Knowledge and Common Knowledge in a Byzantine Environment: Crash Failures

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. " 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.