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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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

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l [ * I- 1 1 . INTRODUCTION AND NOTATIONS Let L(s,1,r,L1,[1) be a non-empty conplete Latlice w:-th partial otde- 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 ) } ) . TarskifT-lrs theorenr stale:r that the set of fired points of f (solutions to the equation X=F(X)) is a non-empty complete lattice with ordering E. Let u be the smallest ol','li nal such that l-he <:lass { 0 : 6e u } has a candinai-ity greater than the cardinality of l,n. Cousotl3] defines corrs'ructively the fixed noints of F by means of the following !-termed transfinite sequences : The I terat'ion sequonce fi,t: I starting uitLt o e Ln is the lr-termecl A secuence <u r 0€l-l> of efernents of Ln deFined by transfinite recursion 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" ) for c'\'ery , t ^ - L " = t_l R- for every c < 6 successor ordinal 6ep limit ordi"nal 6e p urle sry that the sequence is stationary iff {ge ,-u : {VBe u, lgze)={ue=gB}}} in which case the Limit of the sequence denoted by _e jzs(i )(lJl ]-S detlned to .De lJ . A suiticient condition for the iteration sequence for F starting with D to be stationary is that D is a prefixed point of F (Dsf(D)). In this t c a s e < B " , 6 e u > i s a n i n c r e a s i n g c h a i n , i t s l i m i t L i s G ) ( n ) l s t h e f e a s t cf the fixed points of F greater" than D. It is also gr"eater than any i.i.:ort nnr'nr n€ tr |sss than D (if such fixed points exist), (Cousot[3]).

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