fo r th e d is ju n c tiv e
a n d n o n - d is ju n c tiv e d e c o m p o s itio n i s n o t r e s tr ic te d to th e p a r tic u la r
B. T. s y s te m .
T hey c a n a l s o be u s e d fo r th e s y n th e s is o f a n y a rb itra ry
d ig ita l n e tw o rk in m u lti- le v e l fo rm .
6 .2
S cope fo r F u rth e r R e s e a rc h
T here i s s c o p e of w ork on b o th th e o r e tic a l a n d p r a c tic a l
a s p e c t s of m any p ro b le m s in th is a r e a .
C e rta in s p e c if ic p o s s i b i l i t i e s
a n d q u e s tio n s w h ic h a r o s e in c o n n e c tio n w ith th is r e s e a r c h w ill be
d is c u s s e d .
It h a s b e e n m e n tio n e d th a t e a c h o f th e in p u t a n d o u tp u t a rra y s
m ay c o n s i s t o f s e v e r a l c a s c a d e s . 'In s u c h s y s te m s , th e in p u t a rra y m ay
■ possibly r e p r e s e n t th e in p u t d o m ain o f tw o -d im e n s io n a l p ro b le m s lik e
th e g rid s tr u c tu r e of th e b o u n d a ry -v a lu e p ro b le m s o r th e in p u t re tin a
o f a p ic tu re p r o c e s s in g d e v ic e or th e m a tr ic e s in o rd in a ry c o m p u ta tio n s .
U s e fu l p rim itiv e fu n c tio n s c o n n e c te d w ith th e o p e ra tio n s in so lv in g t h e s e
p ro b le m s m ay be d e fin e d a n d im p le m e n ta tio n o f th e s e f u n c tio n s in th e
-1 3 6 B..T. s y s te m m ay be a tte m p te d . ■In p a r tic u la r , i t w ill be of i n t e r e s t to
fin d o u t a n optim um n u m b er in o rd e r to do s u c h p ro b le m s .
T h is problem
h a s a p a r tia l s im ila r ity w ith th e pro b lem o f m in im iz a tio n o v e r B oolean
g r a p h s (!5 )
^ t h a t w h ile th e n u m b er o f lo g ic m o d u le s a t a n y s ta g e of th e
m ap p in g o p e ra tio n re m a in f ix e d , i t i s th e n u m b er o f s ta g e s in m ap p in g
( le v e ls in B oolean graph) w h ic h m u st b e m in im iz e d .
An in te r e s t in g th e o r e tic a l q u e s tio n c a n be r a is e d w ith r e s p e c t
to th e s tr u c tu r e of M a it r a - r e a liz a b le f u n c tio n s .
In s e c tio n 3 .4 th e
lo g ic a l u n iv e r s a li ty o f a B. T. s y s te m c o n s is tin g of a p a ir o f C a s c a d e s
w a s p ro v e d on th e a s s u m p tio n th a t e a c h o f th e s e c a s c a d e s c o n ta in e d a t
l e a s t o n e d ir e c t c e ll in p u t m ore th a n th e n u m b er of o r ig in a l v a r ia b le s
t
, . . .x n } . If e a c h o f th e c a s c a d e s c o n s i s t s of e x a c tly (n-1) c e ll s
w ith n d ir e c t in p u ts , c a n w e r e a liz e a r b itr a r y fu n c tio n o f n v a r ia b le s
w ith th e ty p e of lo g ic - f r e e m ap s th a t c a n o n ly p erm u te th e v a r ia b le s
(in d e p e n d e n t o r d e riv e d ) ?
By th e a rg u m e n t g iv e n in s e c tio n 3 . 4 , th e
s y s te m c le a r ly c a n r e a liz e a rb itra ry fu n c tio n of (n-1) v a r i a b l e s . A lso ,
i t c a n b e e a s i l y s e e n th a t a l l fu n c tio n s b e lo n g in g to th e sam e e q u iv a le n c e
c l a s s a s a n y M a it r a - r e a liz a b le fu n c tio n o f n - v a r ia b le s c a n b e r e a liz e d .
W h e th e r th is s y s te m is lo g ic a lly u n iv e r s a l or n o t is s t i l l a n o p en
q u e s ti o n .
-1 3 7 Some im p ro v e m e n t in th e a lg o rith m p re s e n te d in C h a p te r 5 may.
\
b e a tte m p te d .
T h is a lg o rith m fo r g ro u p in g v a r ia b le s in v o lv e s th e p ro c e d u re
o f ta k in g o n e v a r ia b le a t a tim e an d d e te rm in in g w h ic h s u b fu n c tio n ■
in c lu d e s i t .
For a n y v a r ia b le , s a y
, w h ic h m ay b e in p u t to th e l a s t
le v e l U LM , th e p ro c e d u re sh o w s th a t th e re q u ire d s u b fu n c tio n is th e
sa m e a s th e o rig in a l fu n c tio n a f te r a n e x h a u s tiv e c o m p a riso n h a s b e e n
m ade.
Some f a s t e r w ay of d e te rm in in g th e v a r ia b le s w h ic h a re in p u t
to th e lo w e r - le v e l U L M 's m ay s a v e th is la b o u r.
The t e c h n i c a l a s p e c t of b u lk tr a n s f e r of d a ta a s e n v is io n e d in
t h i s t h e s i s s h o u ld form a n im p o rta n t p a rt of fu rth e r s tu d y a n d w ork in
th is a re a s o th a t a p r a c tic a l b u lk tr a n s f e r s y s te m c a n be r e a liz e d .
APPENDIX
-139-
A ppen d ix A
The n e x t s ta t e e q u a tio n s fo r th e v a r ia b le s to g e n e r a te th e p e rm u ta tio n
c y c le (m p irig , . . .m ^n) a re
x j = X1 , x j = x 2 ® x j ;
Xj1 = XnX^ 1 + x ; _ j
( X ^ 1S x n)
T h e s e a re d e riv e d fo r th e m interm s e q u e n c e of w h ic h T ab le 2 .3 is a
r e p lic a fo r th e c a s e of th r e e v a r i a b l e s .
N o te th a t in a n y rtn s u c h th a t
2 = i = 2 n, th e v a lu e o f x^ i s th e c o m p le m e n t o f th a t in m ^ _ i.
The v a lu e
of X^ in m j i s th e c o m p le m e n t of th a t in nr^n • T h e re fo re , fo r th e c y c le
(m j , mg , . . .m ^n) w e m ay w rite
x^ = x
.
F or x ' , w e s e e th a t x ' c a n be e x p r e s s e d in te rm s o f th e p r e s e n t
I
2
v a lu e of X g, p r e s e n t v a lu e of X j a n d th e n e x t v a lu e X j of x^ .
N ote
th e m interm s h a v in g Xg = I .
0
0
From in s p e c tio n o f th e a b o v e d ia g ra m , w e h a v e , Xg = I i f Xg = 0 an d
x 'j = 0 o r if X g = I a n d X j = I .
H e n c e x ^ = X2 © x^
-1 4 0 -
T hen w e o b ta in th e fo llo w in g tru th ta b le (c o m b in a tio n s fo r w h ic h
CO
X
COx -
x ' 3 i s tru e a re o n ly s h o w n ).
;
'
X 2
X 2
0
I
0
I
I
0
0
I
I
0
I
I
I
I
I
I
From th is t a b l e , w e h a v e
x 3 = X3X2 + X 2 (x3 ® x 2 >
I t i s n o tic e d th a t of th e e ig h t m in term s o f th e th re e v a r ia b le s {X3 ,Xg
fo u r g iv e a tru e v a lu e fo r x l . T his p ro p e rty hold's tru e fo r h ig h e r
)
)
)
I ,
-1 4 1 - .
v a r ia b le s lik e x ^ , x ^ , . . .x ^ .
■ x n = x!,xn - l
T h e re fo re , fo r
" X -l ^ © V l ’
3 , x ^ c a n be e x p r e s s e d a s .
■
To g e t th e e q u a tio n s fo r g e n e r a tin g 0 , n o te th a t th e m ap i s s im ila r to
th e c y c lic p e rm u ta tio n e x c e p t t h a t th e l a s t m interm X 1X0 , . „ .x
i
b e m ap p ed in to i t s e l f .
z
n
An AND g a te i s u s e d to d e te c t th e o c c u rre n c e
of th is m interm an d th e o u tp u t of th e g a te i s u s e d in th a t e v e n t to
o b ta in th e p ro p e r n e x t s t a t e s of a l l v a r i a b l e s .
T h e re fo re fo r g e n e r a tin g 5> , w e h a v e
=
m u st
X l + X 1X2 . , .x n;
=4=
PornS 3 , Xj1 = S ^ l ( x ^ e
(= ^ e
X2H X1X2 . .'..X n
Xr ) + X ^ l Xn + Xl X2 . . .Xr
-142-
A ppen d ix B
.
The p ro o fs of th e g e n e r a liz e d th e o re m s in v o lv e th e sam e k in d of
a rg u m e n t a s in th e s p e c ia l c a s e s .
T h e s e a re s k e tc h e d b e lo w .
•
Proof o f T heorem 4 . 2 . 1 ' : The f i r s t s te p i s to p ro v e t h a t e ith e r
o r Pj^ TL .
u,
A ssu m in g th is to be f a l s e , l e t th e re be m in term s {M p IV^}
o f th e v a r ia b le s o f TL s u c h t h a t M 1 , TV^e Pi a n d M i ^ Ui a n d
Ui .
E x p re s s in g Pq P^. . . P i . . . P^ = HP^ w h e re H r e p r e s e n ts th e p ro d u c ts of
l i t e r a l s o th e r th a n th o s e c o n ta in e d in Pi , i t c a n be sh o w n u s in g th e
sam e a rg u m e n ts a s in th e p ro o f o f 4 . 2 . 1 , th a t H is a n im p lic a n t o f F .
I t i s a c o n tra d ic tio n to th e a s s u m p tio n th a t HPi i s a PI of F .
Pi E Ui o r PjE Ui .
of U j .
T h erefo re
N e x t, a s s u m e W L O G , P c U^ a n d Pi i s n o t a PI
L et P ^ c p * c U j . T h is a g a in le a d s to th e c o n tra d ic to ry r e s u lt
th a t HP^ i s a n im p lic a n t o f F .
Proof of T heorem 4 . 2 . 2 ':
P. c U b u t Q - s U .
1
I
1
i
Q.E.D.
W ith th e g iv e n c o n d itio n , a s s u m e W LO G ,
I t le a d s to th e c o n c lu s io n th a t X i s a n im p lic a n t
o f F w h ic h i s c o n tr a d ic to r y .
Q , E Uj if Pj E U j .
So Pj m u s t b e a -PI o f U j .
T h e re fo re w ith th e c o n d itio n s g iv e n ,
The o th e r p a rt fo llo w s from Theorem 4 . 2 . 1 ' .
.
C o n v e rs e C a s e : WLOG l e t P ,, Q i £ U . . W h en P 1X is t r u e , F = I .
,
1
I . !
i
T h e re fo re F m u s t be tru e w h e n Q jX is t r u e , b e c a u s e F c a n b e e x p r e s s e d
-143-
a.s a fu n c tio n o f IL a n d X .
QX i s
Thus Q.X^ F . F o r.th e s e c o n d p a rt, a s s u m e
n o t a PI o f F an d Q ^X c QjfcXltQ F .
th e c o n tra d ic to ry c o n c lu s io n t h a t
If Q^c Qjfc , th e n w e a re le d to
i s n o t a prim e im p lic a n t of U , . If
X c x * , th e n i t le a d s to th e c o n c lu s io n th a t PX is n o t a PI o f F .
Q^ -
, X = X*. an d Q X i s a PI o f F .
T h erefo re
Q.E.D.
Proof of C o ro lla ry I1: If F is a c o m p le te sum of P i ' s , th e n th e proof
fo llo w s from Theorem 4 . 2 . 2 ' .
L et F b e a n in c o m p le te s e t o f P i ' s .
W L O G , le t M b e a tru e m interm o f
w h ic h i s n o t c o n ta in e d in th e P i's
of Ui o b ta in e d u n d e r Ai . Let PiX be a Pl o f F s u c h th a t P c U . . T hen
M Xe F .
T h e re fo re , MX m u st b e in c lu d e d in som e P i's of F .
n o t a P I, th e re e x i s t s Qi u n d e r A s u c h th a t M q Qi .
Q i i s a PI o f U^.
S in ce X-is
By Theorem 4 . 2 . 1 '
T h e re fo re , th e a s s u m p tio n M q Ui b u t n o t in c lu d e d in
a n y PI o f Ui u n d e r Ai i s f a l s e .
Proof of T heorem 4 . 2 . 3 ':
Q.E.D.
A ssum e th a t Pi Pj . . . P ^ an d Qi Qj • • -Qjc a re
tw o p ro d u c t te rm s of F . A p p ly in g th e tw o c o n d itio n s , m e n tio n e d in th e
p ro o f o f Theorem 4 . 2 . 3 , to th e s e t o f p ro d u c t te rm s o f F to c h e c k if
th e y c o n ta in a n y im p lic a n ts w h ic h a re n o t prim e im p lic a n ts , i t c an be
sh o w n th a t c o n d itio n (i) c a n n o t b e tru e fo r th e p a ir Pi P j. ... Pj, an d
Qi Q j . . .Q k , b e c a u s e i t w o u ld le a d to th e c o n c lu s io n t h a t c e r ta in P^
a n d Qj s a t i s f i e d th e c o n d itio n .
T h is w o u ld v io la te th e a s s u m p tio n th a t
Py a n d Qj. a re prim e im p lic a n ts o f a c o n s ti tu e n t fu n c tio n U j .
C o n d itio n
—14 4 —
(ii) a l s o c a n n o t be tru e b e c a u s e i t w o u ld im p ly th a t c e r ta in P.cz Q 1, or Q.c-P,
J
J
J j
Q.E.D.
w h ic h w o u ld a g a in v io la te th e s a id a s s u m p t i o n ,
Proof o f Theorem 4 . 2 . 4 ':
L et U be u n a te in IL . A ssu m e th a t th e sum o f
th e te rm s u n d e r A. e q u a ls I . T hen e ith e r U. i s tr iv ia l o r P i's of b o th U
1
I
. i
and U
a re p r e s e n t u n d e r A^. The f i r s t p o s s ib il ity is a g a i n s t th e
a s s u m p tio n t h a t F is n o n - d e g e n e r a te .
th a t U i s n o t u n a te in TL .
U , S
i J
The s e c o n d p o s s ib il ity im p lie s
C o n tr a d ic tio n . T h e re fo re if U is u n a te in
P.j =/ I . .
j
C o n v e rs e C a s e : L et D P. ^ I a n d U i s n o t u n a te in T I.. T hen P i's of
__
J 1
1
b o th TL an d TL m u s t o c c u r in th e P i’s of F . By C o ro lla ry I', th e P i's
o f TL a n d TL o c c u rrin g u n d e r A a re s u f f ic ie n t to d e fin e TL a n d TL .
T h e re fo re th e sum of t h e s e P i 's =
= I.
T h is i s a c o n tr a d ic tio n .
H e n c e th e th e o r e m . '
Proof of T heorem 4 . 2 . 5 ' :
Ui (Ui ) .
Q.E.D.
A ssum e P X , Q .s p .
i i
Let P , Q. b e P i's of
i
i
T hen by T h eo rem 4 . 2 . 2 ' , Qi X m u s t be a PI of F .
L et P .£ U 1 a n d Q. £ U .
i i
i i
T hen F c a n b e e x p r e s s e d in te rm s of U. a s i
F = U 1X + Ui + R w h e re R i s som e r e s id u a l fu n c tio n .
T a k in g c o n s e n s u s
b e tw e e n UiX a n d U i , .w e o b ta in X a s a n im p iic a n t of F .
H e n c e th e th e o r e m .
C o n tra d ic tio n .
C o n tra d ic tio n .
Q.E.D.
-145-
Proof o f T heorem 4 . 3 . 1 :
To prove f ir s t t h a t P^X^ c
(U^) and
e U 2 ^ 2 ^ ' C o n s id e r th e p o s s ib le c a s e s a s in th e p ro o f o f'T h eo rem
I
' 4 . 2 . 1 . It c a n be s e e n in a n e x a c tly s im ila r m an n er th a t if P X 4 U 1
I I
I
an d P-jX-^ ^
PI o f F .
or X - j^
£
Ug and X^Pg
£ Ug
, th e n P^Xj Pg c a n n o t b e a
N e x t, to show P 1X^ a s a PI of U 1 (U 1) , a s s u m e P 1X 1 c P*Y*
w h e re P* and Y* a re s u b s e ts of th e l i t e r a l s o f P^ and X 1 r e s p e c tiv e ly .
fo llo w s th a t P 1X^Pg
n Cfc a PI o f F .
C o n tra d ic tio n .
S im ila rly it c a n
b e sh o w n th a t if X1 Pg is n o t a PI of U g(Ug) it le a d s to a c o n tr a d ic tio n .
H e n c e th e th e o re m .
Q.E.D.
Proof o f T heorem 4 . 3 . 2 : W LO G , l e t P 1X 1 S-U1 .
Now if Q^Y 1 e TJ1 , it
le a d s to th e c o n c lu s io n t h a t X1Pg is a n im p lic a n t of F , w h ic h is
c o n tr a d ic to r y . The o th e r p a rt fo llo w s from Theorem 4 . 3 . 1 .
C o n v e rs e p a r ty
W LO G , le t P 1X1 , Q X 1 ^ U 1 and P 1X 1Pg s F'. A ssu m e ,
W L O G z X 1Pg c U g . In a m a n n er s im ila r to T heorem 4 . 2 . 2 , it c a n
b e show n th a t w h en Q 1X 1Pg is t r u e , F = I .
T h u s Q 1X1Pg e F .
To
p ro ve th e o th e r p a rt n o te th a t s in c e P 1X 1Pg is a PI o f F an d P 1X 1
a PI of U 1 , h e n c e X 1Pg i s a PUof Ug (by T heorem 4 . 3 . 1 ) .
Now if
Q 1X I P2 *s n o t a P I o f P an d th e r e e x i s t s c e r ta in Q*X*P^
Q 1X 1 P g ,
th e n ,
it le a d s to th e c o n tra d ic tio n th a t e ith e r Q 1X 1 is n o t a PI of U 1
o r X 1Pg is n o t a PI o f U g .
H e n c e th e r e s u l t .
Q.E.D.
It
“ 146-
G e n e r a liz a tio n o f T h eo rem s 4 . 3 . 1 a n d 4 . 3 . 2
Let P j , Q , IL e t c — le t t e r s w ith s u b s c r ip t i r e p r e s e n t p ro d u c ts of th e.
lite ra l's of th e v a r ia b le s of Di fo r i = 1 , 2 , . . . s ;
fo r th e v a r ia b le s o f Ci , fo r i = 1 , 2 , . . . ( s - 1 ) .
i , jS ^ Y .
e tc .
L et LL b e a fu n c tio n on
th e v a r ia b le s of A. a n d U' b e th e fu n c tio n on Ui 's fo r i = 1 ,2 , . . . , s .
Theorem 4 . 3 . 1 ' :. L et P G ^ P ^ Q !^ i . . . PQh Pj + i . . . Pk + i
Then P.%. i s a PI of U .(U.) a n d « . P . + 1 i s a PI of U
b e a PI of F .
j (U +1) fo r
j = i , . . . k.
Theorem 4 . 3 . 2 ' :
If PiA a n d Q^A b e tw o P i's of F w h e re A, B r e p r e s e n t
p ro d u c ts of l i t e r a l s n o t c o n ta in in g a n y l i t e r a l s of Di , th e n Q p i C Ui (Ui )
if P p , - U (U.) w h e re a . i s a s u b s e t
I *
Jt I
I
QjGii i s a PI o f Ui (Ui ) if P a
P p i , Q a , c Ui (Ui ) ,
of th e l i t e r a l s o f A> A lso ,
is a PI o f Ui (Ui ) •
C o n v e r s e ly if
th e n Q iA c p if Pi Ac F w h e re A c o n ta in s th e
l i t e r a l s O f a i - F u rth e r if P ^ i a n d Q p i a re P i's of Ui a n d Pi A is a PI of
F , th e n QiA i s a PI of F .
The proofs, of t h e s e th e o re m s m ay b e o b ta in e d b y a rg u m e n ts s im ila r
to th o s e in T h eo rem s 4 . 3 . 1 a n d 4 . 3 . 2 .
LITERATURE CITED
-148-
1.
A n d e rso n , L. K. , " H o lo g ra p h ic O p tic a l M em ory fo r Bulk D ata
S to ra g e " B ell L a b o ra to rie s R ec o rd , N o v em b er 19 6 8 . pp 3 1 9 -3 2 5 .
2.
A s h e n h u r s t, R. L. , "The D e c o m p o s itio n o f S w itc h in g F u n c tio n s ,"
P r o c . I n t . Sym posium on th e T h eo ry o f S w itc h in g A nnals, of th e
C o m p u ta tio n la b o r a to r y o f th e H arv ard U n iv e rs ity ,V o l-2 9 ,p p 7 4 -1 1 6 .
3 .
B irk h o ff, G . , a n d M a c la in e , S . , A S u rv ey of M o d ern A lg e b ra , N ew
Y ork, M a c m illa n & C o . , 19 6 5.
;
4.
C u r t i s , FI.A. , The D e s ig n o f S w itc h in g C ir c u its , D , Van N o stra n d
C o . > I n c . , P r in c e to n , N ew J e r s e y , 19 62 .
5.
E l s p a s , B. , e t . a l . , P ro p e rtie s of C e llu la r A rrays fo r L ogic an d
S to ra g e , S c ie n tif ic R eport 3 , SRI P ro je c t 5876, Ju ly 1967
(A FC R L-67-04 6 3 ) .
6.
F r a le ig h , J . B . , A F ir s t C o u rse in A b s tra c t A lg e b ra , A d d is o n -W e s le y
P u b lis h in g C o . , 19 67.
7.
H a w k in s , J . W , , a n d M u n s e y 1 C . J . , "A T w o -d im e n s io n a l I te r a tiv e
N etw o rk C o m p u tin g T e c h n iq u e a n d M e c h a n iz a tio n , " P r o c . 1962
W o rk sh o p on C o m p u te r O rg a n iz a tio n , pp 9 3 -1 2 5 , 1963.
8.
H o lla n d , J . H . , " I te r a tiv e C irc u it C o m p u te rs ," P r o c . E a s te rn
J o in t C o m p u ter C o n fe re n c e , p p . 1 0 8 -1 1 3 , 1957.
9.
M a it r a , K. K. , " C a s c a d e d S w itc h in g N etw o rk s of T w o -in p u t
F le x ib le C e l l s , " IRETEC, V o l. E C -1 1 , N o . 2 , pp 13 6 -1 4 3 , A pril 1962
1 0 . M c C lu s k e y , E J . , In tro d u c tio n to th e T heory o f S w itc h in g C ir c u its ,
M cG raw FIill Book C o . , pp 1 6 5 -1 7 4 , 1965.
1 1 . M in n ic k , R . C . , "A S urvey o f M ic r o c e llu la r R e s e a r c h ,"
V oL 14, pp 2 0 3 -2 6 1 , A pril 1967.
J.A.C.M.,
12 . M in n ic k , R . C . , " C u tp o in t C e llu la r L o g i c ," IEEE T r a n s , on
E le c tro n ic C o m p u te rs , V o l. E C -1 3 , pp 6 8 5 -6 9 8 , D e c e m b e r 19 64.
1 3 . M in s k y , M . an d Pa p e r t, S . ,
P e r c e p tio n s , The MIT P r e s s , 1969 .
14.
—149 “
M u k h o p a d h y a y z A ., "U n ate. C e llu la r L o g ic , " IEEE T ra n s , on
C o m p u te rs V o l. C - 18 , pp 1 1 4 -1 2 1 , F e b ru a ry 1969 .
'1 5 .
R oth, J . P . , an d K arp, R . M. , " M in im iz a tio n o v e r a B oolean G ra p h , "
' I^M- Jo u rn a l of R e s e a rc h a n d D e v e lo p m e n t, V o l. 6, N o . 2 , pp 2 2 7 2 3 8 , A p ril, 19 6 2.
16.
S h o rt, R . A . , " T w o -ra il C e llu la r C a s c a d e s , " P ro c . o f the. AFIPS
F a ll'J o in t C o m p u te r C o n fe re n c e , V ol. 2 7 , P art I, pp 3 5 5 -3 6 9 , 1965 .
17.
U n g e r, S . H . , "A C o m p u ter O rie n te d Tow ard S p a tia l P ro b le m s ,"
P ro c . IRE, 10 (O c t. 1 9 5 8 ), pp 1 7 4 4 -1 7 5 0 .
-150-
RESEARCH REPORTS AND PAPERS ON W H IC H THE THESIS HAS' BEEN BASED.
1.
M in n ic k , R . C . , T h u rb e r, K. J. , M u k h o p a d h y a y , A ., R oy, K.K. ,
C e llu la r Bulk T ra n s fe r S y s te m s , F in a l R ep o rt, MSU P ro je c t
E R L -8 -0 0 0 9 -6 0 1 , M o n ta n a S ta te U n iv e r s ity , B o zem an , M o n tan a
AFCRL 6 8 -0 4 9 7 , O c to b e r 19 68.
2.
M u k h o p a d h y a y , A ., S c h m itz , G . , T h u rb e r, K . J . , a n d R oy, K.K. ,
M in im iz a tio n o f C e llu la r A rrays , F in a l R ep o rt, NSF C o n tra c t G J - 1 5 8 ,
E le c tro n ic s R e s e a rc h L a b o ra to ry , E ndow m ent an d R e s . F o u n d a tio n ,
M o n ta n a S ta te U n iv e r s ity , B ozem an , M o n ta n a , S e p te m b e r 1969.
3.
Roy, K. K. , " D e c o m p o s itio n of L ogic F u n c tio n s fo r R e a liz a tio n in
M u lti- L e v e l ULM N e tw o rk s" - - t o b e p r e s e n te d a t th e T hird H a w a ii
"I n te r n a tio n a l C o n fe re n c e in S y stem S c ie n c e s , J a n u a ry 1970.
libraries
f
D378
R812
co p . 2
J
m
*
m
Roy, Kalyan K1
C e llu la r b u lk t r a n s f e r
system
n a MB:
UAL
An d AODWgaa
IN T E R U 3 R A H Y
ZD6'1$