ASYNCHRONOUS ITERATIVE METHODS FOR SOLVING A ... - DI ENS

METHODS ITERATIVE ASYNCHRONOUS OF MONOTONE FORSOLVINGA FIXED POINTSYSTEM LATTICE IN A COMPLETI EQUATIONS Patr i ck Cousot R .R . B 8

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ou qtrrrvl;:l:;:t::":::l:;;fi:; uoT+ezTuo,rqcuds uaer"rtraq

p o suorlenbe 1o uralsds aq1 Sutnlos Joi uqlr'-ro81e u asaqJ 1a11e"red e o1 puodseJ.roc spoqlau o^rlPJalT '(suorlTpuoc se qcns) sasaql ureqc ,:o dltnulluoc JeqleJ aql TeuorlTppP ou asn s;:oo"rd Tecruqcol q tr suorlenba auulouolu Jo uels^s c u c + e :t ruru{vgv 1 9 3 P

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ANDNOTATIONS 1 . INTRODUCTION Let L(s,1,r,L1,[1) be a non-empty conplete Latlice w:-th partial otde=, Least upper. bounrl"t.l, gneatest Lodev, i>ound n. The inlimun t

rtnq

of L is [1L, the supremumr of L is Ut. l.r

into

from L"(E,r,r,ll,ll) TarskifT-lrs to

theorenr stale:r

the equation

Let

u be the

candinai-ity

X=F(X)) is

corrs'ructively

the fixed

The I terat'ion

the

vX,YeLn, {xcY} set

{F(x)sF(Y)}).

points

of fired

of

a non-empty complete lattice

than the

l,n. Cousotl3]

of

cardinality

F by means of

of

with

E.

the

defines

following

sequonce fi,t: I

o e Ln is

uitLt

starting

the

lr-termecl

A

secuence of rn

ordering

l-he 1s l

1I

' { [ ]= .r-gl Put TPuTpJo Jossarcns P sT I JT 11u olnpour [)+r i=of Pue TPurP.ro lTurTT P Jo Jossaccns sT I 'ro o} }uaTe^lnba s1 Poq}au uor+eJa}T s'Tapras-ssnP9 seeJaqA ' r t } g A 1 S u l s o o q c u T s + s T S L ' o cp c ) q } a u u o T l p J a l T s r r q o c e f T=9 JT {T}=gf "''T {t {tu '

} = o -6 '

'( sacuanbas alrurJ JoJ

;9l1.raeof, ur uanr8 sT uoTfTurJap

JeTrurs

e stsdleup

TPcrJaunu uI) 9>0

Id taJ^9

r n t r r n

[ IJU.LPLTU + !u].1 L

f

- ^ ^ ^

4vJ

1\dd/\o

A.rena puP rlrg [PuTp,ro Jossaccns nerand'luj f

Ol-,T

- ^ ^ ^

^ r r h

d a ^

r b r r r n l n

, { r r a r

r n c e - ^ ^ n -

r n

r

v T 1 rr Dn T

( , ^ X ) ' J = To "x r-Y J

(ol-{u""'t})tT 1'

d.reaa pue rl 39 TeuTp.ro "lossaccns r{;rane .ro3-

X = T' X

r- 9"

9" 0

: sr4oTToJ Se UorsJncaJ acuanbes pout.rel-rt oql sT ( {( a:]ruTJSuPr+ l{q pauTJap rl

' rt r g eur p.to l,rulT T f.rane T

.rossaccns

^v x t l = y^ x

.ro3

rt?g TeurPJo d.rana pue {ut ' ' 't T }-tI .{.rar'a ,-ro;.

T

( -_ n X) ' J T_ \

n

T

= :x Y -

v

-

0

aeuenLas uoTlP"ralT aq.I

^ ^a, ,u - L +J^a^ P u a q ] s T :. ^r ^q p I q l T ^ , r3 u 1 f . r e l s I r o i

'( gt ecuanbas aql Jo uraf q1-g eq+ g:o luauoduoc qt-T aqt =r lx ^ X t Y v t e c u a n b a s p a u . r a e l - r te s r a"ToJadaql'LuJal q]-9 s+T solouep

we have

tiorrs that F is continuou.s (for ' ^-L\ -'C') l|. and Lhat the )ength of t( ri L ) i

the periods of.J0r6ep>

i. I: i I u r , i f o r n r l y b o u n d e d{ g m e o : { Y 6 e u , V i e { 1 r . . . , r } , / '

( i . " 1- 0- r) jr j r. A

.,u

Also

in

the model of

X: (irJ-)

are evaluated

zed Darallel

in tenm of

to

rtion

this

account

bet'nieen cooperating

An important conlained

in

opt-imization

paper

alf

components r . , u - Iiterate X and this

o n DhJr r sD qt vr svr eo lr : l

nrocess

srrnr-hroni-

vt

plv\srs-

oyr19l

compiete

lattice.

is

previous

implementation

Most often

iterative

point

to

wj-thout major revision

global

these

of

restrictions

that

are based on sequenti,rl implementation

of

the the

evol-ution

classical

iterative

processes. of

the

in

a

can be solved program

algorithms

on a multiprocessor

between cooperating follow

techniques

equations all

T

I ,and

analysis

monotone equations

such systems of

t

l t

the results

program

I

iterative

synchronization

of

conpiling

system of

methods implies

a paraIIel

synchronization

application

the one of

Showing that

wr"itters

the

computen system without

a fixed

methods which

are amenable to

eliminate

domain of possible

solving

by asynchronous

to

processes.

techniques.

in

is

the parallel

for

paper

this

consist

compiler

the previous

r uiltPuLqLrutr

methods on a muftiprocessor

without

iterations

processes.

The purpose of and mainly

g o e u : ( 6 < o < 6 + m )a n d

I

m h( r: . uduor lns e uhyr r q- D^ l-r 'r -y ^, ! -u r u f urrrr s l u

analysis

chaotic

A

,u \

was

t t t

This

hardware

system enables

technology

known techniques.

the

t t"

t' t" I

t'

2 . ASYIICHRONOUS ITERATIONS 2 . 1 D E IFNI Ti O N

l ,

A

L e t < J - , 6 e O r d > b e an Ord-termed such that

sequence of

subsetsof {1r...rn]

:

( , r ) { V 6 e O r d , V j € {,1. . . r n ) :

f,o)6 : ieJ*]

A L e t < S - , 6 e O r d > b e an Ord-termed +L-+ A

( b ) { v i € { 1 , . . . , n .} V 6 e O r d . S : < 6 ] a

sequence of

elements

of

Ord^' such

I I I lI.

uoTlpnrela SulpuodsaJJoo

s T q . [ ' T x , { . r o u . r a ua q + u t aql

aJoJaJaqI'gfrT

struauoduroc TTp Jo uoTlenTp^a E g aulT+ 1P +eql aql

aql

ua}+rJn

enTp^ dlsnoauefue}sur "T ;x uialsds aql Jo T

qcTL$a.roJ suorlenba;:o aleuTui.ral sdossaaoJd

satretrs 1 '3 uo]lTuTJaP

uaql

g:o ,:aqlnu

'suoT lenba

urp1.rac

3:o ura1s.,{s

aql o1 pau8tsse sT .rossacoud e Jo +uauoduroc fue g:o uoTlpnfela 'aceld aJaq/'1 a{e1 X g:o luauoduoc a{ros 3:o 3u111.rr'r ,.ro Sutppal

aTpT ueq{ sluplsur

aurT+ Jo acuanbas Sutspa.rcuT up sp rl Y N x).'rr

T e u T p J o + T t l r T T. { ; a n a . r o g ;

pul] g Teurpdo Jossacilns

d"rar'a .f:T J AJaAe .roJ

t, l

I, I I, l

1 I: I I I I

gX

{ ( . . u) . . " " ' u.

(^f-1u" "eT))11 f"raaa v pup g TeurpJo Jossacons d.raaa .ro3;

S''

I t

'X)'J = :VX

T

T

L

L

JS V

T. X - T' X r- 9" 9

o: ox

: sr"roIToJ sp ucrs.rnca.r alruTJSueu:l ,{q peuTJap .p.rgrg'^Xt J

^ ^,,^^T- ^d Judrrue>

^ ^,

youlda]-pclO

aq+

pue Y

sT

(

nq pau'LJ';,r

d J

'TlnTA] =

5 rrtS

t*

'pror9'BAl (tr)

e sT !] t 9 > € P u e T P u r p ; 1 e+ T t i l T T T

{tis>9

' g < D A }: g < g E ' { u ' " ' ' T } - T A ' p ' r o r g A(J' )

I l l: - / I - l

I ]

,

s

in neading the value X1

of X: consisted ,

t

of the memory X1 at time

A

^6 r s ., n 'An anq writing

o6

, c b the va.Iue Xl=fr(X1 rA

which jdJ"

The components X. of X for Notice

l ,...

ane nrt

.6 b r*rt)

time 6 in Xt'

at time 6 .

modified

of every" value ) ,li ^ """U 6 in the^ computation 6 >l --or -6 , " ',Xr.,n) is takes place before the value x!=rrtxr'

I

wr-itten in X.. This justifies Acconding to definition

condition 2.1-.(b). between the coope-

2.1 no synchnonization

pnocesses is needed, and the scheduting policy

the scheduring poJ-icy must be fair

so that

(specifying opened. However

which processors evaluate which components) is left condition

imposes

2'1'(a)

no component can be abandoned foreven

that

W e a l s o a s s u m et h a t

Sg S6 the evaluation of every fi(Xlt r...r*r,t) bounded) peniod of tirne and that

(but not necessarily

takes a finite if

at

the reading

that

A of X] necessanily

rating

i

i n - n e a d i n g Xrnr a t t i m e S' i , i n a p p r y i n g t h e o p e r a t o r F t t o

si:..., -o 5 ,, r AI :...

'

il il

a pnocesson collapses

will

task perfonmed by the remaining

onder to have its ^

the scheduling policy

'

Thenefor"e

processons.

A

every 6, max(6-Sl) must be bound (but not necessarily

for

in

be modified

uniforrnly).

r - f

such that

states that for any 6 thene exists

2.1.(c)

Consequently condition

can terminate

time B no processor

aften

B

that

a computation

began at time 6. A

In pnactive is

after

stationary

that

for

we should also nequine that

example this

chain condition. termination

of the computations

limit

(that

It

is

we state that condition

t t t L

t t

the ascending

ane not unique it practical

from a

is better limitation

and

I

L-

ordinals.

is the evaluation

I

hypotheses guaranteeing the

of view to ignone this

our plausible

t t

immediat fnom what follows

would be the case when L satisfies

The considenation eliminate

time e.

Since the possible

mathematical point consider

a finite

the sequence

r r1

of an infinite hypothesis

computation that

the elementany computations

of a conponent) must last 2.L.(d)

sequence does not

a finite

time.

Hence

is necessany to take account of this

:

sJ"lOrToJ Sp uoTS(Inca(I

a+TUTJSUEJf

fq paul3:ap s+uauaTe Jo d'rouraurq l T A 1u1od paxt;e.rd p q+TA 8ur1.rels ,r'l t u ' u - ' acuanbas uoTle.ra+T snouo,:qcudse ue JapTsuoc al'l 3u1no11og: a q + u r v

t' l3 U0 lHI l3 NI9 U3 A N0 t3' t 'I'Z aTqpqrdcsap

lou

suotlelnduoc

aql

(9x)r= r

u o T + 1 u 1 g i a{ pq

J o uolllsodurocap TeJnleu p sT ricTqr4

= grt-l( ox)B r+exI r= o xI

,*sI i

r= ;^l : suor+PJe+r

TPda+ETToc ofi+ aq+

o+

r r* l -: rLp-a-r 'n l r a +

er

.

Y ) r -\( Y c 91\ I+9"'*

uJnf

a - rr rQ l

uI

X z+g' ,t ' X

T =

T=

ox

sr q}Til

JoJ AJoueur

I

u o r l e , r a + Ta l q l s s o d v . ( l ) q n ( x ) B = ( I . x ) J a r a q m( ( ) ( o o J ) s ? l = r c ) ( J ) s ? . t ueql '{(X)qn(X)3=(X)J ',,T3XA} asoddnsalduexa aog + E q + '(a)(t)s'tt

alnduroc o+ J o+ Sutpuodsa.r.roc .r{uouiaurq+T$ poq+au aAr+pJa+T snouc,.rqc -udsp dup esn uec a $ u a q J ' ( 0 ) ( o o J ) s ' I T = ( 0 ) ( J ) s ' 1 7 + e q + q a n s

, r T < -u r ( u T ) e J

l. I

10

tl hypothesis Dr=Z] and Vjei1r...,m), i m p l i e s F - ( o ( D ) ) = F .(.Z l , . . . , 2 * )

3y transfinite

D E Z J . B y m o n o t o n yo f F . t h i s

so that by transitivity

we have

End of Proof.

Proof -

{ V 6 eO n d , X 6 = L i s ( F o o ) ( D ) i

:

Let us recat-t-(cousotl3]) and is a fixed

point

tnat since D=F(o(D)), Lis(t"o)(D)

of Foo greater

exists

than D. Hence the leruna is tnue

for 6=0 since X0=D. -

Assume that -

If

the l_emmais true

6 is a limit

fon all

8 acuanbas aql

ST e

Q

rr

D

T 9JT

T+(

un

: u 9

^ t t) V U =

II Il Il I -

i

t t II I

t t t t t

9>p

TPuTP,ro ITuITT e TEUrpdo Jossacsns P

U

9 = o u

T-V

U

: s A o T T o J s P u o T S J n c a J a + T u T J s u E . r + dq paulg:ap sTpurpJo ;o

acuanbas pau;al-pd6

'g'€'t

aq+ sT 9'(9)Y [ r t : € i u l u t= ( 9 ) Y t \D>/>v : dq paulJap (g)y rpuTpro ue s-r adaqf 'p.ro;g d.raaaroJ +pq1 salldurr (c)'T'e uoTtTpuoJ 'n'e'e NOIJII,II-{X0

{

IT

I

I

t t t t t t t t

T2

Crzse 2. is true

Assume that

every 6' ecuanbes uot1e,-ralT aq+ +pL{} ([e]+osnoC) l,lou>iay : ]ooaa1 '(0)( ooJ)s'L/ sT +TurTTslT ..6.reuor1e1s sT J Jo C +urod paxr3:a.rd e qlT$ 3u11.re+s puE **(,r1 )3J Jo+p.rado aq1 o1 Surpuodsa.r,roc d.rouaur u1 q+T$ acuanbas uol+E,ro+r snouo.rqcu.fse uV . OT. t. e i,,lgUOgHJ 3CN3OUSANOJ

'Joo,za . Jo puA {'X=OA} .: {gu, g f.ra,re JoJ anlTl sr 6. g. 0 euuaf JT +pq+ peao.rd alpq er,l ( g ' e ' Z ' g ' T ' e s a s e r ) I u o u o T + c n p u Ta + r u r J s u p . r +f g 'gx5gg apnTcuoo an d11,r1+Tsuerl ,{q 1eq1 os g>D

gto>gU

= = gg saTTdul s1q1 .o* =gg aleq a/rl grojgu n X = o X [ 1 pX n d,:aaa JoJ +pql pup TeuTpro +TrIlrT e sT g ]pq+ aurnssv .g.g asDS

'jx = (r2" "'

,z)rt = ((gslolTr=le

duolouourdq 1aE afi ([e]]osnoC; Eulsea.rcuTsT e1ru6....tlr[6 slsaqloa,cq a c u a n b e sa q + a c u r s ' t z = g s os + p q + -r .' (r :- sl * = l tY ) . eJr { f . { u . . . . 6 T } r ) t 1. u o T + c n p u fTq a . r o 3 : a . r a q {rx.....tirlrt fSir'u suorlTpuoc aq+ ueq+ ,releaa8 ou TpuTpdolrrurr leQ+ ''fTdurT (p)-(q)T'g

. 6 1 1 c 1 us1Ts g e c u r s ' ( * z " " - r r l T , r = j x p u e ""r" jx=r_jx=ia nf>T

'}x=ia uer{+ u6 ...'T=TA a,loid eA .oxi'g alpq a^l gro>gu gf/T JI .,{.raaaJoJ }eq+ pue TpuTpJo Jossaccns p sT g +pq+ aunssv .A.t asDJ

l l

I l l l l:

I I I I

T {

t t

'gx398 guro apnTcuoc am i{11a1+Tsupr+ z{q 1eq1 os g

utlLl n

I

uX

=

9

g>,l =

lux

eroJa.raql

'{OLr>^tr

oX

,

rg'gA 'C'f fq P a U T P + q oa s o q + J o u o - t l P z t 1 e , . r a u a 3 1'7 uollTuTJap uT : nOTTSII,,I

p se papJeEa.r aq upc s+Tnsa.r dpau otr uirou{un sem qerq$ (1161 aunp 3ecue.rg 6alqoua.rg 6alqoua.rg ap afecrpg!,1 +a anb13:1+uaTcsatrTSJaArun tL'V'-i 'BLaou anbl.raung asr{leuyrp a.rreururgs t auo+ouoa aouabaaauoo ap sV4VVzdarc ap sawt41t,to677) +Tnsal srfloTTSII,i 'f,'l o+ uorluatr+E dur 8ut.^ae,-rp : uo'?.+DtD7ao(.

I 1 T I I I I

T

t {

t l

t; (

tl

r

dtpuT>t .roJ IUggOg sto5ue.rJ o+ apn+Tle;8 du ssa.rdxa o+ +ue& I : (lt6t

'97 aaqaaaoN) ulnpu'dppv

{

I,

t I,