Knowledge and common knowledge in a Byzantine environment: Crash failures

Abstract

By anaizng the stares of novledge Li 'he processors attain 19 an urre as.e SYS Tern of a sImp.e tYpe. we capture some o: :rs;c uxder!ying struc t ire of such svsams. particu ar. we study s hat facts become 'on mu kit 0 uiedgc at var:ous ponts in the execu ion of protoco!s n an un reliable system. This characterizes the sirnu Itaneous ac ions hat can he carHed 0:': in s:ch systems. For et ne. we obtain a ccco!e:e charac erizat 0:1 of the nurnSer of ro-iLds rccuired o reach ." ''uUu,eou.s By:anne ,Lq€rr:ent. giver. the pa: tert in hich fa:ares 0CC U. From ri we derive a new protcico: for th: p rbern t ha is optimal in a(l ru7. rarher than just alavs matching the worst-case lower bound. Er some cases this pro? oco a, ains S rnulta ii togs Rvza nt me Agreenient In as few as 2 rounds. e also present a non-trivial simu [tarieou agreement problem called b ualerfl agreement for which there is a protocol that aiways halts in two rounds. Our analysis applies to simultaneous actions in general, and not just to Byzantine agreement. The lower bound proofs presented here generalize and siniplifv ihe previously known proofs.

Uniform consensus is harder than consensus

“…However, the lower bound in Theorem 4.1 and the worst case lower bound t + 1 as well, do not hold anymore when considering the very weak validity condition in [14] that only stipulates that there are at least two possible decision values. Indeed, Dwork and Moses [12] devised a two rounds synchronous algorithm, which solves this very weak agreement problem. Coming back to the proof of Theorem 4.1, we observe that this is due to Lemma 4.2 which is no more true for this latter agreement problem.…”

Common Knowledge and State-Dependent Equilibria

“…The role of common knowledge has been studied in the fields of distributed computing and artificial intelligence [11,7,12]. This line of work suggests that knowledge is an important abstraction for distributed systems and for the design and analysis of distributed protocols, in particular for achieving consistent simultaneous actions.…”

Revisiting simultaneous consensus with crash failures

“…Let C r] be the set of the processes that are seen as crashed by (at least) one of the processes that survive (i.e., do not crash before the end of) round r. For any r, let d r = max(0 jC r]j ; r); as we shall see, d r represents the number of rounds that could be saved with respect to the worst case t + 1 bound, thanks to the failures that occurred and were seen by at least one process that terminate round r. Let D = m a x r 0 (d r ); D represents the best saving in terms of rounds. Notice that the values of d r and D in a given run are determined only by its failure pattern F. It is shown in [3] that the smallest round number at the end of which a common decision can be simultaneously taken is RS t F = ( t + 1 ) ; D (where D = D(F)). The quantity D is considered in [3] to be the waste inherent in the failure pattern F, since it specifies the number of rounds the adversary has "lost" in its quest to delay decision for as long as possible.…”

“…This intuition is formalized in [3] as follows. Let C r] be the set of the processes that are seen as crashed by (at least) one of the processes that survive (i.e., do not crash before the end of) round r. For any r, let d r = max(0 jC r]j ; r); as we shall see, d r represents the number of rounds that could be saved with respect to the worst case t + 1 bound, thanks to the failures that occurred and were seen by at least one process that terminate round r. Let D = m a x r 0 (d r ); D represents the best saving in terms of rounds.…”

“…The previous question can now be translated as follows: "Which are the failure patterns that force a simultaneous decision consensus algorithm to decide in k rounds for 2 k t + 1 ". This question was posed and answered in [3] where a bound is stated and proved (this bound is denoted RS t F in the following; F stands for the failure pattern that occurs in the considered run).…”