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Binary Decision Diagrams for Random Boolean Functions

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„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„ Ÿ›

› „› ¤fˆ

q

œbzh

lFt}

£

kmlFqy¡npNq zƒ#a6v

kmlFn

h

qso v9‰

qso Í6‚f††

l

„Š„|„Š„Š„|„|„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„ ŸŸ

› „Ÿ ¤fˆ

q

œbzh

lFt}

£

kmlFqy¡npNq zƒ#

qso v9‰

qso Í6‚f††

l

„|„Š„Š„|„Š„|„Š„Š„|„|„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„ Ÿ

› „0 q w k h Ž z

ryt

ˆ q u

nplTt z|h



zƒ¤fˆ

qslFq

‚fz|v

r5o9l

„|„Š„Š„|„Š„|„Š„Š„|„|„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„ Ÿ42

Ÿ A#B(CEb(cXd(C)/"!CX&eOQ*,RCS%Tf-\]^9^9Ggh*,"!B7MiX*,cjCX&7kMN$*,MNlmWCh\V$H&(C$*, #n „Š„|„Š„Š„|„Š„|„Š„„Š„ 0*—

Ÿ „— gh

q h qsÙ

v

nplTnptql

„„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„ 0›

Ÿ „› ¤fˆ

qHù5Ïm§¯q

‰

tqso®¡npKq zƒ

t ˆ q û qs€q

{l zƒ

k kmr5o

z|w

Ù

Í6‚f†† „Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„ 0 3

Ÿ „Ÿ ¤fˆ

qHù5Ïm§¯q

‰

tqso®¡npKq zƒ(

kmr9o zw

Ù

Í6‚f††

l  npt

ˆ

k¨‘|npÏmqsoy÷mk

h

npk x{

q

Í6h

o9q

h

npr

™"„Š„ 1Ÿ

0 A#B(C7O"+$H%- (noOQB(MN # #% 7b(TpTpCX)"[Tp%$9f\V]^9^9Gqgr*W"!B7\d("!*Ws7MNm(kXMN$H*WMtl-m,CE\$&-CX$H*W #ntG „ 141

0 „—   

v9w

k ª l’Jr9qsÙ

v k {

npt

„|„Š„Š„|„Š„|„Š„Š„|„Š„„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„ 141

0 „› û k

h:™

q † qs€snpkmtn

z

r5l

ƒJh:z|w

t ˆ qùÏm§q

‰

tqsoyÙ Í6‚f††

lFnpKq

„Š„|„|„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„ 1 3

0 „Ÿ Í

§tn w k { Ù

Í6‚f††

l

ƒJz|h

kmr9o

zwþ‚fz|z{

qskmr%‘

v r ‰

tn

z

r9l

„Š„|„|„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„ 1

1 A#B(CEb(TpTpC)/"%-T"+B(Ch^9Cm,CX"!*W% <Y['(mWC „Š„Š„|„Š„„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„ 1¦

1 „— ¤fˆ

qHù5Ïm§¯q

‰

tqso

qso v9‰

tn z r x

Qt

ˆ q † q {

qstn z r

v9{

q

„|„Š„Š„|„|„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„ 1¦

1 „› û k

h:™

q † qs€snpkmtn

z

r5l

ƒJh:z|w

t ˆ qùÏm§q

‰

tqso

qso v9‰

tn z r x

't

ˆ q † q {

qstn z r

v5{

q

„Š„„Š„u3 ›

1 „Ÿ Í

§tn w k {

÷mk

h

npk x{

q

Í6h

o9q

h

npr

™

lkmr9oyt

ˆ q † q {

qstn z r

v9{

q

„]„|„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„u3 0

1 „0 ¤fˆ

q œ qskmŽ%¡

ˆ

kmr5r

z

rbù

ƒJƒ

q ‰ t

ƒJzh)Í6‚f††

l

„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„u3 1

3

K%-s7d-MN$*, #nv"!B#ChYC&('#)"+*,%- <Y'#m,CXG

„Š„|„Š„„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„u3

3*„— £Cz|w

§k

h

npr

™ tˆ

q'ù5Ïm§¯q

‰

tqso  *wyzv

r9t zƒ#

qo v9‰

tn z r

„Š„Š„|„|„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„u3

3*„›   r z t ˆ q h[“

npr5o z|ƒfû

k

h:™

q † qs€snpkmtn

z

r%’r5qÙ v k {

nptnpqsl

„Š„|„Š„Š„|„|„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„ Ÿ

\"+B(CX$9^wCX)*WGI*,%- <^9*,MNnN$HMNs7Gqgh*,"!B7MkXMN$H*WMtl-m,Ch\$&-CX$H*W (n „¯„Š„|„|„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„ 4

„— x q

h:z

}F¡

v

§¯§

h

qlTlFqso

‚

npr9k h  † q ‰

nplFn z r †

npk

™|h

k w l Êx ‚f††

lFË

„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„ 4

„› Í0h

o5q

h

qso

“h:z

r5q

‰

ŽŠq h ‘ v r ‰

tn

z

r9k {9†

q ‰ nplFn

z r †

npk

™Šh

k w l ÊÍ6“

‘

††

lFË

„|„Š„Š„|„Š„|„Š„„Š„ 4

(6)

2 yZ*W #*,s7MNm#it]^w^9G

„Š„|„|„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„|„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„

2*—

2 „— †

qÎ5r9nptn

z

r5l

„Š„|„|„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„|„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„ 2*—

2 „›

¡t h:z

r

™r

qso v9‰

tn z r

v9{

qsl

„|„Š„Š„|„Š„|„Š„„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„|„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„ 2Ÿ

¦ U'#MNGI*,$C&('()/C&7iN]^9^9GMN #&7"!B#Ch^wCXm,CX"!*,%- <Y'#mWC „Š„|„Š„|„Š„Š„|„Š„|„|„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„ 2

¦ „—  C{p™Šzh

npt

ˆ5w

’{zt|#}Q~4}

†

}Q€}#‚Wƒz

„|„Š„„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„|„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„ 2

¦ „›

 ÕghVz

§¯q

h

’{zt|}Q~4/}

†

}Q€„} (‚…ƒz

„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„|„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„

¦ „Ÿ gh:zx

k x n{ nplFtn

‰0 

r9k

{

slFnpl zƒ

’{zt|#}Q~4}

†

}Q€}#‚Wƒz

„„|„Š„Š„|„Š„|„|„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„ ¦*—

—/.†it]^w^9G‡MN (&P"+B(CEyPC$nN*, #nvY'#mWC „Š„Š„|„Š„|„Š„„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„|„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„ ¦41

—/. „—  C{p™Šzh

npt

ˆ5w

’{zt|#}Q~4}K‹ˆ}Q~4‰(‚…z‰

„|„Š„„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„|„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„ ¦41

—/. „› q {

kmtnpr

™

Ù|‘

‚f††

lt z Ù

Í6‚f††

l

„|„Š„„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„|„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„ ¦ 3

—/. „Ÿ ¤fˆ

qH‹%npr5n w k { ‘

‚f††

¡npKq zƒ#

kmr9o

z|wö‚fzz|{

qskmr%‘

v r ‰

tn

z

r9l

„|„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„ —/.4.

—/.

„0 ¡ zw

q q w k h

ŽŠl

z

r%‹%kmnpr

¤ˆ

q

zh

q w › Ê

npË

„Š„Š„|„Š„|„Š„Š„|„Š„|„|„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„

—/.›

——†Os7MNm,m#iN]^9^wGVMN (&PmWMt$nNCS\]^9^9G

„|„Š„|„Š„„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„|„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„

—/.41

—— „—  C{p™Šzh

npt

ˆ5w

¡Š‚,‹Œ/€}

¤#Ž

Œ}

„|„Š„Š„|„Š„|„Š„„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„|„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„ —/.41

—— „› ¤fˆ

qHù5Ïm§¯q

‰

tqso®¡npKq z|ƒ

qy¡qst‡

„„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„|„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„ —/.

—— „Ÿ û k hV™

q † qs€snpkmtn

z

r9l

ƒJhVzw

tˆ qHù5Ïm§¯q

‰

tqsoÕ¡npKq zƒ

q”¡qst

„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„ ———

—— „0 ¡ w k {p{

‘

‚f††

l€snpk4¡Š‚,‹Œ/€}

¤#Ž

Œ}

„„Š„„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„|„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„ ——›

—— „1 u6z‘ÆmÔ

tz ’w § h:z

€sq  *{p™Šzh

npt

ˆ9w

¡Š‚,‹qŒ€„}

¤Ž

Œ}

„„Š„Š„|„Š„|„|„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„ ——Ÿ

—›†’[d(d-CX #&-*Wc

„„Š„|„Š„|„Š„Š„|„|„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„|„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„

——/1

—› „—  z

tkmtn z r

„|„Š„Š„|„|„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„|„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„ ——/1

—› „›

’Jr9qsÙ v k {

nptnpqsl

„|„|„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„|„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„ ——/1

—› „Ÿ  

lF

w

§¯t

z

tn

‰ l

„|„|„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„|„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„ ——/1

—Ÿ†YCT!C$C #)CXG

„|„Š„|„Š„Š„|„|„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„|„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„

——4

û

nplFt

zƒfgv9x|{

n‰ kmtn

z

r9l

„|„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„Š„|„|„|„Š„Š„|„Š„|„Š„Š„|„Š„|„Š„Š„|„|„|„ —›41

(7)

npr5k  q nplFn r npk k lHk qykmrÕq Î npqsr9to9kmtkylFt t q qskmr r tn r5l ¡npr q

‚zz{

qkmr

ƒv

r ‰ tn

z

r5l Ê l

zw

qstn

w

qslk {l

k

{p{

qso l  npt

‰jˆ

npr

™SƒJv

r ‰ tn

z

r9lFËbk

h

q¢kmr k x lFt

h k ‰

tn z r

‰ n hV‰jv

nptl

šfºXÆmÆmÓp÷¾Š¼Õɾ|¼»õ-Óp¾|Ô»½Æm¼

tkmlFŽŠlHk

h

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