A Distributed First and Last Consistent Global Checkpoint Algorithm

Transcription

1 A Dstrbuted Frst and Last Consstent Global Cecpont Algortm Yosfum Manabe NTT Basc Researc Laboratores 3-1 Mornosato-Waamya, Atsug-s, Kanagawa Japan Abstract Dstrbuted coordnated cecpontng algortms are dscussed. Te frst global cecpont for a cecpont ntaton s a set contanng te cecpont for eac process n wc any cecpont before te element s not consstent wt te ntaton. Te last global cecpont for a cecpont ntaton s a set contanng te cecpont for eac process n wc any cecpont after te element s not consstent wt te ntaton. Ts paper presents dstrbuted algortms tat mae te frst and last global cecponts consstent wt a mnmum number of cecponts taen n eac process. 1 Introducton Dstrbuted coordnated cecpontng obtans a set of states as a consstent global cecpont [8], n wc no message s recorded as receved n one process and as not yet sent n anoter process. It can be used for process rollbac 1. Wen a process ntates cecpontng, addtonal cecponts must be taen n oter processes n order to obtan a consstent global cecpont tat ncludes te ntaton. Dfferent global cecponts mgt be obtaned dependng on te addtonal cecponts taen by eac process. Ts paper consders two cases and defnes two nds of global cecpont. Te frst case s recovery from falure. Wen process p experences a falure, all processes roll bac to eac state n a consstent global cecpont. Te addtonal rollbac for processes oter tan p must be as small as possble n order to mnmze te overead of reexecuton. Tus, t s better for te oter processes to roll bac to latter cecponts. Te second case s rollbac n debuggng. Assume tat p rolls bac to a state wen an error s observed. Te bug mgt be n process p and a wrong 1 In order to roll bac, te messages wc ave been sent but not receved must be restored. Te message restoraton metod s smlar to tat n [9] and te detals are gven n [6]. Ts paper tus dscusses obtanng a consstent global cecpont. message from p mgt ave caused te error. Tus, te debugger user wants to observe eac process n a consstent global state. If a latter cecpont s used for p s rollbac, te bug mgt be dden by furter executon of p ; for example, extng from a subroutne and deletng all varables tat decded te content of te wrong message [4]. Tus, t s better for te oter processes to roll bac to former cecponts. Ts paper tus defnes two global cecponts: te frst and last global cecponts [5]. Te frst (last) global cecpont for a cecpont ntaton s a set contanng te cecpont for eac process n wc any cecpont before (after) te element s not consstent wt te ntaton. Ts paper ten gves two dstrbuted algortms tat mae te frst and last global cecpont consstent wt te mnmum number of addtonal cecponts taen n eac process. Toug ndependent cecpontng algortms, suc as tat n [9], do not need consstent global cecponts, tey can be used only for systems n wc all non-determnstc events can be recorded durng executon and replayed durng re-executon. For systems n wc records of nondetermnstc events can be very large or replayng nondetermnstc events s dffcult, a consstent global cecpont s necessary. Candy et al. s dstrbuted snapsot algortm [] obtans a consstent global cecpont (neter te frst nor te last) for concurrent ntatons. Te autor [7] extended ter algortm and te extended verson mnmzes te number of addtonal cecponts. Venates et al. s algortm [10] obtans a last global cecpont tat s consstent, and Baldon et al. s algortm [1] obtans a frst global cecpont tat s consstent, but tey dd not defne te concept of frst and last global cecponts. Ter algortms do not mnmze te number of addtonal cecponts. Te frst and last global cecpont Te dstrbuted system s modeled by a fnte set of processes fp 1 ;p ;:::;p n g nterconnected by pont-to-pont cannels. Cannels are assumed to be error-free, non-fifo, and ave nfnte capacty. Te communcaton s asyn-

2 cronous; tat s, te delay experenced by a message s unbounded but fnte. p s executon s a sequence of p s events wc nclude cecpont ntatons. Cecpont ntatons are done ndependently by eac process. System executon E s te set of eac process s executons. p s executon wt cecpontng algortm A s p s executon nterleaved wt te addtonal cecponts taen by A n p. System executon wt A, E(A), s te set of eac process s executon wt A. Te followng assumptons are common for dstrbuted cecpontng algortm A [1][10]. A as no pror nowledge about executon E. All nformaton for A s pggybaced on program messages between processes. Wen p receves a message m, A can get te nformaton pggybaced on m and tae an addtonal cecpont before p executes te receve event. Te appened before(!) relaton between te events n E(A) s defned as follows [3]. Defnton 1 e! e 0 f and only f (1) e and e 0 are executed n te same process and e s not executed after e 0. () e s te send event s(m) and e 0 s te receve event r(m) of te same message m. (3) e! e 00 and e 00! e 0 for event e 00. Wen e and e 0 are executed n dfferent processes and e! e 0, tere s a sequence of events e; s(m 1 );r(m 1 );s(m );:::;s(m );r(m );e 0 n wc e! s(m 1 ), r(m )! s(m +1)( = 1;:::; 0 1), r(m )! e 0, every par of events s executed n te same process, and every s(m ) s executed n a dfferent process. Ts sequence s called a causal sequence from e to e 0. s te lengt of te causal sequence. Two specal events,? and >, are defned for p.? s an magnary event wc s p s ntal state. > s p s current event f p s not termnated. If p s termnated, > s an magnary event wc s p s termnal state. For any p event e,?! e and e!> old. Ts paper consders > and? as cecponts n E. For p s event e n E(A), two events on p, causal-past event, cp e (), and causal-future event, cf e (), are defned as follows. Defnton cp e ()=cf e ()=e. cp e () s last event e n p tat satsfes e! e.if tere s no event e satsfyng e! e, cp e ()=?. cf e () s frst event e n p tat satsfes e! e.if tere s no event e satsfyng e! e, cf e ()=>. Intutvely, cp e () s p s last event wc s nown to p at e. cf e () s p s frst event wc nows e. In Fg. 1, cp c1 (1) =s(m 1 ), cp c1 () =c 1, cpc1 (3) =s(m ), cp c1 (4) =? 4, cf c1 (1) = > 1, cf c1 () = c 1, cf c1 (3) = r(m 5 ), and cf c1 (4)=r(m 6 ). Defnton 3 A par of cecponts (c; c 0 ) s consstent f and only f c 6! c 0 and c 0 6! c. Defnton 4 A global cecpont (c 1 ;c ;:::;c n ) s n-tuple of cecponts were c s p s cecpont. A global cecpont s consstent f and only f all dstnct pars of cecponts are consstent. Defnton 5 Te frst global cecpont for p s cecpont ntaton c n E(A), FG c (E(A)), s defned as follows. -t element, FG c (E(A);), s te frst cecpont n p wc s not before cp c (). E(A) s omtted f t s obvous. Defnton 6 Te last global cecpont for p s cecpont ntaton c n E(A) LG c (E(A)), s defned as follows. -t element, LG c (E(A);), s te last cecpont n p wc s not after cf c (). FG c (E(A);) (or LG c (E(A);)) = > means tat p need not roll bac at all wen p rolls bac to c. Any cecpont of p before cp c () or after cf c () s not consstent wt c c. Tus, FG c (E(A)) and LG c (E(A)) are te best possble former and latter global cecponts for c. In Fg. 1, FG c1 (E) = (c 1 1 ;c1 ;c1 3 ;? 4) and LG c1 (E) = (> 1 ;c 1 ;c1 3 ;c 4). FG c1 (E) s not consstent snce c 1 1! c1 3. LGc1 (E) s not consstent snce c 1 3! c 4. Toug FG c (E) and LG c (E) mgt not be consstent, FG c (E(A)) and LG c (E(A)) can be consstent by te addtonal cecponts taen by A. In Fg. 1, f A taes an addtonal cecpont at e 1, FG c1 (E(A); 3) =e 1 and FG c1 (E(A)) s consstent. If A 0 taes an addtonal cecpont at e, LG c1 (E(A 0 ); 3) =e and LG c1 (E(A 0 )) s consstent. If algortm A 0 taes an addtonal cecpont ust before every receve event, FG c (E(A 0 )) and LG c (E(A 0 )) s consstent for any cecpont ntaton c. However, te overead of A 0 s very large. Ts paper sows two dstrbuted cecpontng algortms, FA and LA. Among algortm A wc maes every FG c (E(A)) (LG c (E(A))) consstent, FA (LA) taes te mnmum number of addtonal cecponts. Our algortms deal wt cecpont ntatons and addtonal cecponts dfferently. If a user wants to deal wt an addtonal cecpont as an ntaton, ts can be done by executng te cecpont ntaton procedure for te addtonal cecpont.

3 3 Algortm FAfor FG c In te rest of te paper, a sequence number s assgned for (bot of ntaton and addtonal) cecponts n eac process n E(A).? s p s 0-t cecpont. Let c x be p s x -t cecpont. It s sometmes denoted as x n subscrpts f t s not ambguous. p mantans a varable c (). c ()=xf p currently nows p s cecpont c x. c () = 01 fp currently nows no cecpont n p.if c ()=x( 0) at event e, c x! e and c ()=x at cp e () s satsfed. c () s p s newest cecpont number. Updatng c can be done by sendng ts current value on every message. Ts s sown n detal n Fg. 4. For any algortm A, FG c (E(A)) for p s cecpont ntaton c can be represented usng c as follows: Let CK() be te value of c () at c. FG c () = c CK()+1 ( 6= ). If te (CK()+1)-t cecpont does not exst n p, FG c () =>. In order to mae t consstent, FA taes addtonal cecponts. Now consder te case wen a message m from p arrves at p. Assume tat p as taen (x 0 1) cecponts before te arrval of m. Te frst case wen te x -t cecpont must be taen before r(m) s sown n Fg.. p nows p s cecpont ntaton c x and cx 01! c x. Snce p nows te ntaton, c x! r(m) s satsfed. If p does not tae te x -t cecpont before r(m), c x! r(m)! cx (= FG x ()) and FG x s not consstent. Ts condton s represented as follows. Consder a varable n (). n () =true f p nows a cecpont ntaton c tat satsfes c x! c for p s current cecpont c x. Te followng s p s rule for tang a cecpont before r(m). (Rule F1) n ()=true. Te update rule of n s sown n Fg. 4. Te second case s sown n Fg. 3. In ts case p mgt ntate a cecpont after e = cp r(m) (). Toug ts cecpont s unnecessary f p actually does not ntate after e, p taes t snce p cannot predct p s executon after e at r(m). Ts case s dvded nto two subcases. Te frst subcase s wen p nows p s current (te (x 0 1)-t) cecpont at e. p ntates cecpont c x ust after e. Assume tat tere s a cecpont c x tat satsfes cx! r(m) and c x 6! e. Snce c x 6! e, FG x from te decson ()! c x rule of FG.Ifp does not tae te x -t cecpont before r(m), FG x ()! c x! r(m)! cx (= FG x ()) and FG x s not consstent. Te second subcase s wen p does not now p s (x 0 1)-t cecpont at e. In order for p to ntate a cecpont tat satsfes FG x ()=x, p must frst receve a message tat carres te nformaton of c x01 and ten ntate. Assume tat tere s a message m 0 sent to p but not receved before e, wc carres te nformaton about c x01. Assume also tat p receves m 0 ust after e and ten ntates a cecpont c x. Furter assume tat tere s a cecpont c x tat satsfes cx! r(m) and cx 6! r(m0 ). Snce c x 6! r(m0 ), FG x ()! c x. If p does not tae te x -t cecpont before r(m), c x! r(m)! cx and FG x s not consstent. Ts condton s represented as follows. Introduce boolean varable cr (; ) and ad (; ; ). cr (; )=true f p nows tat p nows p s current (te c ()-t) cecpont. Note tat cr (; ) s always true for every. ad (; ; )=true f p nows tat p wll now p s current (te c ()-t) cecpont f p receves any message sent to p tat carres p s current (te c ()-t) cecpont. If p already nows p s current cecpont or suc a message does not exst, ad (; ; )=false. (Rule F) (cr (; ) = true and cr (; ) = false) or (ad (; ; ) = true and cr (; ) = false and ad (; ; )=false) for some par of (; ). Te algortm FA, wc ncludes te update rule of te above varables, s sown n Fg. 4. Teorem 1 Every addtonal cecpont taen by FA s necessary. Teorem 1 s obvous from te above dscusson. (E(FA)) s consstent for any cec- Teorem FG x pont ntaton c x. (Proof) Assume tat FG x s not consstent and c x FG x ())! c x (= FG x ()). From te FG decson rule, c x 01! c x and cx 6! c x f 6=. (Case 1: = ) c x! cx contradcts te above fact. (Case : = ) Let te last message on te causal sequence from c x to c x be m. Snce c x 01! c x and c x (=! r(m), n () = true at r(m). From Rule F1, p must ave taen te x -t cecpont before r(m). contradcts te asserton tat c x s after r(m). Ts (Case 3: 6= and 6= ) Tere s a causal sequence CS from c x 01 to c x. Let te sequence of messages n CS be M 1 ;M ;:::;M l. Let te process tat executes r(m a ) be p za (a = 1;:::;l). Wtout loss of generalty, p za satsfes c za () <x 0 1 before r(m a ) (a = 1;:::;l). Note tat c x 6! e for any event e n CS. Oterwse, cx! cx and ts contradcts te FG decson rule. Let te last event e n CS tat satsfes e! c x be e 0 on p. Suc an event always exsts because s(m 1 )! c x.if s(m 1) s after c x, c x! c x and ts contradcts te FG decson rule. Let e = cp x (), tat s, let e be p s last event nown to p at c x. From te defnton, e0! e. (Case 3-1: e 0 s receve event r(m)) From te assumpton, te next send event n CS s after e. Tere s a causal

4 sequence from e to c x. Let r(m0 ) be te last receve event n te causal sequence. Snce c () = x 0 1 and c () <x at e, cr (; ) =true and cr (; ) =false at r(m 0 ). Tus, p must ave taen te x -t cecpont before r(m 0 ) from Rule F. (Case 3-: e 0 s send event s(m)) Let te recever of M be p g and let e g = cp x (g). From te assumpton, r(m) s after e g. ad (g; ; ) = true, cr (g; ) = false, and ad (g; ; ) =false are satsfed at r(m 0 ) snce c g () < x 01, c g () <x at e g and c g ()=x 01, c g () <x at r(m). Tus, p must ave taen te x -t cecpont before r(m 0 ) from Rule F. Te nformaton pggybaced on eac message and ept n eac process s O(n) nteger and O(n 3 ) boolean values. 4 Algortm LA for LG c Consder te case wen p realzes p s ntaton c x at receve event cf x (). Let te message be m and ts sender be p. Cecpontng algortm A sets LG x (E(A);) as p s newest cecpont or taes a new addtonal cecpont before r(m) and sets LG x (E(A);) as te new one. In eter case, LG x (E(A);) s te last cecpont tat s not after cf x (). LA must decde weter t taes a new addtonal cecpont before r(m) n order to mae LG x consstent wt te mnmum number of addtonal cecponts. Assume tat p as taen (x 01) cecponts before te arrval of m. Te cases wen te x -t cecpont must be taen before r(m) are sown n Fg. 5. Te frst case s wen tere s a cecpont c x suc tat cx01! c x and cx! r(m) are satsfed. Tere s an ntaton c x tat satsfes LGx ()= c x. LGx s not consstent f cx s not taen before r(m) because c x01! c x. Tus, an addtonal cecpont s necessary. Ts condton s represented as follows. Consder a varable see (). see () =true f p nows a cecpont c tat satsfes c x! c for p s current cecpont c x. (Rule L1) see ()=true. Te second case s wen p sends a message m 0 to process p after c x 01 and neter LG x () nor LG x () as been decded. Ts addtonal cecpont s necessary f p sends message m 00 to p after r(m) and p executes r(m 0 ), taes a cecpont c x, and ten executes r(m00 ). Now, snce cf x () = r(m 00 ), LG x () s c x or an addtonal cecpont tat s taen ust before r(m 00 ). In eter case, c x01! LG x () and LG x s not consstent f LG x () =x 0 1. Terefore, p must tae an addtonal cecpont before r(m). Ts case s represented as follows. Varable num ()= x f p nows te newest ntaton by p s c x. Varable dc (; )=true f p nows tat p as decded te element of LG x (), were x = num (). Varable st () =true f p as sent a message to p after p s latest (te c ()-t) cecpont. (Rule L) dc (; ) = false, dc (; ) = false, and st ()=true for some par of (; ). Algortm LA, wc ncludes te rule for updatng tese varables, s sown n Fg. 6. LG x () s stored n varable lg. lg (; x )=yf LG x () =c y. Note tat lg (; x ) mgt be undefned for two reasons. Frst, p does not now ntaton c x x, tat s, cf ()=> at a gven state. In suc a case, LG x ()=>. Te second case s wen cf x ()= cf x0 (), tat s, te nformaton for two dfferent ntatons by p arrves at p at te same tme. Te nformaton about old ntatons s dscarded because LG x () = LG x0 (). Tus, f lg (; x) s undefned for every x(x 0 <x<x ) and lg (; x ) s defned, lg (; x) = lg (; x ) for any ntaton c x (x 0 <x<x ). Teorem 3 Every addtonal cecpont taen by LA s necessary. Teorem 3 s obvous from te above dscusson. Lemma 1 [7] If a global cecpont (c 1 ;c ;:::;c n ) s not consstent, tere s a par (c ;c ) suc tat tere s a causal sequence from c to c wose lengt s 1. Teorem 4 LG x (E(LA)) s consstent for any cecpont ntaton c x. (Proof) Assume tat LG x s not consstent. From Lemma 1, assume tat c x (= LG x ())! c x (= LG x ()) and let te message n te causal sequence be m. Let e x be te event wen p decdes LG x (). Note tat LG x () mgt be decded by decdng LG x0 () for x 0 > x as descrbed above. In suc a case, e x = e x0. e x s an ntaton, a receve event, or > (n ts case, LG x () = > ). If e x s a receve event, LG x () s before e x. In te oter cases, ex = LG x (). Tus, LG x ()! e x s satsfed. Note tat LG x () s p s newest cecpont at e x. num () <x s satsfed before e x and num () x s satsfed after e x c x. 6= > snce tere s an event s(m) after c x (Case 1: e x! e x ) Snce cx s p s newest cecpont at e x and cx! cx! e x! e x, see ()=true at e x. Tus, c x must be te newly taen cecpont ust before e x from Rule L1. Ts contradcts te noton tat tere s an event s(m) between c x and ex. (Case : e x s before s(m)) num () x must be satsfed at r(m). Tus, e x must be equal to or before r(m). Ts contradcts te noton tat c x s after r(m). (Case 3: e x 6! e x and ex s after s(m)) Snce ex 6! e x, dc (; )=false at e x. Snce tere s an event s(m) between c x and ex, ex s a receve event. Let X = num () at e x. X x s satsfed. Snce st ()=true.

5 and dc (; )=false at e x, cx must be te newly taen cecpont ust before e x from Rule L. Ts contradcts te noton tat tere s an event s(m) between c x and ex. Te nformaton pggybaced on eac message and ept n eac process (oter tan te output) s O(n) nteger and O(n ) boolean values. Here, te rule for removng old cecponts s sown. Te amount of stable storage usage becomes large f old cecponts are not removed. Assume tat p rolls bac to te newest ntaton by p. p mgt be forced to roll bac to lg (; num()). Tus, p does not need te cecponts before mn lg (; num()) and tese cecponts can be removed. 5 Concluson Ts paper sowed two dstrbuted algortms tat mae te frst and last global cecpont consstent wt a mnmum number of addtonal cecponts taen n eac process. Remanng problems nclude reducng te amount of nformaton used by te algortms. Acnowledgments Te autor would le to tan Dr. Hrofum Katsuno of NTT for s encouragement and suggestons. References [1] Baldon, R., Helary, J.M., Mostefaou, A., and Raynal, M.: Consstent Cecpontng n Message Passng Dstrbuted Systems, INRIA Tecncal Report No. 564 (June 1995). [] Candy, K.M. and Lamport, L.: Dstrbuted Snapsots: Determnng Global States of Dstrbuted Systems, ACM Transacton on Computer Systems, Vol. 3, No. 1, pp (Feb. 1985). [3] Lamport, L.: Tme, Clocs, and te Orderng of Events n a Dstrbuted System, Communcatons of ACM, Vol. 1, No. 7, pp (July 1978). [4] Manabe, Y. and Imase, M.: Global Condtons n Debuggng Dstrbuted Programs, Journal of Parallel and Dstrbuted Computng, Vol. 15, No. 1, pp (May 199). [5] Manabe, Y.: A Dstrbuted Frst and Last Consstent Global Cecpont Algortm, IPSJ SIG Notes, AL (Oct. 1996). [6] Manabe, Y.: A Dstrbuted Consstent Global Cecpont Algortm wt a Mnmum Number of Cecponts, Tecncal Report of IEICE, COMP97-6 (Apr. 1997). [7] Manabe, Y.: A Dstrbuted Consstent Global Cecpont Algortm wt a Mnmum Number of Cecponts, Proc. of 1t Int. Conf. on Informaton Networng (Jan. 1998). [8] Netzer, R.H. and Xu, J.: Necessary and Suffcent Condtons for Consstent Global Snapsots, IEEE Trans. on Parallel and Dstrbuted Systems, Vol. 6, No., pp (Feb. 1995). [9] Strom, R.E. and Yemn, S.: Optmstc Recovery n Dstrbuted Systems, ACM Trans. on Computer Systems, Vol.3, No.3, pp.04 6 (Aug. 1985). [10] Venates, K., Radarsnan, T., and L, H.F.: Optmal Cecpontng and Local Recordng for Domno-Free Rollbac Recovery, Informaton Processng Letters, Vol. 5, No. 5, pp (July 1987). process p p p p Ý Ý Ý Ý m 1 c 1 m 1 e 1 m 1 c 3 c 1 c 1 4 m 4 3 tme c e c 1 m5 4 m 6 Ý 1 Ý Ý Ý 3 4 : message : cecpont : ntaton Fgure 1. System executon E. c x p m p p p p c x 1 c x r(m) cp r(m) () c x x c c x cp r(m) m r(m) :ntaton or addtonal cecpont :ntaton :addtonal cecpont Fgure. Rule F1. c x 1 m p p p c x 1 cp r(m) () c x x c x c cp r(m) m r(m) Fgure 3. Rule F.

6 program FA; /* program for p.*/ const n = :::; /* number of processes */ var c(n): nteger; n(n), cr(n; n), ad(n; n; n): boolean; procedure cecpont begn tae a cecpont; c() := c()+ 1; n() :=false; for eac (6= ) do cr(; ) :=false; for eac (6= ); do ad(; ; ) :=false; for eac (6= ); do ad(; ; ) :=false; end; /* end of subroutne */ /* man */ ntalzaton begn for eac (6= ) do c() := 01; c() := 0; for eac do n() :=false; for eac (6= );do cr(; ) :=false; for eac do cr(; ) := true; for eac ; ; l do ad(; ; l) :=false; end /* end of ntalzaton */ wen p ntates a cecpont begn cecpont; for eac 6= do n() :=true; end /* end of cecpont ntaton */ wen p sends m to p begn send(m, c, n, cr, ad)top ; for eac do f not(cr(; )) and not(ad(; ; )) ten for eac do ad(; ; ) :=true; end /* end of message sendng */ wen message (m, mc, mn, mcr, mad) arrves from p begn for eac do begn f c() < mc() ten begn n() := mn(); for eac (6= ) do cr(; ) := mcr(; ); for eac (6= );ldo f mc(l) c(l) ten ad(; ; l) := mad(; ; l) else ad(; ; l) :=false; end /* end of case c() < mc() */ else f c()=mc() ten begn n() := n() _ mn(); for eac (6= ) do cr(; ) := cr(; ) _ mcr(; ); for eac (6= );ldo f mc(l) >c(l) ten f ad(; ; ) ten ad(; ; l) :=false else ad(; ; l) := mad(; ; l) else f mc(l)=c(l) ten f (cr(; l) or mcr(,l) or (ad(; ; ) and not(ad(; ; l))) or (mad(; ; ) and not(mad(; ; l))) ten ad(; ; l) :=false else ad(; ; l) := ad(; ; l) _ mad(; ; l) else /* mc(l) <c(l) */ f mad(; ; ) ten ad(; ; l) :=false; end /* end of case c()=mc() */ else /* c() > mc() */ for eac (6= );ldo f mc(l) >c(l) ten ad(; ; l) :=false end; /* end of f statement */ end; /* end of loop by. */ for eac (6= ) do c() := max(c(); mc()); f n() or 9(; ), ((cr(; ) and not(cr(; ))) or (ad(; ; ) and not(cr(; )) and not(ad(; ; )))) ten cecpont; execute r(m); end /* end of message arrval */ Fgure 4. Algortm FA. c x LG x p p c x m p c x 1 c x r(m) p p p x c c x 1 c x m c x x LG m r(m) m" Fgure 5. Rule L1 and L. program LA; /* program for p.*/ const n = :::; /* number of processes */ var c(n), num(n), lg(n; 3): nteger; dc(n; n), see(n), st(n): boolean; procedure cecpont begn tae a cecpont; c() := c()+ 1; for eac do st() :=false; for eac (6= ) do see() :=true; see() :=false end; /* end of subroutne */ /* man */ ntalzaton begn for eac 6= do c() := 01; c() := 0; for eac do num() := 0; for eac ; do dc(; ) :=true; for eac do see() :=false; for eac do st() :=false; end /* end of ntalzaton */ wen p ntates a cecpont begn cecpont; num() := c(); for eac (6= ) do dc(; ) :=false; end /* end of cecpont ntaton */ wen p sends m to p begn send(m, c, num, see, dc) top ; st() :=true; end /* end of message sendng */ wen message (m, mc, mnum, msee, mdc) arrves from p begn for eac do begn f num() < mnum() ten for eac do dc(; ) := mdc(; ); else f num()=mnum() ten for eac do dc(; ) := dc(; ) _ mdc(; ); end /* end of loop by */ for eac do num() := max(num(); mnum()); for eac do f c() < mc() ten see() := msee() else f c()=mc() ten see() := see() _ msee(); for eac do c() := max(c(); mc()); f see() or 9(; ), (not(dc(; )) and not(dc(; )) and st()) ten cecpont; for eac do f not(dc(; )) ten begn dc(; ) :=true; lg(; num()) := c(); end; execute r(m); end /* end of message arrval */ Fgure 6. Algortm LA.