In Serh of Lost Time Bernette Chrron-Bost 1, Mrtin Hutle 2, n Josef Wier 3 1 CNRS / Eole polytehnique, Pliseu, Frne 2 EPFL, Lusnne, Switzerln 3 TU Wien, Vienn, Austri Astrt Dwork, Lynh, n Stokmeyer (1988) n Lmport (1998) showe tht, in orer to solve Consensus in istriute system, it is suffiient tht the system ehves well uring finite perio of time. In shrp ontrst, Chnr, Hzilos, n Toueg (1996) prove tht filure etetor tht, from some time on, provies goo informtion forever is neessry. We explin tht this pprent prox is ue to the two-lyere struture of the filure etetor moel. This struture lso hs impts on omprison reltions etween filure etetors. In prtiulr, we mke expliit why the lssi reltion is neither reflexive nor extens the nturl history-wise inlusion. Altough not tehnilly iffiult, the point we mke helps unerstning existing moels like the filure etetor moel. Keywors: istriute omputing filure etetors synhronous system Consensus 1 Introution The im of the pper is to eepen the unerstning of one of the most lssil moel in istriute omputing, nmely the filure etetor moel y Chnr n Toueg [2]. Roughly speking, in the filure etetor pproh, istriute system is represente y two lyers: lyer tht orrespons to the system itself formlly efine in terms of proesses, utomt, n ommunition links tht runs in totlly synhronous moe, n seon lyer onsisting of the filure etetor efine with respet to time essile to the proesses t ny time. From moeling perspetive, this two-lyere pproh rilly hnge from the lssi pprohes (e.g., [3, 4, 7]) where the timing ehvior of system is restrite in orer to irumvent funmentl impossiility results [5]. Bsilly, the new ie in the filure etetor moel [2] ws to onsier istriute omputtions with unrestrite timing while the moel is ugmente with n orle, expresse y time-epenent lyer tht provies informtion from outsie. In this pper, we explin how some hrteristi fetures of this moel tully originte from the interply etween the two lyers, n the isrepny etween their timeless vs. time epenent efinitions. One of these fetures ws proven in [1], nmely, tht Consensus nnot e solve without (impliit) permnent greement on leer from some time on. It seems in ontrition with erlier positive results y Dwork, Lynh, n Stokmeyer [4] n y Lmport [7] who showe tht suffiiently long finite goo perio mkes Consensus solvle. Another peulirity of this iprtite moel is the wy filure etetors re ompre: Chnr n Toueg [2] introue omprison reltion whih we show to e not reflexive. This is not just forml prolem tht n e esily fixe. Atully, the prolem gin 1
origintes from the interfe etween the system lyer n the filure etetor lyer, n the lk of timing ontrol y the filure etetor on the synhronous system lyer. 1.1 Permnent vs. intermittent limittion of synhrony The entrl result in fult-tolernt istriute omputing is tht Consensus nnot e solve in n synhronous system in the presene of rsh filures [5]. Severl pprohes hve een propose to irumvent this impossiility result. In prtiulr, Dwork, Lynh, n Stokmeyer [4] showe tht Consensus n e solve if the synhrony of the system is intermittently limite uring polynomil (in the size of the system n their timing prmeters) mount of time. Similrly, Lmport propose the Pxos lgorithm [7] tht solves Consensus if proesses n elet ommon leer for suffiiently long finite perio (the so-lle goo perios ). These two positive results re very importnt s they oth provie restritions whih re quite resonle for implementing Consensus in prtie. Menwhile, Chnr n Toueg [2] introue the notion of filure etetors tht re efine s externl evies to the system: filure etetor is istriute orle whih is suppose to provie informtion out filures, n so ugments the unerlying synhronous system. Besies, Chnr, Hzilos, n Toueg [1] prove tht ertin filure etetor enote Ω n e extrte from ny filure etetor tht mkes Consensus solvle. Roughly speking, Ω provies the following informtion: There is time fter whih ll the orret proesses lwys trust the sme orret proess (the leer). Contrry to the orretness onitions of the Consensus lgorithms in [4, 7] relle ove, Ω is very strong onition, lerly not hievle in rel systems. In fe of [4] n [7], it seems proxil tht n eventully forever leer is neessry to solve Consensus. The eventully forever requirement ws isusse in [1]: it ws stte tht in prtie it woul suffie tht some properties hol for suffiiently long, until n lgorithm hieves its gol, n tht [...] in n synhronous system it is not possile to quntify suffiiently long, sine even single proess step or single messge trnsmission is llowe to tke n ritrrily long mount of time. [1] Unfortuntely, the rgument of the impossile quntifition of the length of goo perios is frgmentry. Inee, the point is to speify in terms of wht urtion my e quntifie. For instne, if one onsiers the numer of messges in uslity hin n the numer of steps tken y proesses, one my quntify suffiiently long internlly to n synhronous system. The pproh tken in [4, 7] n e interprete in suh wy. In this pper, we mke expliit tht the origin of this neessity lies in the two-lyere struture of the moel. On the one hn, filure etetors re totlly inepenent of the exeution of the system, n filure etetor histories re efine only with respet to time. On the other hn, the system is not orle-triggere : proess my query its filure etetor moule t ny time, ut nnot e inite y the filure etetor or y some other orle (e.g., lok) to mke step. Hene, the system runs totlly synhronously: steps my our t ritrry times, n so my e ritrrily elye with respet to time. In orer to e of ny help to solve prolem, filure etetor must therefore e insensitive to suh elys. In this pper, we formlize this intuition, n more generlly we stuy the impt of two-lyere struture on the moeling. The point is not tehnilly iffiult, ut we elieve tht more thorough isussion on the si properties of omputtionl moels for istriute systems is essentil. In prtiulr, it my help in etter unerstning the existing moels suh s the filure etetor moel. 2
To o tht, we introue for eh filure etetor D, filure etetor D whih represents ll the possile smplings of the histories of D. More preisely, D n e unerstoo s the olletion of ll the possile time-ontrte versions of D, inluing D itself. Clerly, D is stronger thn D in the sense tht D provies more preise informtion thn D. The entrl point of our isussion is to show tht given ny (non rel-time) prolem P in synhronous systems, D is tully suffiient to solve P if P n e solve using D. The si reson lies in the ft tht our time-ontrtion hs no visile effet in n synhronous system. In other wors, the only informtion tht n e exploite y n synhronous system from filure etetor D is its time-ontrte version D, i.e., time is lost t the interfe etween the synhronous system n filure etetor moules. As onsequene, the wekest filure etetor W P to solve P, if it exists, is timeontrtion insensitive. In prtiulr, this hols for the filure etetor Ω, whih extly ptures the neessity of n eventully forever leer to solve Consensus. 1.2 Comprison reltion etween filure etetors The reltion introue in [2] to ompre filure etetors is efine from n opertionl viewpoint: D D if there is n synhronous istriute lgorithm tht trnsforms D into D. As expline ove, timing informtion tht my e given y the filure etetor lyer is efinitely lost t the interfe etween the two lyers. Therey, the oupling etween filure etetors n time whih y the wy, my e very strong nnot e hnle y suh n opertionlly efine reltion. In other wors, s soon s D is weker thn some other filure etetor D with respet to, D is in the restrite lss of time-free filure etetors. In prtiulr, non time-free filure etetors re not omprle to themselves, n is not reflexive. We shll explin why this shortoming nnot esily e fixe in two-lyer moel, n propose new efinition of preorer (reflexive n trnsitive) for ompring filure etetors, whih is quite ifferent in essene: it ompres filure etetors just with respet to their pilities to solve prolems. This reltion extens oth the originl reltion n the inlusion reltion etween filure etetors. 2 Filure etetors, runs, n time In this setion we rell the efinitions of the filure etetor moel to the extent neessry. 1 We enote y Π the finite set of proesses whih onstitute the istriute system. 2.1 Filure etetor lyer Centrl to the efinition of filure etetors is isrete glol time T = IN. A filure pttern is funtion F : T 2 Π. In the se of rsh filures, p F (t) mens tht p hs rshe y time t, n so F (t) F (t + 1). We sy tht p is orret in F if p never rshes, i.e., p orret(f ) = t T Π F (t), n we only onsier filure ptterns F suh tht orret(f ). An environment E is efine s non-empty set of filure ptterns. A history H with rnge R, where R is ny non-empty set, is funtion H : Π T R. A filure etetor with rnge R is then efine s funtion tht mps eh filure pttern F E to non-empty set of histories with rnge R. Chnr n Toueg [2] introue the notion of filure etetor s mpping from filure pttern to histories in orer to pture the notion of n orle tht provies some 1 For omplete n forml efinitions, the reer is referre to the originl ppers [2, 1]. 3
hints out filures. In prtiulr, nturl hoie is R = 2 Π, n q H(p, t) for some history H of filure etetor D mens tht D tells p to suspet q to hve rshe y time t. Alterntively, the filure etetor Ω [1] hs rnge R = Π, n q = H(p, t) enoes the ft tht Ω tells p tht q is orret. The key property of Ω is tht from some time on, Ω provies the sme informtion to ll orret proesses, n this informtion is orret. The Ω property is invrint uner time trnsltion, llowing lrge egree of freeom regring time. This is not generl property shre y ll filure etetors. Inee, filure etetor my exhiit strong reltion to time: s n exmple, onsier the history H C with rnge T efine y n the onstnt filure etetor C: p Π, t T : H C (p, t) = t, F E : C(F ) = {H C }. In shrp ontrst to Ω, C gives no hint on filures ut provies perfet informtion on time. Hene the term filure etetor my e somewht misleing s in the se of C. 2.2 Asynhronous system lyer A istriute lgorithm A over Π is olletion of eterministi utomt (A p ) p Π. Computtion proees in steps of A. In eh step, proess reeives or not messge sent to it, my query its filure etetor moule tht replies some vlue to p, unergoes stte trnsition epening on its urrent stte, m, n n then sens messge to ll other proesses. We n thus ientify this step with the triple (p, m, ). A sheule of A is sequene of steps tken y proesses exeuting A; the i-th step of S is enote y S[i]. The notion of the ppliility of step to glol stte is then nturlly 2 introue n generlize to sheule. A run of A using filure etetor D is tuple F, H, I, S, T, where F is filure pttern, H D(F ) is history of D, I is n initil glol stte of A, S is sheule of A pplile to I, n T is sequene of inresing vlues in T (suh tht the i-th element of T, T [i], represents the time t whih the step S[i] ours), n F, H, I, S, n T stisfy the following onsisteny onitions: S = T n for ny i S, if S[i] = (p, m, ), then p F (T [i]) n = H(p, T [i]). In ition, R must stisfy the following firness properties: (1) eh orret proess tkes n infinite numer of steps in S, n (2) eh messge sent to orret proess is eventully reeive. The two lyers of the moel re thus onnete solely vi the onition whih speifies tht if proess p tkes step t time t, then p hnges its stte epening mong others on the vlue of D t time t, whih is efine externlly to the synhronous system. 3 Compring filure etetors Numerous ppers re evote to etermining wekest filure etetor to solve given prolem. All of them use the sme omprison reltion whih hs een introue in [2]. Roughly speking, is efine s follows: D D if there exists n lgorithm A tht trnsforms D into D. Thus, the reltion rosses the orer etween the two lyers of the moel in the sense tht two filure etetors re ompre through the synhronous 2 Ensuring onsisteny s requiring tht reeive messge hs een sent n hs not een reeive efore. 4
system lyer. In this setion, we show tht the efinition of les to some meningless inomprility results, n we stuy how to fix the prolem y extening the originl reltion. 3 We explin why trivil extensions re not stisftory, n propose new omprison reltion whih is preorer, n whose efinition oes not resort nymore to the synhronous system lyer. Let us rell wht it mens for n lgorithm A to trnsform filure etetor D into nother filure etetor D : lgorithm A uses D to mintin vrile out p t every proess p. This vrile, reflete in the lol stte of A, emultes the output of D t p. Let O R e the history of ll the out p vriles in run R, i.e., O R (p, t) is the vlue of out p t time t in run R. Algorithm A trnsforms D into D if for every run R = F, H, I, S, T of A using D, O R D (F ). If suh n lgorithm A exists, then D D. The output history O R highly epens on the time list T in R. More preisely, if D is suh tht D D for some D, then D neessrily llows finite ut unoune stuttering: if proess p oes not tke step t time t in run R, then O R (p, t) = O R (p, t 1). This onition on D to e weker thn some filure etetor D strongly restrits the grph of the reltion, n in prtiulr it prevents to e reflexive, ontrry to wht n e inferre from the nottion. 3.1 Extening inlusion We oul esily fix the prolem of non-reflexivity y onsiering the reflexive losure of. Unfortuntely, the resulting preorer is still too restritive s we explin elow. Consier the nturl reltion etween filure etetors: D D F E : D(F ) D (F ). In other wors, D provies itionl possile histories ompre to D, n so is less urte thn D. Hene, nturl requirement for omprison reltion over filure etetors is to e preorer tht extens. Unfortuntely, this is the se neither for nor for its reflexive losure. To see tht, let us efine the instntneous strong ompleteness whih ensures tht every rshe proess is immeitely suspete y every orret proess: F E, H D(F ), t T, p orret(f ), q Π : q F (t) q H(p, t), n onsier the two filure etetors P + n S + tht oth stisfy instntneous strong ompleteness, ut iffer on their ury properties: P + stisfies strong ury [2], i.e., F E, H D(F ), t T, p, q Π F (t) : q H(p, t), wheres S + stisfies only wek ury [2] F E, H D(F ), p orret(f ), t T, q Π F (t) : p H(q, t). Then one n esily show tht: Proposition 3.1. P + S +, ut P + S +. 3.2 Time ontrtion To fix the ove prolem, we n onsier the reltion efine s the union of n. Clerly, the reltion is preorer, ut it ppers to e still too restritive in the sense tht it oes not relte filure etetors tht shoul e omprle from the stnpoint of synhronous systems, nmely, filure etetors tht only iffer in their reltion to time. 3 In orer to preserve ll the wekest filure etetor results, new omprison reltion must ontin. 5
H(p) H(q) H(r) t 2 3 6 11 t Θ.H(p) Θ.H(q) Θ.H(r) Figure 1: The history Θ.H with rnge {,,, } n the sequene Θ = 2, 3, 6, 11, To see tht, for ny filure etetor D, we onstrut new filure etetor D similr to D exept the oupling to time tht is weker in D thn in D. More preisely, eh history of D is otine y smpling history H of D, i.e., tking t inresing times the vlues from H while utting out the remining vlues (see Fig. 1). In other wors, D is time-ontrte vrint of D, where the ontrtion is homogeneous etween proesses. Let Θ = (Θ t ) t IN e n inresing sequene in T, n S e the set of ll suh sequenes. The ontrte vrints of history H n filure pttern F with respet to some Θ S re efine y: Θ.H(p, t) = H(p, Θ t ) n Θ.F (t) = F (Θ t ). Beuse filure ptterns re non-eresing funtions of time, orret(f ) = orret(θ.f ). In the following, we ssume tht E is suh tht eh Θ-ontrtion F Θ.F is surjetion from E to E. 4 Then, for every Θ S, we onstrut the filure etetor Θ.D whih is otine y the Θ-ontrtion of filure ptterns n orresponing histories of D: Θ.D(F ) = { Θ.H : F E, F = Θ.F H D(F ) }. We now efine D(F ) s the union of ll possile Θ-ontrtions of the histories in D(F ): D(F ) = Θ S Θ.D(F ) Speilizing Θ t = t, we otin D D, n so D D. In generl the onverse inlusion oes not hol, i.e., D n D my e not equivlent with respet to. Proposition 3.2. C C Proof. Let Θ S e suh tht t IN : Θ t > t. Thus, for ny F E there is history H C(F ) suh tht p Π : H(p, t) = Θ t > t. Suh history H C(F ) is not in C(F ), n thus C C. Moreover, C oes not llow stuttering, n so C C. It follows tht C C. However, we now show tht D n D re lwys equivlent from the viewpoint of the synhronous system lyer in the sense tht ny synhronous lgorithm nnot istinguish D from D. In prtiulr, we prove tht D n D hve the sme ility to solve (non rel-time) prolems in n synhronous system. 4 This ssumption hols in the se of the lssil environment where t most f proesses my rsh. 6
Let V in n V out e two non-empty sets suh tht V out n / V in. An initil onfigurtion is funtion C 0 : Π V in, n n output smpling is funtion Σ : Π IN V out. 5 A prolem P is preite over triples (F, C 0, Σ), where F E, C 0 is n initil onfigurtion, n Σ is n output smpling. To remove ny time epeneny, we only onsier prolems suh tht F, F E : orret(f ) = orret(f ) P (F, C 0, Σ) = P (F, C 0, Σ). Let A = (A p ) p Π e n lgorithm with the sets of sttes n of initil sttes enote y Sttes p n Sttes 0 p, respetively. We sy tht A solves P if there exist two mppings σ 0 : Sttes 0 p V in n σ : Sttes p V out suh tht for ny run R = F, H, I, S, T of A, P hols t (F, σ 0 (I), σ(i, S)), where σ 0 (I) is the initil onfigurtion orresponing to I y σ 0 n σ(i, S) is the output smpling tht nturlly erives from I n S vi σ. If D D n A solves P using D, then A solves P using D sine eh run of n lgorithm A using D is lso run of A using D. In prtiulr, eh run of A using D is lso run of A using D. Conversely, run of A using D my e not run of A using D. However, for eh run R of A using D there is run R of A using D with the sme initil stte n the sme sheule s in R, n with equivlent filure ptterns: Proposition 3.3. For every run R = F, H, I, S, T of n lgorithm A using D there is run R = F, H, I, S, T of A using D suh tht orret(f ) = orret(f ). Proof. For every history H D(F ) there is some Θ S suh tht H Θ.D(F ). It follows tht there exists some filure pttern F E n some history H D(F ) suh tht F = Θ.F n H = Θ.H. Let T enote the inresing time vlues efine y i IN : T [i] = Θ T [i]. We re going to show tht R = F, H, I, S, T is run of A using D. We know tht F E, n I is n initil stte of A. We enote y S[i] = (p, m, ) the i-th step tken in S. Sine R is run of A, = H(p, T [i]). By efinition of H n T, we hve = H(p, T [i]) = Θ.H (p, T [i]) = H (p, Θ T [i] ) = H (p, T [i]). Hene, = H (p, T [i]). Furthermore, y efinition of F n T, we eue tht F (T [i]) = F (Θ T [i] ) = F (T [i]). It follows tht if p tkes the i-th step in S, then p / F (T [i]) s S is the sheule of run R, n thus p / F (T [i]), s neee. Finlly, we know tht eh messge sent in R to proess in orret(f ) is eventully reeive, n eh orret proess in F tkes n infinite numer of steps in R sine R is run. As F n F re equivlent, R lso stisfies these two firness properties. Consequently, R is run of A using D. Theorem 3.4. An lgorithm A solves prolem P using D iff A solves P using D. 5 We use IN inste of T to insist on the ft tht Σ(p, i) enotes the i-th output vlue n not the output vlue t time i T. 7
3.3 Time regine Hene, D n D re equivlent in the synhronous setting s they n e use to solve prolem with the sme lgorithm. However, Proposition 3.2 shows tht there re filure etetors D suh tht D D. We propose omprison reltion tht keeps internl to the filure etetor lyer, n thus oes not erse the possile timing informtion tht filure etetors n ontin. This new reltion ompres filure etetors just with respet to their pilities to solve prolems. Formlly, D s D if ny prolem solvle using D is lso solvle using D. Clerly, the reltion s extens, n s is reflexive. Moreover, s is trnsitive, n so it is preorer. As lime in [2], if D D then ny prolem solvle with D is lso solvle with D, i.e., s extens the originl reltion. Finlly, Theorem 3.4 shows tht ny filure etetor D is equivlent to D with respet to the reltion s. 4 On wekest filure etetors We re now in position to explin the pprent prox etween the positive results of [4, 7] n the impossiility result in [1]. At some point, we will give only intuitive rguments, s we o not wnt to go into the etils of the forml hrteriztion of eventul forever properties in temporl logi. Let R e non-empty set n let e the set of filure etetors with rnge R. Given some prolem P, we enote y P the suset of filure etetors in tht n e use to solve P. Let e ny nturl preorer to ompre filure etetors in, i.e., s extens whih in turn extens. Rell tht W P is wekest filure etetor to solve P with respet to if W P is wekest element of ( P, ), i.e., (1) W P P, n (2) for ny D in P, D W P. We now show tht W P, if it exists, is neessrily time-free filure etetor. Theorem 4.1. If there exists wekest filure etetor W P in to solve P, then W P is lso wekest filure etetor to solve P, n W P is equivlent to W P. Proof. Theorem 3.4 shows tht W P P, n so W P W P. Sine W P W P n extens, it follows tht W P W P. Hene, W P n W P re equivlent. Moreover for ny D in P, we hve D W P, n y trnsitivity D W P. This shows tht W P is wekest element in ( P, ). For non-empty time intervl τ, we efine φ to e onition on the olletions of type (F t, (D p,t ) p Π ) where for ny t τ n ny p Π, we hve F t Π n D p,t R. We sy tht time intervl τ is φ-goo perio for history H n filure pttern F if φ is stisfie y (F (t), (H(p, t)) p Π ) t τ. For instne, φ-goo perio my express tht the sme orret proess is truste y ll proesses. Let P e prolem for whih there is wekest filure etetor W P to solve P tht ensures φ-goo perios in eh of its histories n tkes ritrry vlues outsie φ-goo perios. Formlly, for eh filure pttern F n ny history H W P (F ), () there is t lest one φ-goo perio for H n F, n () ny history H with rnge R, tht oinies t τ 8
Θ.H D t H D φ φ t Figure 2: Finite goo perios for H my le to no goo perio for Θ.H. with H outsie the φ-goo perios, is lso in W P (F ). Hene φ-goo perios re neessry n suffiient in eh history for solving P. To seek ontrition, suppose now tht φ-goo perios hve only finite urtion in some history H 0 W P (F 0 ). Let D 0 e the filure etetor in whih oinies with W P for ny filure pttern F F 0, n D 0 (F 0 ) = {H : p Π, t T, H(p, t) R}. Sine W P onsists of ll smplings of the histories of W P, it follows tht there is some history H W P (F 0 ) without ny φ-goo perio (f. Fig 2). Even, H my e totlly ritrry euse of property () of W P. Hene, D 0 W P. Sine extens, it follows tht D 0 W P. Moreover, Theorem 4.1 shows tht W P P. As s extens, we erive tht D 0 P. This shows tht for F 0, φ-goo perio is not neessry to solve P, ontrition. Therey, W P ensures infinite φ-goo perios in eh of its histories. In onlusion, φ-goo perios must hve infinite urtion in orer to e useful to solve prolem. In other wors, φ must eventully hol forever, n this forever requirement stems solely from the moeling n the eomposition of the moel into two lyers. 5 Disussions In this pper we pointe out the restritive nture of the lssil omprison reltion etween filure etetors, n we expline why in the filure etetor moel, Consensus nnot e solve without n eventully forever greement on proess tht will never rsh, non-relisti requirement in prtie, wheres there re Consensus lgorithms tht run orretly in rel systems. Both moeling prolems originte from the interfe etween the filure etetor lyer n the synhronous system lyer, n from the lk of timing ontrol y filure etetors on the synhronous system: the time-ontrte version of ny filure etetor D is equivlent to D from the viewpoint of the synhronous system. For ompring filure etetors, we propose new reltion whose efinition keeps internl to the filure etetor lyer, n thus preserves the possile timing informtion ontine in filure etetors. On the ontrry, there is no hope to weken the properties on filure etetors neee for solving Consensus sine the neessity of n eventully forever greement is inherent to the nture of the moel. This is the reson why we elieve tht the filure etetor moel, n more generlly ny moel of synhronous systems ugmente with externlly efine orles, is not stisftory s it les to lower oun results tht re invlite y the experiene of DLS or Pxos lgorithms in rel systems. In this respet, system moels whose properties re efine internlly, i.e., in terms of system trnsitions, pper more pproprite. 9
Jynti n Toueg [6] reently estlishe tht every prolem hs wekest filure etetor with respet to yet nother reltion. While they mentione tht the lssi reltion in [2] is not reflexive, they i not isuss the prox highlighte in here. Their new reltion is reflexive n extens the inlusion reltion. As our reltion s, this new reltion lso seems to extens the reltion of [2]. Unfortuntely, Jynti n Toueg [6] employe new moel, ifferent from the lssil filure etetor moel. Therefore it is elite to estlish preise n forml omprison of these two ifferent pprohes of repiring the originl reltion. Note tht the new moel in [6] is still struture in severl lyers efine seprtely, n thus oes not espe the prox. Referenes [1] Tushr Deepk Chnr, Vssos Hzilos, n Sm Toueg. The wekest filure etetor for solving onsensus. Journl of the ACM, 43(4):685 722, June 1996. [2] Tushr Deepk Chnr n Sm Toueg. Unrelile filure etetors for relile istriute systems. Journl of the ACM, 43(2):225 267, Mrh 1996. [3] Dnny Dolev, Cynthi Dwork, n Lrry Stokmeyer. On the miniml synhronism neee for istriute onsensus. Journl of the ACM, 34(1):77 97, Jnury 1987. [4] Cynthi Dwork, Nny Lynh, n Lrry Stokmeyer. Consensus in the presene of prtil synhrony. Journl of the ACM, 35(2):288 323, April 1988. [5] Mihel J. Fisher, Nny A. Lynh, n M. S. Pterson. Impossiility of istriute onsensus with one fulty proess. Journl of the ACM, 32(2):374 382, April 1985. [6] Prs Jynti n Sm Toueg. Every prolem hs wekest filure etetor. In Proeeings of the 27th ACM symposium on Priniples of Distriute Computing (PODC), pges 75 84, 2008. [7] Leslie Lmport. The prt-time prliment. ACM Trnstions on Computer Systems, 16(2):133 169, My 1998. 10