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Boolean a n d M ultiple-V alued Functions in C o m b in ation al Logic Synthesis by
E lena V ladim irovna D ubrova Dipl. Eng., H IM E E , Sofia. B ulgaria. 1991
A D issertation S u b m itte d in P a rtial Fulfillm ent o f th e R eq uirem en ts for th e Degree of
D O C T O R O F PH IL O SO PH Y in th e D e p a rtm e n t of C o m p u ter Science We accept th is d issertatio n as conform ing
to th e required sta n d a rd
Dr. J . C. Vluzio, Supervisor (D e p a rtm e n t of C o m p u ter Science)
Dr. J .W Æ llf e ^ e p a r tm e A ta l M em ber (D ep artm en t o f C o m p u ter Science) Dr. D oM . M iller, D ep artm ei^al M em ber (D ep artm en t o f C o m p u te r Science) Dr. C. M organ, O utside M e rfi^ r (D e p artm e n t of Philosophy)
Dr. I. G. Tabakow , E xternal E x am in er (H IM EE. Sofia, B ulgaria) 0 Elena V ladim irovna D ubrova, 1997
U niversity of V ictoria
.All rig h ts reserved. This d isse rta tio n m ay not be reproduced in w hole o r in p a rt, by photocopy or o th e r m eans, w ithout th e perm ission of th e a u th o r.
11 Supervisor: Dr. J .C . M uzio
.A B STR A C T
T h e su b ject of th is d issertatio n is th e th e o ry o f Boolean and m u ltip le-v alu ed func tion s. T h e m ain a re as considered axe: fu n ctio n al com pleteness, canonical form s, m in im izatio n o f fu n ctio n s, discrete differences a n d functional decom posability. T he resu lts o b tain ed axe used as a foundation for th e developm ent o f several new algo rith m s for logic sy n th esis o f com binational logic circuits. T hese in clu d e an efficient alg o rith m for th ree-lev el A N D -O R -X O R m in im iza tio n for B oolean functions, an algo rith m for gen eratin g th e com position trees for B oolean and m u ltip le-valu ed functions in a c ertain class, a n d an algorithm for c o m p u tin g a new canonical form of m u ltip le valued functions. S everal o th e r problem s, re la te d to logic sy n th esis, such as te st g e n eratio n for co m b in atio n al logic circuits a n d synthesis of easily te s ta b le circuits axe also addressed. Possible directions for fu tu re research are discussed.
E xam iners:
D r. . E lliQ É e p artm en ^ al M em ber (D e p a rtm e n t of C o m p u ter Science)
D r. D. M. M ille r,^ e p a x tm e n ta l M em ber (D e p a rtm e n t of C o m p u te r Science) D r. C. M organ, O u tsid e ^ e m b e r (D e p a rtm e n t o f Philosophy)
C O N T E N T S iii
C o n te n ts
C on ten ts
iii
L ist o f Tables
vi
L ist o f Figures
vii
A cknow ledgm ent
ix
D ed ica tio n
x
1
In trod uction
1
2
Background
7
2.1 N o t a t i o n ... 7 2.2 Basic n o t i o n s ... S 2.2.1 R elations ... 8 2.2.2 F u n c t i o n s ... 8 2.2.3 B inary o p e r a tio n s ... 9 2.3 C hain-based Post a lg e b r a ... 93 A Canonical Form o f M V L F unctions
12
3.1 R eed-M uller canonical form an d its generalizations ... 133.2 The a l g e b r a ... 15
3.3 Functional com pleteness o f {0 , • } ... 17
3.4 P roperties of th e o p eratio n s o f th e algebra B ... 19
3.5 D ecom position th e o r e m ... 2 1 3.6 C anonical form of m ultiple-valued f u n c t i o n s ... 23
3.7 -^.n algorithm for c o n stru ctin g th e canonical f o r m ... 27
3.8 C o n c lu sio n ... 32
4 A N D -O R -X O R M in im ization o f B o o lea n F unctions
33
4.1 Logic m in im i z a tio n ... 344.2 N otation and d e f i n i t i o n s ... 36
4.3 U pper bound on th e n u m b e r of p ro d u ct-term s in th e A N D -O R -X O R e x p a n s i o n ... 38
C O N T E N T S iv
4.4.1 D ividing th e cubes in to equivalence c l a s s e s ... 42
4.4.2 O b tain in g Ti éind T2 ... 45
4.4.3 C o n stru ctin g g l J n i t a n d g ' I J n i t ... 46
4.4.4 D eterm ining com m on d o n ’t cares in g I J n i t an d g2 J n .it . . . . 46
4.4.5 M u ltip le-o u tp u t p r o b l e m s ... 48
4.5 E xperim ental R e s u l t s ... 48
4.6 C o n c lu sio n ... 52
5 Test G eneration for M u ltip le-V alu ed C ircu its
53
5.1 Test generation for logic c i r c u i t s ... 535.2 Boolean difference for te st g e n e r a t i o n ... 55
5.3 Definition of full sen sitiv ity ... 56
5.4 C alculation of full s e n s i t i v i t y . ... 57
5.5 Full sensitivity in te st g e n eratio n ... 62
5.5.1 Test g en eratio n for p rim a ry in p u ts ... 62
5.5.2 Test g en eratio n for in te rn a l lin e s ... 63
5.6 T otal n um ber o f m -valued functions fully sensitive to all th e ir variables 6 6 5.7 C o n c lu sio n ... 71
6 C om p osition Trees in Logic S y n th esis
73
6.1 D isjunctive decom position o f f u n c t i o n s ... 736.2 C om position t r e e s ... 77
6.3 .^.n algorithm for c o n stru ctin g com position t r e e s ... 81
6.4 C o n c lu sio n ... 87
7 S yn th esis o f E asily T estable C ircu its
89
7.1 Im p lem en tatio n of m odulo m sum -of-products canonical fo rm ... 907.2 T estability of in tern al lines... 92
7.3 T estability of p rim ary in p u ts ... 95
7.3.1 A procedure for te st g e n eratio n ... 96
7.3.2 E valuation of th e effectiveness of th e procedure... 98
7.4 T estability by hardw are re d u n d a n cy ... 1 0 1 7.5 C onclusion... 102
8 C onclusion
104
A p p en d ices
A C urrent-M ode CM OS M u ltip le-V alu ed C ircuits
107
.4.1 Basic operations an d sy m b o ls... 108A .1.1 C o n sta n t-cu rren t s o u r c e ... 108
A .1.2 C u rren t m i r r o r ... 109
A .1.3 C u rren t c o m p a r a to r ... 110
.A. 2 Im p lem en tatio n of basic logic g ates using current-m o d e C M O S circu its 1 1 1 A .2.1 M IN and M AX c i r c u i t s ... 113
A.2.2 A d d itio n m odulo m c i r c u i t ... 114
C O N T E N T S ^
L I S T O F T A B L E S vi
L ist o f T ables
4.1 B enchm ark resu lts... 49
4.2 AOXM IN results for (481... 51
5.1 T ru th ta b le for th e function from e x am p le ... 59
L I S T O F F I G U R E S vii
L ist o f F ig u res
3.1 D iagram illu stra tin g th e m ultiplication o f /th row o f th e m a trix by
th e m a trix ... 30
4.1 K arnau g h m ap o f th e function from th e e x a m p le ... 37
4.2 M ap o f th e ex am p le fu n ctio n... 43
4.3 Im p lem en tatio n of th e su b ro u tin e D i v i d e E q C I a s s e s ( ) ... 44
4.4 A function w ith all p ro d u ct-term s in th e sam e equivalence class. . . . 46
4.5 F unctions g i J n i t a n d g 'lJ n it for th e fu n ctio n from th e exam ple. . . . 47
4.6 Im p lem en tatio n of th e subroutine S p e c i f y B o t h ( ) ... 47
5.1 Full sensitivities for th e exam ple fu n c tio n ... 58
5.2 E x am p le c irc u it... 63
5.3 A m ultiple-valued logic c irc u it... 64
5.4 Set U w ith subsets .4t. .Ag 4 „ ... 6 6 5.5 P lo ts for e (m .n ) as a function of n for fixed m = 3 .5 and 10... 6 8 5.6 P lo ts for e (3 .n ) a n d e“(3 .n ) as functions of n for m = 3... 70
6.1 Sim ple disjunctive d e c o m p o s itio n ... 74
6.2 D ecom position c h a rt for an exam ple fu n ction in E ... 75
6.3 E x am p le of a com position t r e e ... 78
7.1 R eed-M uller c irc u it... 91
7.2 C irc u it realizing m odulo m SO P form ... 91
7.3 C irc u it im plem enting th e function from th e e x a m p le ... 92
7.4 C irc u it realizing m odulo m SOP form ... 94
7.5 C irc u it w ith an e x tra m ultiplication m od m g a te G "... 101
.A.l C u rre n t source: (a) circuit configuration, (b) sy m b o l... 109
A .2 N -ty p e and P -ty p e cu rren t m irrors: (a),(c) circu it configurations, (b).(d) sy m b o ls... 1 1 0 A 3 C lassical cu rren t c o m p a ra to r... 110
A .4 T h resh o ld cu rre n t co m p arato r of O nnew eer a n d K e rk h o ff... I l l .A..5 M in im u m -m ax im u m circuit of O nnew eer an d K e r k h o f f . ... 113
A. 6 (a) M inim um an d (b) M axim um circu its o f Z hijian an d H ong... 114
A .7 T h e sim ulation resu lts for th e M IN /M A X circu it of O nnew eer and Kerkhoff: (a) lA , (b) IB, (c) M IN (IA ,IB ). (d) M .AX (IA .IB), (e) power d issip a tio n ... 115
L I S T O F F I G U R E S viii A.S T h e sim u la tio n results for th e M IN and M A X circ u its o f Zhijian and
Hong: (a) lA , (b) IB. (c) M IN (IA .IB ). (d) M A X (IA .IB ), (e) power d issip a tio n ... 116 A .9 (a) A b solu te difference circu it (b) C orrection c irc u it... 118 A. 10 A dd itio n m odulo m circu it (a) circuit configurations, (b) sym bolic
schem e... 119 A .11 T h e sim u la tio n resu lts for th e ad d itio n m o d u lo m circu it: (a) lA. (b)
A C K N O W L E D G M E N T ix
A c k n o w le d g m e n t
I w ould like to th a n k Dr. Jo n M uzio for his guidance a n d su p p o rt th ro u g h o u t m y g ra d u a te stu d ies. Being a d o cto ral s tu d e n t and a new m o th e r sim ultaneously vvasn t e asy for m e. a n d I w arm ly ap p re cia te his patience a n d kindness.
I would also like to express m y g ra titu d e to Dr. M ichael M iller for th e m any help fu l discussions and for sh arin g w ith m e his tech n ical e x p e rtise , w^hich inspired m e in o b ta in in g a n um ber of resu lts co n tain ed in this d isse rta tio n .
I am th a n k fu l to m y c o m m itte e m em bers and m y e x te rn a l ex am in er. Dr. Ivan T ab ak o w . for careful review ing o f th is dissertatio n an th e ir valuable com m ents and suggestions.
I a m obliged to Dr. M ichaela S erra and my fellow s tu d e n ts in th e VLSI group for a tte n d in g m y talks on m ultiple-valued logic a n d h elp in g m e to becom e a b e tte r le c tu re r, .\s h r a f Hafez. Bill G ard n er. Claudio C osti a n d K evin C attell read an d c o m m e n te d on th e in tro d u ctio n of m y d issertation. S te p h e n G oglin and Ken K ent su g g este d m e th e nam e of th e tool described in C h a p te r 4. E nyu W ang helped m e to p re p a re th e slides for my defense. I g reatly ap p reciate th e ir kind help.
M y special th an k s to m y husband Dr. Dilian G urov for careful proofreading and n u m e ro u s suggestions for im provem ents and corrections in th e d issertatio n .
F inally, I w ould like to acknow ledge th e N atu ral Sciences an d Engineering Re sea rc h C ouncil o f C an ad a a n d th e C an ad ian M icroelectronic C o rp o ratio n for p ro v id in g research g ra n ts m aking m y g ra d u a te studies possible.
D E D I C A T I O N
D e d ic a tio n
C h a p ter 1
In tr o d u c tio n
T h e m ain objects o f s tu d y o f this d issertatio n are discrete functions. W hile th e th e o ry o f discrete functions is an in terestin g area of research on its own. it also has a d irect p ractical ap p licatio n to logic synthesis. We stu d y th e pro p erties o f d iscrete fu n ctio ns an d use th em to d evelop several new algorithm s for logic synthesis. Som e problem s re la te d to logic sy n th esis, such as test generation for logic circuits an d sy n th esis of easily testab le c ircu its, are also addressed.
D iscrete functions are m appings relating finite sets. In general, th e y m ay be
heterogeneous, w here th e variables of th e function do not take values in th e sam e
set. T his d isse rta tio n , however, considers only the case of homogeneous fu n ctio n s of ty p e —)• M on a fixed set .V/ := { 0 .1 , m — I}. T his is a com m on re stric tio n for logic sy n th esis-related work. Such functions are usually called multiple-valued or
m-valued, and. for th e special case of m = 2. Boolean or switching functions.
Logic synthesis is a step in th e design process for dig ital circu its. G enerally, th e design process d e p en d s heavily on th e ta rg e t technology'. In te g ra te d circuit technology progresses very q u ick ly an d th e design m ethods used to d a y m ight not be efficient in 10 years. However, logic synthesis is technology independent and th erefo re m ost of its techniques can b e m ap p ed into any underlying technology.
Logic synthesis s ta r ts w ith a description of a discrete function (by m ean s o f tr u th ta b le , decision d ia g ra m , hardw are description language) an d produces a logic d ia g ra m
C H A P T E R L I N T R O D U C T I O N 2 o f th e c irc u it im plem enting it. Such a diagram is u su ally called a logic circuit. logic c irc u it has several in p u ts a n d one o r m ore o u tp u ts , w hich tak e discrete values. It is com posed o f building blocks called logic gates from a selected set. T h e g a tes realize logic o p e ra tio n s such as A N D . O R and N O T. hence th e n a m e logic. U sually, th e goal o f logic sy n th esis is to find a m in im al circuit re a liza tio n o f th e function in te rm s of a given set o f g a tes, u nder som e c rite ria of m inim ality. T h e c rite ria m ight be red u cin g th e n u m b e r a n d size of gates th a t are needed to b u ild th e c irc u it, reducing th e n u m b e r o f in terco n n ectio n s betw een th ese gates.
T h e re a re two types o f logic circuits - c o m b in a tio n al a n d sequential. In a com b in a tio n a l c irc u it, th e o u tp u t value depends only on th e c u rre n t value of th e in p u ts. In a seq u en tial circuit, th e o u tp u t depends on th e c u rre n t value o f th e in p u ts an d on th e p a st in p u t values. A sequential circuit can be rep resen ted as c o m b in a tio n al c irc u it w ith ad d ed m em ory devices or feedback loops. T herefore, a co m b in atio n al c irc u it is a m ore fu n d am en tal building block. T h is d is se rta tio n deals only w ith com b in a tio n a l logic circuits, a n d we use the term "logic c irc u it" to m ean "co m b in atio n al logic c irc u it" .
.A d iscrete function models a com binational logic circ u it by m apping th e possible in p u t assig n m en ts onto th e values assum ed by th e o u tp u t. T h e properties o f d iscrete fu n ctio n s th erefo re provide a foundation for th e m eth o d s o f synthesis of c o m b in a tio n al logic circu its.
If a logic circuit is com posed o f gates realizing B oolean functions, th e n such a c irc u it is called a Boolean o r two-valued logic c irc u it. Likewise, a logic circu it b u ilt of g ates realizing m ultiple-valued functions is called multiple-valued or m-valued. T h is d isse rta tio n , w ith the ex cep tio n o f C h ap ter 4. add resses th e p o ten tial pro b lem s asso c ia te d w ith m ultiple-valued logic circuits.
O u r in te re st in m u ltip le-v alu ed logic circuits is tw ofold. F irst, we found th a t s tu d y in g a problem in th e g eneral m -valued case gives us a b e tte r u n d e rs ta n d in g o f th e u n d e rly in g stru c tu re in th e two-valued case b ecau se som e p ro p erties, e v id e n t in
C H A P T E R 1. I N T R O D U C T I O N 3 th e richer m -v alu ed s tru c tu re , often degen erate w hen re stric ted ju s t to two values. E xam ples s u p p o rtin g th is claim are shown in C h a p te r 5 and C h a p te r 7. Second, m ultip le-v alu ed logic circu its offer several p o te n tia l o p p o rtu n ities for th e im prove m ent o f p resen t v ery -larg e scale integrated (V L SI) circu it designs. Serious difficulties w ith lim itatio n s on th e n u m b e r of connections o f an in teg rated c irc u it w ith th e ex te rn a l world (p in o u t pro b lem ) as well as on th e n u m b e r of co n n ectio n s inside th e c ircu it (in terc o n n e ctio n problem ) encountered in som e VLSI circu it sy n th esis could be s u b sta n tia lly red u ced if signals in th e circu it a re allowed to a ssu m e four o r m ore sta te s ra th e r th a n o n ly tw o. If. for exam ple, each connection carries tw ice as m uch inform ation, th e n o n ly h a lf as m any connections are required. M any lab o rato ries w orld wide p re sen tly inv estig ate possibilities for electronic fab ric a tio n of m u ltip le valued logic c irc u its. A recent achievem ent is th e IN T E L 16 M bit flash m em ory chip w ith each cell o f th e m em o ry capable of sto rin g four discrete values [67]. Em ploying 4-valued logic allow ed IN T E L to drop th e cost o f th e chip to S20 p e r M b y te. IN T E L also declared th a t th e ir longer term target is a 16-valued flash m e m o ry w ith a cost o f 50 cents p er M b y te. W e believe th a t th e dev elo p m en t of synthesis tech n iq ues for m ultiple-valued logic c irc u its is essential to fa c ilita te th e ir electronic fa b ric a tio n . T his m o tiv ated us in o u r research.
fu n d a m e n tal role in logic synthesis is played by complete sets o f function s, A set o f functions is said to be functionally complete if any function c an be defined as a com position of fu n c tio n s from this set. T h eoretically , logic sy n th esis can be based upon any set of g a te s realizing a com plete set o f functions. In p ra c tic e , how ever, th e choice of gates to b e used in logic synthesis is n o rm ally d ic ta ted by th e cost of th e ir im p lem en tatio n , w hich changes rapidly w ith progress in circuit technology. O th e r issues, influencing th e choice o f th e basic set o f g ates, w hen realizing a given function, are:
• th e ex isten ce o f a sim p le expression for th e fu n ctio n as a co m p o sitio n o f functions from th e basic set (im plying the existence o f an efficient c irc u it im p le m e n ta tio n
C H A P T E R L I N T R O D U C T I O N 4 for th e function), a n d
• th e existence of a fast alg o rith m for c o m p u tin g this expression.
T h e search for a m in im a l expression for a given function (u n d e r c ertain crite ria of m in im ality) w ith o u t a n y restrictio n s on its s tru c tu re , in te rm s of any practically m eaningful com plete set o f functions, is known to be an ex trem ely difficult ta sk in te rm s o f co m p u tatio n al com plexity. To be feasible, th e p ra c tica l algorithm s for logic sy n th esis norm ally put som e restrictio n s on th e p ro b lem a n d seek for th e solution to th is re stric ted problem . T w o com m on approaches are:
1. re strict th e expression to be o b tain ed to a p a rtic u la r ty p e (e.g. two-level .A.ND- O R expression)
2. re strict th e fu n ctio n s for which th e solution is sought to a p a rtic u la r class (e.g.
sym m etric functions, m onotonie functions)
In th is d issertatio n we stu d y b o th approaches to logic synthesis. C h ap ter 3 and C h a p te r 4 follow ap p ro ach ( 1 ) for two different ty p e s of re stric te d expressions. Chap>- te r 6 follows approach (2 ) for a clziss of functions w hich is form ally defined using a
d iscrete difference in tro d u ce d in C h ap ter 5. C h a p te r 5 also shows how th is difference can be advantageously used for g eneratin g tests for m ultiple-valued logic circuits.
In c ertain cases, finding a m inim al circuit realizatio n for a given function is not th e p rim ary goal of logic synthesis. Such a situ a tio n m ay arise in specific applications w here som e o th er p ro p e rtie s o f th e circuit are m o re im p o rta n t, like fault to leran ce or safety. In C h ap ter 7 we develop a technique for logic sy n th esis, suitable for applica tio n s in which th e ab ility to te st circuits easily a n d quickly is critical.
T h e m ore d etailed s tru c tu re o f th e d issertatio n is as follows.
C h a p te r 2 describes th e m a th e m a tic a l background for th e d issertatio n .
In C h ap ter 3 we prove fu n ctio n al com pleteness o f th e set consisting o f th e o p era tions of a d d itio n m odulo m , m inim um , and th e set o f all lite ra l o p erato rs, where m is
C H A P T E R 1. I N T R O D U C T I O N o a positive integer. We show th e ex isten ce of a canonical form for th e m u ltip le-v alu ed functions in term s of th e se o p eratio n s, an d give th e alg o rith m for th e c o n stru c tio n such a form . For th e case m = 2, th is canonical form reduces to a fixed p o la rity R eed-M uller canonical form , w hich is known to provide a bcisis for econom ical im ple m e n ta tio n s of some p ra c tic a l B oolean functions [52]. We also show in th e .Appendix how th e basic o p erato rs of th e alg eb ra can be im plem ented a t th e tra n s isto r level by CM OS cu rren t-m o d e technology.
In C h a p te r 4 we consider th e realization of functions as th e X O R o f tw o .AND- O R expressions, which is usually called A N D -O R -X O R expansion. We develop an a lg o rith m for m inim izing .AND-OR-XOR expansions. We also show th a t such an expansion has a sm aller u p p e r b o u nd on th e n um ber of p ro d u c ts th a n th a t of th e A ND -O R an d .AND-XOR expansions an d . therefore, for som e functions, re su lts in sim pler circuits.
C h a p te r 5 in tro d u ces a m ultiple-valued discrete difference, w hich we call fu ll sen
sitivity. an d show its a p p lic atio n to generating tests for m ultiple-valued logic circuits.
Full sen sitiv ity is also used to define a class of functions X. stu d ie d in C h a p te r 6.
T his clciss of functions was also indep en d en tly considered by B ern h ard von Stengel in [57j. He proved th a t all functions in th is class have g. u n iq u e re p re se n ta tio n , called a composition tree, w hich, if n o n -triv ial. suggests th e circuit realizatio n o f th e func tio n a t a cost close to m inim al. In C h a p te r 6. we present an efficient a lg o rith m for
g en eratin g such a re p resen tatio n .
In C h a p te r 7 we in v estig ate th e te sta b ility of circuits realizing m odulo m sum -of- p ro d u cts form s. T his canonical form has been extensively s tu d ie d by m a n y a u th o rs: however, its ap p licatio n s to logic synthesis have only been considered for th e case m = 2. T h e circuits, realizing m odulo 2 sum -of-products form s, are proved by R eddy [6] to be easily te stab le . We e x te n d R eddy's result for m > 2. G en eralizin g from
th e two to th e m -valued case, however, is shown to be a n o n -triv ia l p ro b le m , since for m > 2 several new p h en o m en a occur which allow us to red u ce th e u p p e r b o u n d
C H A P T E R 1. I N T R O D U C T I O N 6 on th e n u m b er of te sts re q u ire d for fault d etectio n , b u t m ak e th e generation of te sts h ard er.
C h a p te r 8 su m m arizes th e d isse rta tio n and suggests fu rth e r work th a t could be
C H A P T E R 2. B A C K G R O U N D
C h a p ter 2
B a ck g ro u n d
T his c h a p te r p resen ts th e necessary m a th e m a tic a l background for the d issertatio n . M ost of its m a te ria l is classical and is beised on [2|. [6]. [14] an d [44].
For convenience, th is ch ap ter includes all th e general background. Background m aterial th a t is specific to a single c h a p te r is included w ith th e chapter.
2.1
N otation
T h ro u g h o u t th e d isse rta tio n we use “ • ” for th e m inim um o p e ra tio n (also called MIN or, for th e tw o-valued case. .\N D ): " “ for th e m ax im u m o p eratio n (M.A.X or. for th e tw o-valued case. O R ): ’’ ‘r " for th e ad d itio n m odulo m o p eratio n (X O R for th e tw o-valued case): " I " for th e m u ltip licatio n m odulo m o p eratio n : and " ' " for th e com plem ent o p e ra tio n (N O T ). ’’ an d *’ £ " are o m itte d betw een adjacent variables, when th is does not lead to any am biguity.
We let A/ := { 0 .1 m — 1} be a finite set of values. We use early lower-case letters a ,6, c. o i, Ug, e tc to denote elem en ts over A/, an d low er-case letters f . g . h . g i . g2,
etc to d en o te fu n ctio n s. We use Xi.X2i to d en ote variables of th e functions.
an d use :V = { 1 ,2 , . . . , n } to denote th e set of indices o f th ese variables. We use cap ital le tte rs A , B . C. e tc for vectors o r sets, an d usually d e n o te th e elem ents of th e set by indexed lower-Ccise letters. For e x am p le, th e elem ents of a set .4 are d enoted as 0 1,0 2, __ W e use bold cap ital le tte rs A .
B,
C . e tc to d en o te m atrices.C H A P T E R 2. B A C K G R O U N D 8
2.2
B asic notions
This section describes briefly th e fundam ental notions o f relatio n , fu n c tio n an d oper ation.
2.2.1
R e la tio n s
Let .4 and B be sets. binary relation R between .4 and B is a subset o f th e C artesian
product .4 X B . We use th e n o ta tio n aRb to denote th a t ( a .6) € R.
B inary relatio ns rep resen t relationships betw een th e elem en ts of tw o sets. .A. m ore general ty p e o f relatio n is th e n -ary relation, which expresses relatio n sh ip s am ong elem ents o f m o re th a n two sets. However, this d issertatio n uses only b in a ry relations, and therefo re we do not in tro d u ce n-ary relations. In th e following, we use th e te rm ” relation" to m ean " b in a ry re la tio n '’.
R elations from a set .4 to itself are of special in terest. .A relation on th e set .4 is a relation from .4 to .4. i.e. a subset of .4 x .4.
Let R be a relatio n on .4 an d let P be a p ro p erty of binary re la tio n s (such as reflexivity, sy m m etry , or tra n s itiv ity ). The closure of R w ith respect to P is th e least relation c o n tain in g R th a t has P.
A relatio n on a set .4 is called an equivalence relation if it is reflexive, sy m m etric, and tra n sitiv e . Let R be an equivalence relation on .4. T h e set o f all elem en ts b of A such th a t bRa for an elem ent a € .4 is called th e equivalence class of a. T he equivalence classes of R form a partition of .4.
2 .2 .2
F u n ctio n s
A fu n c tio n f : A B from .4 to P is a relation, which h as the p ro p e rty th a t every
elem ent a € .4 is th e first elem ent of exactly one ordered p air (a. 6) o f th e relatio n .
So, a fu n ctio n f : A B assigns to each elem ent a E .4 a unique e lem en t 6 = / ( a )
in B , called th e image of a. .4 is called the domain of / a n d B is called th e codomain of / . T h e range o f / is th e set of all images of elem ents o f A .
C H A P T E R 2. B A C K G R O U N D 9 A Function f : A B can be specified by using a rule a / ( « ) • assigning to each elem en t a G .4. its im ag e f { a ) in B .
A function / : A B is called injective when different e lem en ts o f .4 always have
different im ages or. in o th e r w ord, if an d only if a 7^ 6 im plies th a t f { a ) 7^ f {b). A function f : A B is called surjective when th e range is th e w hole codom ain B or. in o th e r w ords, if an d o n ly if for every elem ent b E B th e re is an elem en t a in A
w ith f { a ) = 6.
.4 function is called bijective w hen it is b oth injective and su rjectiv e.
In th is d isse rta tio n we d e al only w ith discrete functions o f th e ty p e / : V/" —)• M on a fixed set .V/ := { 0 .1 m — 1}. where .V/” denotes th e th e C a rte sia n p roduct
M X A/ X . . . X \ I of n sets M . VVe say th a t / ( j * i... x„) is an n-variable un
valued function. Such fu n ctio n s are called homogeneous, as o pposed to heterogeneous functions, w here th e variables z , o f th e function / ( x i x„) do not tak e values in th e sam e set. T h e re are homogeneous n-variable m -valued functions. For th e special ca.se of m = 2. m -valued functions are called switching o r Boolean.
2 .2 .3
B in a ry o p e r a tio n s
.4 binary operation # on .4 is a function of ty p e .4 x .4 ^ .4. So. a b in a ry o p eratio n assigns to each ordered p a ir of elem ents (a, 6) from .4 x .4 a u n iq u ely defined th ird
elem ent c = a • 6 in th e sam e set .4.
2.3
Chain-based P ost algebra
In th is section we describe a chain-based Post algebra, com m only u sed for rep resen tin g m u ltip le-v alu ed functions. T h is alg eb ra is a generalization of S h a n n o n 's switching
algebra to th e m u ltip le-v alu ed case.
D efin itio n 2.1
A chain-based Post algebra is an algebra A = :0 .m —1).
whereC H A P T E R 2. B A C K G R O U N D 10
{ii) J := {Jo, J i , . . . . J m - i } is a set o f literal operators such that
liX :=
I
' ^ 0 otherwise
where x is a multiple-valued variable and i € .\I is a constant. For convenience, we write J ix as x ;
(Hi) " + ~ a n d " • " are the binary operations m a x im u m ( \ I . \ X ) and m in im um (i\{f.N) , respectively:
(iv) 0 and m — 1 are constants o f the algebra.
An alg eb ra is fu n ctiona lly complete if it is based on a fu n ctio n ally c o m p le te set of o p eratio n s. If c o n sta n ts need to be a d d ed to a set o f o p eratio n s to o b ta in a com plete set, th en such an alg eb ra is called fu n c tio n a lly complete with constants. T h e chain- based Post a lg e b ra is known to be fu n ctio n ally co m p lete w ith c o n sta n ts [44].
T h e com plem ent of a m u lti pie-valued variable x is defined in ch ain -b ased Post algebra as x ' := (m — I) — x . where “ — " is th e usual a rith m e tic s u b tra c tio n .
Functional co m p leten ess of A im plies th a t every m u lti pie-valued fu n c tio n can be expressed in te rm s o f its operations. T h e n ex t th e o re m shows a canonical f o r m of any m ultiple-valued function in A . This form is said to be canonical b e ca u se it gives a unique re p re se n ta tio n for m ultiple-valued functions. T h ro u g h o u t th e d is se rta tio n , we refer to this form as th e M I N - A IA X canonical fo r m . T h e sign used in th e theorem stan d s for M.AX.
T h eorem 2.2
.Any m-variable functio n o f n variables has a unique expansion in A o f typem" —1
f ( X i . . . X„ ) — / , C,' X1 X2 . . . Xn.
1 = 0
where c, € M are constants, and (<1 / 2 • - • in) is the m - a r y expansion o f i with ii being
the least significant digit.
In th e tw o-valued case, th e MIN-M.AX can o n ical form reduces to th e .AND-OR canonical form . N otice, th a t in the tw o-valued case, x = x ' an d x = x . .An .AND-OR
C H A P T E R 2. B A C K G R O U N D 11 canonical form is often referred to as a “ sum -of-product" form , b u t we prefer not to use this nam e to avoid confusion w ith th e m odulo m sum -of-products form , cited in several chapters of th e d issertatio n .
. \ function can be p u t into a M IN-M AX canonical form by a successive a p p lic atio n o f generalized S hannon decom position to su b fu n ction s of f { x i j „ ) . Generalized
Shannon decomposition is an expansion of ty p e:
fU) = Yi
(
2
.
1
)
J & M
w here x := ( x i x „) an d ^ is th e v ector x w ith x, = j . i.e. := (J^i x , _ i . j . x , + i , Xn).
w ith j € .Vf. i E .V. So. in te rm s of these n o ta tio n s, /( x ^ ) denotes th e su b fu n ctio n of th e function / ( x ) w ith th e variable x, being fixed to th e value j E M .
C H A P T E R 3. A C A N O N I C A L F O R M O F M V L F U N C T I O N S 12
C h a p ter 3
A C an on ical Form o f
M u ltip le-V a lu ed F u n ctio n s
W hile com plete sets of functions are widely stu d ied for B oolean functions, less is know n ab o u t th e fun ctio n ally com plete sets for m ultiple-valued functions. In this c h a p te r we show th e functional com pleteness of th e set co n sistin g o f th e o p eratio n s o f a d d itio n modulo m . m inim um , and the set o f all literal o p e ra to rs, w here m is a positive integer. We prove th e existence of a canonical form over th is se t. a n d give an a lg o rith m for co n stru ctin g th is form. For th e case m = 2. th is can o n ical form reduces
to a fixed polarity R eed-M uller canonical form, which is know n to provide a su ita b le basis for th e im p lem en tatio n of som e practical Boolean functions [52].
T h e ch ap ter is organized as follows. In Section 3.1. we define th e R eed-M uller canonical form and give a su m m ary of previous work on its g e n eraliz a tio n to th e m ultiple-valued case. In Section 3.2. an algebra béised on th e o p e ra tio n s of a d d itio n m odulo m , m inim um , and th e set of all literal o p erato rs is in tro d u ce d . Section 3.3 presen ts a proof of th e functional com pleteness (w ith co n stan ts) o f th e set consisting of a d d itio n m odulo m and m in im u m operations. Section 3.4 describ es th e p rop erties o f th e operations of th e alg eb ra needed in the proofs o f th e m ain re su lts o f th e ch ap te r. In Section 3.5, a decom position, allowing a function of n variables to be expressed th ro u g h n functions of n — 1 variables, is developed. U sing th is deco m po sitio n , in
C H A P T E R 3. A C A N O N I C A L F O R M O F M V L F U N C T I O N S 13 a tio n s of th e alg eb ra is derived. An a lg o rith m for co n stru ctin g th e c an o n ic a l form is p resented in Section 3.7. Section 3.8 co n tain s conclusion an d su g g estio n s for fu rth e r research. In .Appendix A. we show a C M O S transistor-level re a liz a tio n o f th e g ates, im p lem en tin g th e basic operatio n s of th e a lg e b ra and sim ulation o f th e s e g a tes using th e H SP IC E pro gram .
Som e of th e resu lts in th is ch ap te r are c o n tain ed in [22].
3.1
R eed-M uller canonical form
cind
its general
izations
In 1954 Reed [49] a n d M uller [43] proved th a t an y n-variable B oolean fu n c tio n has a canonical form in te rm s of .AND and X O R o p era tio n s of type:
2 " - l
/(•T i JT„) = ^ c, j 'i ‘ x ÿ . . . x],". (3.1)
1 = 0
w here th e sign sta n d s for XO R. c, € {0. 1} a re constants, (i i i i ■ u ) is th e b in ary
expansion of i w ith ii being th e least significant digit, an d x° = 1 a n d x j = Xj
for j € N . T h e form (3.1) is usually called Reed-Muller canonical fo r m , a fte r its inventors. .\11 p ro d u c t-te rm s in (3.1) consist of uncom plem ented v ariab les only.
If th e restric tio n th a t all th e variables a p p e a r u ncom plem ented is rem oved, a n d variables are allowed to a p p e a r co m p lem en ted as well, then th e R eed-X Iuller canonical form exten d s to fixed polarity Reed-Muller canonical form , w hich is u n iq u e for a fixed p o la rity ^ € (0 , 1 ...2" — 1} and is given by:
/ ( x i . . . x j = ' ^ \ , ^‘x [‘ ""xir (3.2)
1 = 0
w here c, E {0,1} are c o n sta n ts, (ziz? . . . /„) a n d (&i&2 • • • ^n) are th e b in a ry ex p an sion s
of i a n d k, respectively, w ith ii and A:i b ein g th e least significant d ig its. T h e te rm € N is defined as follows: °x} = Xj, ^xj = x'- and, for any k j. *Tx° = 1. W hen
C H A P T E R 3. A C A N O N I C A L F O R M O F M V L F U N C T I O N S 14 T h e concept o f a R eed-M uller canonical form can b e ex ten d e d to m -valued logic in several ways, d ep en d in g on how the AND a n d X O R o p eratio n s a re generalized. T h e first g en eralizatio n , based on th e operations o f a d d itio n a n d m u ltip licatio n m odulo m , w here m is a p rim e n u m b er, was proposed by C ohn in 1960 [9]. He proved th a t any function o f n variables has a unique m o d u lo m sum -o f-p rod u cts form of th e ty p e:
m " — 1
/ ( j i , ---j:„) = ^ c, x'i‘ x ^ . . . (3.3)
1 = 0
w here th e sign 22 s ta n d s for m ultiplication m o d u lo m . c, € :VI a re co n stan ts. («1 / 2 - • - in)
is th e m -ary ex p an sio n of i w ith being th e least significant d ig it, and th e te rm x j ’ denotes th e ijth pow er of th e variable Xj. j 6 N . M odulo m ad d itio n and m u ltip li
catio n form a G alois field of o rder m.
L ater th is g en eralizatio n was fu rther e x te n d e d by P ra d h a n [47] for the case when m is a power o f a p rim e, i.e. m = p* (p - a p rim e n u m b er. A - a positive integer).
K od an dap an i a n d S etlu r [34] proposed a g en eralizatio n of (3.2). based on th e o p eratio n s of a d d itio n a n d m ultiplication m o d u lo m (m - a p rim e num ber) a n d th e set o f all literal o p e ra to rs, which is unique for a fixed p o la rity k € {0 . 1 m " — 1}.
T h e form is of ty p e:
m " —1
/ ( x i . . . x „ ) = Y . A " x ^ . . . '" x t" (3.4)
1 = 0
w here c,- € M are c o n sta n ts, (iiig - . . in) an d { k i k2 . . . kn) are th e m -ary expansions of
i a n d k, respectively, w ith ii and ki being th e least significant dig its, and th e te rm 'JXj-' equals m — 1 w henever ij = 0, and eq u als otherw ise.
H arking and M oraga [27] introduced axi ex ten sio n of C o h n 's form (3.3). w here an a d d itiv e tra n sfo rm x j kj is perform ed on each variable Xj. according to a fixed
p o la rity k £ { 0 , 1 . . . — I}. T he form is of ty pe:
m " —1
f ( x i + k i , X2 + k2, --- x „ -|-A :„ )= Y c,-x'l* Xj^ . . . xJ," (3.5)
C H A P T E R 3. A C A N O N I C A L F O R M O F M V L F U N C T I O N S 15 w here c, € M are c o n sta n ts. ( i i i2 - - - i n) an d (AriAra. . . A;„) a re th e m -ary expansions
o f I an d k. respectively, w ith an d ki being th e least significant digits. W hen k is fixed, this form is u n iq u e for a given function.
- \ 1 1 of th e ab o v e described generalizations are only applicable for th e algebras
w ith m being a p rim e or a pow er of a p rim e num ber. In this c h a p te r we in tro d u ce a generalization o f th e fixed p o larity R eed-M uller canonical form , based on th e op e ra tio n s of a d d itio n m odulo m . m in im u m an d th e set of all literal o p e ra to rs, w ith m being any positive integer. An n-variable m -valued function has m " such form s, each ch aracterized by a fixed p o la rity k G {0 . 1 m" — 1} and a corresp on d in g v ector
o f coefficients [ c q c i . . . C m n - i ] . Cj G M . T he form is unique for a fixed k. W e present
a p rocedure for c o m p u tin g th e coefficients of such forms, based on m a trix m u ltip lica tio n . T he vectors o f coefficients for different polarities are o b tain ed sim u ltan eo u sly , w hich makes it possible to choose th e canonical form w ith th e m inim al n u m b e r of non-zero coefficients.
3.2
The algebra
T h e work in th is c h a p te r is based on a m ultiple-valued algebra B defined as follows: D e f in itio n 3 .1 .4 multiple-valued algebra B is an algebra B = { M ; ~ . - . J : Q . m — 1). where
[i] M := (0, 1 ,m — 1} is the totally ordered carrier o f B: (ii) “-5;" is the binary operation addition modulo m :
(Hi) is the binary operation m in im u m ( MI N) : (iv) J := (J o , J i . . . . , J m - i } is a set o f literal operators: (v) 0 and (m — 1) are constants o f the algebra.
T h e o p eratio n s “0 " a n d are co m m u tativ e an d associative. T h ey do not dis
C H A P T E R 3. A C A N O N I C A L F O R M O F M V L F U N C T I O N S 16 th e c o n sta n t (m — 1) in th e u n it elem ent of T h e co n stan t 0 is th e u n it elem ent
o f “0 ". Recall th a t, for convenience, we w rite as x .
E very elem ent a o f .M has an inverse —a (w ith respect to th e o p e ra tio n ), defined as
~ a := a 0 a 0 . . . 0 n .
^ V m -i tim e s
In o rd er to simplify th e derivations below, we define th e o p eratio n s o f com plem ent a n d s u b tra c tio n m odulo m . All o p eratio n s a re e x te n d e d to functions aa usual.
D e f i n i t i o n 3 .2 The com plem ent o f a multiple-valued variable x is defined by x ' := (m — 1) 0 ( —x)
O bviously x 0 x ' = m — I since for any x ra n g in g in M . x 0 ( —x ) = 0.
D e f i n i t i o n 3 .3 Subtraction modulo m " 0 " is defined by
X 0 y : = X 0 ( - y )
where x and y denote multiple-valued variables.
U sing su b tra ctio n , th e com plem ent of an x can be rep resented as x ' = (m — 1) 0 x.
T h e chain-based Post alg eb ra (D efinition 2,1). b ased on th e o p e ra tio n s M IN . .M.A.X an d th e set of all literal o p e ra to rs, is w ell-know n to be functionally c o m p le te w ith c o n sta n ts [44]. Since M AX can be expressed th ro u g h M IN and co m p le m en t using de M organ's law x -t- y = (x ' • y ') ', and since c o m p le m en t is defined th ro u g h ad d itio n m o d ulo m an d the co n stan t (m — 1) (D efinition 3.2), we can conclude th a t th e algebra
B is also functionally com p lete w ith co n stan ts.
W hile th e functional com pleteness of th e set o f o p eratio n s {0 , -. J } is q u ite obvi
ous, a m ore interesting fact is th a t B rem ains fu n c tio n a lly com plete a t th e suppression o f lite ra l op erato rs J from th e basic set, i.e. B is co m p lete (w ith c o n sta n ts) over th e set {0 , •}. T h is is proved in th e next section.
C H A P T E R 3. A C A N O N I C A L F O R M O F M V L F U N C T I O N S 17
3.3
Functional com pleteness of
{ 0 .
•}
T h e following p ro p e rty shows th a t th e lite ra l o p erato rs can be expressed in te rm s of th e o p e ra tio n s “0 " a n d and th e c o n sta n t (m — I), w hich proves th e functional com pleteness (w ith co n stan ts) of {-S.-}.
P r o p e r t y 3 .4 The literal x can be erpressed in terms o f and ' as follows:
x = { g ' [ x . i ) ^ g { i . i ) ■ g'{x.i))'.
where g ( x . i ) := {x ~ i') ■ { x ' — i).
P r o o f : I) Let x = i. T h en x = m — I. O n th e o th e r hand:
g i i . i ) = {i ri-F) ■ { i ' ~ i)) (d e fin itio n of p (x ./) } = (m — I ) - ( m — I) (Va € .V/ : a - 3 a ' = m — 1}
= ( m — I) (id e m p o te n c y of •} T herefore
{g'{i.i)-i: g { i . i ) g ' i i . i ) ) ' = ((m - I ) ' 0 (m - I ) • (m - I { g (L /) = m - I } = ( 0 -f-(m — I) • 0 )' (D efinition 3.2} = (0 - 9 0)' ( 0 is th e null elem ent of •}
= O' (0 is th e u n it elem ent of -9} = m — 1 (D efinition 3.2}
H ence, for X = 9 x = {g'(x. i ) - r g{x. i) ■ g'{x. i))'.
2) Let X ^ i. T h e n x = 0. O n th e o th e r h a n d , we show below th a t (a) for each x an d (, X ^ i im p lies g { x . i ) < [ y j , a n d fu rth e r, (b) for any a < [ y j . it is tru e th a t
(a ' t9 aa'Y = 0.
a) We prove p a r t (a) by showing th a t
C H A P T E R 3. A C A N O N I C A L F O R M O F M V L F U N C T I O N S 18 I) x 0 / ' > LyJ {hypothesis} 2) LyJ < X 0 /•' < m - 1 {(x ^ i) (x 0 i' ^ m — I)} 3) [x 0 0 X = m - 1] {(:V/. 0 ) is a group} 4) 1 < -- < L f J {(2).(3 )} Ô) (x 0 x ') 0 (i 0 i') = (m - I) 0 (m - I) {Va € A/ : a 0 a ' = m — I } 6) (x 0 i') 0 (x ' 0 i) 0 X = (m - I) 0 (m - I) 0 X {reordering} 7) x ' 0 i = ( m — 1) 0 X {(3).(6 )} 8 ) x ' 0 < = X 0 I {Definition 3.3. — I = m — 1} 9) x '0 : < LyJ { (4 ).(8 )}
H ence, for j # j -9 i' > [ y j im plies i ' i < [ y j and so (.r -9 9 i) < [ y j -C onsequently. g { x . i ) < [ y j .
b) For an y a < [ y J it is tru e th a t:
( a ' 9 - a a ' ) ' = { a ' a ) ' { ( a < [ y J ) ^ ( a a ' = a ) }
= ( m — 1 )' (Va € M : a a' = m — 1}
= 0 {D efinition 3.2} H ence, for J - ^ i. x = {g'{x. i) — g ( x , i ) • g'iJ^A))'.
a
F u n ctio n al com pleteness of an algebra m eans th a t every m ultiple-valued function can be ex p ressed in term s of its o p eratio n s. In Section 3.6. we derive a canonical form , which gives a u n iq u e represen tatio n o f any m ultiple-valued function in th e algebra B.
It hough o u r canonical form can be expressed in term s of {0 . •} only, we use literal
o p erato rs as well because this sim plifies th e form . It is easy to see th a t expanding th e lite ra l o p e ra to rs by applying P ro p e rty 3.4 can n o t result in fu rth e r sim plification o f th e form , since “0 " is not d is trib u tiv e over
T h e p ro o f o f existence of th e canonical form is based on a n u m b e r of properties e stab lish in g relatio n sh ip s betw een th e o p eration s of B. These p ro p erties are presented an d proved in th e n ex t section.
C H A P T E R 3. A C A N O N I C A L F O R M O F M V L F U N C T I O N S 19
3.4 Properties of the operations o f th e algebra B
Let / . g d en o te m ultiple-valued functions a n d i . j . i ^ j . den ote co n stan ts over M . T h e sign used in th e properties and elsew here th ro u g h o u t th e c h ap te r denotes a d d itio n m odulo m .
P ro p erty 3.5
The following properties hold:(-^) — ^ b) / - x = / - ± - [ ( - / ) - ( x ) q c) / - f + g - x = / ■ X ~ g - X d) f ■ {jr ^ x ) = f - X - ^ f - X e) -r = -r •/■S' f g f ) r (/Sâr) = X / e i
-P ro o f (a):
I) Let x = i. T hen clearly^
x = 0. O n th e o ther hand (x ) ' = (m — 1)^ = 0.f.
2) Let X 7^ i. T h e n th e re exists exactly one k in M such th a t x = k an d so x = m — I.
C onsequently ^ x = m — I. On the o th e r hand (x )' = 0' = m — I.
j6 A / - { , }
Hence for b o th caaes ( x f = ^ x.
j € A / - { . } ( b ) : I) Let X = i. T h en / • x = / . On th e o th e r hand: / 'x '[(—/ ) • (•z’)'] = / -S-[(—/ ) • (m — I)'] = / e [ ( - / ) - 0 ) ] = / A’ O = f
C H A P T E R 3. A C A N O N I C A L F O R M O F M V L F U N C T I O N S 20
= / e [ ( - / ) • O']
= f ^ [ { - / ) ■ i m - l ) ] = f e f
= 0
H ence for b o th cases / • x = / - 9 [(—/ ) • (-r)']
( c ) : Since i ^ j . x cannot be equal to b o th i and j at once, it is alw ays th e case th a t e ith e r x ^ i or x ^ j or b o th . Let x ^ i. T hen th e left h a n d side is / - I + g ■ x = f - 0 + g ■ 0 + y • x = g- x . an d th e right hand side is f - x - i r g - x =
y . 0 9 gr • i*= 0 9 gr - x = g ■ x.
H ence for x ^ f - x + g- x = f - x — g- x. For th e o th e r cases th e p ro o f is sim ilar.
( d ) : Since i / j . x cannot be equal to b o th i an d j a t once, it is alw ays th e case th a t e ith e r x ^ i or x ^ j ot b o th . Let x ^ i. T hen th e left h an d side is
f ( x 9
i )
=/-(O
9 i ) = / • X. and th e right h an d side is / • x 9 / • x = /■ Ü 9 / • / • x.H ence for x ^ L f ■ {x x ) =■ f - x ~ f - x. For th e o th e r cases, th e p ro o f is sim ilar.
( e ) : 1) Let X = /. T h en th e left hand side is x - { f ir g) = {m — I) ■ ( f ^ g) = f g,
a n d th e right hand side is x - / 9 x - ^ = ( m — l ) - /9 ( m — l ) - ^ = / 9 5r.
2) Let X ^ i. T hen th e left hand side is x • ( / 9 fir) = 0 ■ ( / "5 5^) = 0. a n d th e right
h a n d side is x • / 9 x - ^ = Q- / 9 0 - f ir = 0.
H ence for b o th cases x -{ f g) = x x -g.
( f ) : 1) Let X = i. T h e n th e left hand side is x - { f S g) = {m — I) ■ ( f — g) = f ^
g-a n d th e rig h t hg-and side is x - / 9 x - g = ( m — l ) / 9 ( m — l ) g r = / 9
fir-2) Let X 7^ i. T hen th e left hand side is -r -(^ 9 fir) = 0 • ( / 9 S') = 0. a n d th e right
C H A P T E R 3. A C A N O N I C A L F O R M O F M V L F U N C T I O N S 21 H ence for b o th cases i • ( / 0 g) = x - / 0 x -g.
□
3.5
D ecom position theorem
In this section we present a decom position allow ing a function of n variables to be expressed th ro u g h m functions of n — I v'ariables. T h is decom position can be con sidered cis a g e n eraliz a tio n of th e positive a n d n eg ativ e decom positions of Boolean functions to th e m ultiple-valued case. R ecall from C h a p te r 2 that f { x ^ ) denotes a subfunction o f th e function / ( x ) w ith th e \^ariable x, being fixed to th e value j . i.e.
fiîi)
= / ( x i X , x ,+ i x „). T h e n , th e p o sitiv e and negative decom posi tions of B oolean functions are of form [36]:f [ i ) = / ( £ ° ) -T x „ ( / ( x ° ) 0 /( x ^ ) ) p o sitiv e decom position
(3.6)
= ^ / ( ^ a ) ) n eg ativ e decom position
T heorem 3.6 is th e general decom position th e o re m for a function / ( x ) a b o u t som e variable x,. H owever, for n o tâ tional convenience, th e th e o re m is stated a n d proved for decom positions a b o u t th e least significant v a ria b le x„.
T h eorem 3.6 (D eco m p o sitio n T h eo rem )
Every m-valued function f { x ) can be decomposed with respect to the variable x„ a nd a given i G M in the following way:m—1
j=i
Proof:
Using generalized S hannon decom position (2.1) we can express the fu n ctio n / ( x ) as follows:
C H A P T E R 3. A C A N O N I C A L F O R M O F M V L F U N C T I O N S f i x ) = / ( x “ ) + / ( x ‘ ) + . . . + x^V U lî^ {(2.1)} - Z,. /(X ^ ) e i n / ( ^ ) © . . . © ) { P ro p e rty 3.5(c)} = i f U l ) © i - f i ^ ) ■ © E f U i ) J=l { P ro p e rty 3.5(b)} m —1 m—1 = ( f U l ) e ( - / ( i S ) • E ^»l) ® E /( £ < ) *=1 j=l { P ro p e rty 3.5(a)} m —1 m — I -= / ( £ S ) * j= l J=l { P ro p e rty 3.5(d)} m —I = H î ° ) i E / ( ^ i ) J= 1 { c o m m u ta tiv ity o f 0 } m —I = / ( £ : ) - » E < / ( ^ ) a ( - / ( £ : ) ) ) £ . J= 1 { P ro p e rty 3.5(e)} m— I = E i / i z ^ ) - e / ( £ : ) ) L j=i (D efin itio n 3.3} 99
In th e above deriv atio n we ex p an d ed using P ro p erty 3.0(b ). If a ltern ativ ely
we expanded -/(x jj) for som e i ^ 0. th en th e derivation gives th e proof for th e
corresponding value o f i.
□
For exam ple, a 3-valued 2-variable function / ( x i . x y ) can be decom posed w ith respect to th e \'ariab le x-2 an d a given i € {0. 1.2} in th e following way:
f U u i
2)
= / ( £ i ) * [ ( /( £ ? ■ ) e / ( x i ) l % 'i * [ ( / ( £ ? " ) e / ( 4 )) % | .T h e decom positions for all th re e possible values of i are:
For i = 0: / ( x i , X2) = fiî .1 ) 0 [ i f U \ ) ^ f { x ^ ) ) ^2] "5 [{/{ xj) 6 / ( x " ) ) ^2
]-For i = 1: / ( x i ,X2) = / ( x ‘ ) 0 [ ( / U2) © / U 2)) h ] © [(/(i® ) © f i î .2)) ^2
]-For i = 2: / ( x i , X2) = /( x ^ ) 0 [(/(x ^ ) © /( x ^ ) ) Xg] 0 [{ f{ x\ ) 0 /( x ^ ) ) X2].
C H A P T E R 3. A C A N O N I C A L F O R M O F M V L F U N C T I O N S 23 T h en we have X 2 0 I 2 0 0 I 2 X l I I 1 2 2 2 0 0 X l /( z ^ ) f U Ï ) / ( d ) 0 0 I 2 1 1 I 2 2 2 0 0
an d . for th e decom position w ith respect to xz w ith i = 0
= /( £ ° ) efiri(j*i) X2-?g2(-ri) ^2 w here 5^1 ( J i) and ^ i( x i) are functions, defined as follows:
X l m ( - T i ) 5 ' 2 (x i)
0 1 2
1 0 1
2 I 1
Obviously, if each o f th e subfunctions /(x ;^ ). j 6 -V/. in th e deco m p o sitio n of / ( x )
is successively decom posed ab o u t th e rem ain ing variables, we finally get an expression in which / ( x ) is ex p an d ed in all its variables. Since for each su b fu n c tio n the decom position can be m ade w ith respect to som e c o n stan t i Ç M . th e re are m" different ways to expand th e function / ( x ) in all n variables. In th e next sectio n we prove th a t, for a fixed i. each of these m " expansions is a canonical form u n iq u ely representing a m ultiple-valued function, an d show how to find th ese expansions d irectly , i.e. w ithout applying step-by-step decom position.
3.6
Canonical form of m ultiple-valued functions
In a two-valued system , any Boolean fu n ctio n of n variables has 2" fixed p olarity R eed-M uller canonical form s of ty p e (3.2). In such an expansion, each variable x,- is
C H A P T E R 3. A C A N O N I C A L F O R M O F M V L F U N C T I O N S 24 e ith e r in a c o m p le m en te d , o r in an u n co m p lem en ted form , a cc o rd in g to som e p o la rity vector At = (A:„ ...AraAri). If A:, = 1 th e v ariab le x, a p p ea rs in co m p lem en ted form .
otherw ise x. a p p e a rs in uncom plem ented form . For e x a m p le , th e p o larity v ecto r Ar = (O il) im plies th a t x i a n d x? a p p ea r co m p lem ented in th e R eed -M uller canonical form, an d X3 a p p e a rs uncom plem ented (x i is th e lowest o rd e r v ariab le). A p o larity can be given n o t only as a binary vector (A:„ . . . A^aAri ). b u t also as a decim al n u m b e r A: 6 { 0 .1 ...2” — 1}. w hose binary exp ansio n is th is b in ary v e cto r. For ex am ple, for
A: = (O il) th e p o la rity can be given as A: = 3.
We gen eralize th e n o tio n o f fixed p o la rity for m ultiple-x'alued logic, assu m in g th a t in a fixed p o la rity form each variable x ,. i € {1, ---- n}. is re p re se n te d by all literals
fe
except X,. w here (Ar„ . . . ArgAzi ) is the m -ary ex pan sio n of a p o la rity A: 6 ( 0 . 1 m " — 1}. given as a d e cim al num ber. For e x am p le, if m = 3. p o la rity v ecto r k = (0 2 1)
im plies th a t x i is re p re se n te d by literals x^ an d Xi in th e c an o n ical form , x-2 by literals
x-2 and X2. an d X3 by literals X3 and X3. S im ilar g e n eraliz a tio n o f p o la rity was used
in [34].
T he following th e o re m shows th a t th e re exist m " can o n ical form s o f a m -valued n-variable fu n c tio n , each characterized by a p o larity k € (0 . 1 m " — 1} an d a
corresponding v e c to r of coefficients [cq Ci . . . Cm»-i]. Cj € M . T h e n o ta tio n ‘x-' used in th e th eo rem below is defined as follows:
m — 1 if i = 0
X o therw ise
w here x is a m u ltip le-v alu e d variable a n d i . j € M are c o n sta n ts . W e d en o te by
j € { 0 ,rrC — 1}. th e coefficients from th e tr u th ta b le for / ( x i x „). w ith Xi
C H A P T E R 3. A C A N O N I C A L F O R M O F M V L F U N C T I O N S 25 T h e o r e m 3 .7 A n y m-valued n-variable function can be expressed in a canonical f o r m
with a fixed polarity k G {0 . 1 m " — 1} as: m " —1
f ( X i , . . . . X n ) = ‘"•^1
j=0
. Jl J2 - r h Jn ~ k n J.2 . .
-where cj € M are constants, and {jn ■ ■ ■ j2j i ) ^re the m-ary expan
sions o f f and k. respectively, with j i and ki being the least significant digits.
P r o o f : B y in d u ctio n on n.
1) Let n = 1. A ccording to T h e o re m 3.6. any function o f o n e variable x can be decom posed w ith respect to th is variable and a given i € M as:
m — 1 . _ . / ( x ) = f i ~ Y . (/"TJ ^ / . ) {T h eo rem 3.6} j=i m — 1 = Co 9 C; T (w here cq = / . a n d Cj = / .^ j -r /.} J = l m — I = Co ° x ‘ -5 ^ Cj ^i ‘ (° x ' = m — 1. and -'x' = x . for j ^ 0} j = i m — I = Y . J = 0
which is th e canonical form for n = I an d p o larity i G ( 0 ,1 m — I}.
2) H ypothesis: .Assume the resu lt for all functions o f n variables. .According to T h eo rem 3.6. any function of n + 1 variables can be d ecom posed w ith respect th e variable x„+ i and a given i G M as:
TTl— 1 . .
/ ( T , ... X „ „ ) = / ( £ : + , ) ^
p = i
By th e in d u c tio n hypothesis, w hich assum es th e result for th e fu n ctio n s of n variables, we can express each of th e subfunctions / ( x |, ^ i ) , f{x^-^+i), p € -V/ in the canonical form for som e p o larity k = . . . ^2^1). We use th e n o ta tio n cfj to denote th e j t h
C H A P T E R 3. A C A N O N I C A L F O R M O F M V L F U N C T I O N S 26 m' * —I ...- T n + l ) = 5 1 C* J ' X i * . . . j = 0 m —I / m " —I m " —1 \ e Z Z <=?'' • • ■ '" -5 " e E c; ■. ■ " 4 " p = l \ J = 0 j = 0 /
To sim plify th e ex p o sitio n , we use th e n o tatio n to s ta n d for th e term ■'‘Xi* . . x*” . T h e n th e ab o v e expression becomes: m " —1 m —1 / m " —1 m " —1 \ / k i = E 4 a E Z f ' ^ - Z 4 ' . ^ 4 j = 0 p = l \ j = 0 J = 0 ) m ” —1 m —l / m " —I / m ” —I \ \ . . = E 4 t E Z f * - Z J = 0 p = l \ J = 0 \ j = 0 / / {D efinition 3.3} m " —1 m —l / m " —I m " —1
'
'- n+ 1 = Z ^ X " ^ ' e z a E j = 0 p = l \ J = 0 j = 0 ){ d istrib u tiv ity o f " — “ over**—^'}
m " —I m —l / m " —1 \ . . = z 4 ^ z z ^ ( - 4 " : + i j= 0 p = l \ j= 0 / (c o m m u ta tiv ity o f"—"} m'* —I m —l / m ” —I \ = Z 4 ^ Z Z ( f ^ 4 J —0 P=1 \ J=0 / (D efinition 3.3} m " —1 m —l / m " —1 \ = z 4 ^ z z ( f ' ^ 4 ) J = 0 p = l \ j = 0 / (P ro p e rty 3 .5 (f)} m " —1 m —l m " —1 = z 4 Z Z ( f ^ 4 ) J=0 p = l J=0 (P ro p e rty 3 .5 (f)} m " —1 m —l m " —1 = Z < ° < + . ® Z Z ( f 6 c ') j = 0 p = l j = 0 ( ° x ‘ = m — 1, a n d ^x' = x^.for p 0}
