Patrick Cousot. Asynchronous iterative methods for solving a fixed point system of monotone equation

Patrick Cousot. Asynchronous iterative methods for solving a fixed point system of monotone equation

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Unformatted text preview: ASYNCHRONOUS ITERATIVE METHODS FOR SOLVING A FIXED POINT SYSTEM OF MONOTONE EQUATIONS IN A COMPLETI LATTICE Patri ck Cousot R . R . B 8 lanl- omlrno L Y I I l l RAFPORT DE REGT-!ERCHE l 6ITt0 anbTleulroJuI-dIV +uPr3 raPun 's ' L o u o r c o s s v a d r o l e J o q E ' I ' U ' N ' J d q p a p o d d n s s E M 1 r o m s r q l ' ' S ' U ' N ' C n p a q c " r a q c a g a P o q c e + l v l l I , I ; I l l I ' /i,-rouauL qlT^4 sPoqleul aArleJe+T snouoJt{cudse 6spoql.l'll a^T+e'ralT snou- o a q c u d s e s a c T + + e T a l e l d u o c e u r s u o T l e n b e o u o l o u o u l go uralsds '1u1od paxTJ : SaseJqd pue spJon /tEl uaer"rtraq uoT+ezTuo,rqcuds ou qtrrrvl ;:l:;:t::":::l:;;fi:; p uo suorlenbe 1o uralsds aq1 Sutnlos Joi uqlr'-ro81e 1a11e"red e o1 puodseJ.roc spoqlau o^rlPJalT asaqJ ' ( s u o r l T p u o c u r e q c , : o d l t n u l l u o c s e q c n s ) s a s a q l-oddq TeuorlTppP ou asn s;:oo"rd Tecruqcol JeqleJ aql : r r { g 1 9 3 P u r s u o r l e n b a a u u l o u o l u J o u e l s ^ s e J L + + E L c + e t u u v v c q t lurod pexTJ e anTos o] pasn aq uec A'lououl qlr^l spoq+eu anrfp.ratrT snouoJqcu{se pue spoq}au e^T+e"ra+T snouoJr{cuz{se 3:o ssPTc aqf, treq} u!\oqs st }I 'lf,PJlSqV ( L L 6 I a e q u a l d a S l a c u e J , f o x a p a c a T q o u e J e T h 0 8 t 0 S ' d B ' ' 3 ' 1 , , 1 ' S ' n ' a n b r l p u r . r o J u l r P e J T o l P J o q P T * l O S n O J I l L J l e d lSlrlvl lrlldhol v NI sN0lrvnbS lNol0Noll l0 HIISAS INI0d olxll V 9N:iA10S UoJ S0oHIlll lAIIvusII SnoilouHlllAsv I I I I , ' - 1 , I I I I I I I I l [ * I- 1, 1 . INTRODUCTION AND NOTATIONS L e t L ( s , 1 , r , L 1 , [ 1 ) b e a n o n - e m p t y c o n p l e t e L a t l i c e w : - t h p a r t i a l o t d e - rtnq =, Least upper. bounrl" t.l, gneatest Lodev, i>ound n. The inlimun t of L is [1L, the supremum r of L is Ut. Let F be a monotone operator l.r f r o m L " ( E , r , r , l l , l l ) i n t o i t s e f f ( i . e . v X , Y e L n , { x c Y } - { F ( x ) s F ( Y ) } ) . T a r s k i f T - l r s t h e o r e n r s t a l e : r t h a t t h e s e t o f f i r e d p o i n t s o f f ( s o l u t i o n s t o t h e e q u a t i o n X = F ( X ) ) i s a n o n - e m p t y c o m p l e t e l a t t i c e w i t h o r d e r i n g E . Let u be the smallest ol','li nal such that l-he <:lass { 0 : 6e u } has a c a n d i n a i - i t y g r e a t e r t h a n t h e c a r d i n a l i t y o f l , n . C o u s o t l 3 ] d e f i n e s c o r r s ' r u c t i v e l y t h e f i x e d n o i n t s o f F b y m e a n s o f t h e f o l l o w i n g ! - t e r m e d t r a n s f i n i t e s e q u e n c e s : The I terat'ion sequonce fi,t: I starting uitLt o e Ln is the lr-termecl A s e c u e n c e < u r 0 l - l > o f e f e r n e n t s o f L n d e F i n e d b y t r a n s f i n i t e r e c u r s i o n r n i l r e l o I L o w : n g w a y : I I I t I I I 1' t' t' I-- r . 0 - n A A - I- I = F ( L " ) f o r c ' \ ' e r y , t ^- L " = t_l R- for every c < 6 s u c c e s s o r o r d i n a l 6 e p limit ordi"nal 6e p...
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Patrick Cousot. Asynchronous iterative methods for solving a fixed point system of monotone equation

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