1. Failure Detectors motivation failure detector properties
4/28/2019
Failure Detectors
motivation
failure detector properties
failure detector classes
detector reduction
equivalence between classes
consensus
solving with S
solving with S
corollaries and other results

2. 4/28/2019
Why Failure Detectors
consensus in asynchronous systems is impossible even if a single process crashes
(pure) asynchronous systems are not useful for fault tolerance studies
asynchronous system is a generic model for reasoning about distributed algorithms
how can asynchronous systems be augmented to enable consensus?

3. Notation T – state numbers in a computation (logical clock ticks)
4/28/2019
Notation
T – state numbers in a computation (logical clock ticks)
failure pattern is a function F(t) that denotes the set of processes that have crashed so far
F: T2P
F is monotonic: (p  F(t))  (p  F(t' > t))
crashed(F) are the processes that crash at some time correct(F) = P - crashed(F)
once the process crashes it does not recover
failure detector is a module of a process that outputs the set of processes that it currently suspects to have crashed
failure detector history H is the output of a failure detector
H: P  T  2P
H(p, t) is the set of processes that p suspects at time t.
q  H(p, t) means "p suspects q at time t".
failure detector D maps F to a set of H.

4. Failure Detector Properties
4/28/2019
Failure Detector Properties
completeness
strong – every process that never crashes eventually suspects every process that does crash
F, HD(F),tT, pcrashed(F), qcorrect(F), t'  t: p  H(q, t')
weak – some process that never crashes eventually suspects every process that does crash
F, HD(F),tT, pcrashed(F), qcorrect(F), t'  t: p  H(q, t')
(perpetual) accuracy
strong – no process is suspected before it crashes
F, HD(F),tT, p, qP-F(t): p  H(q, t)
weak – some correct process is never suspected
F, HD(F),pcorrect(F), tT, qP-F(t): p  H(q, t)
eventual accuracy
eventual versions of (weak and strong) accuracy require that the property holds only eventually
ex: eventual strong accuracy:
F, HD(F),tT, t’>t, p, qP-F(t): p  H(q, t’)

5. Failure Detector Classes
4/28/2019
Failure Detector Classes
the properties define eight detector classes

6. Detector Reduction reduction algorithm TDD’ transforms D into D’
4/28/2019
Detector Reduction
reduction algorithm TDD’ transforms D into D’
T uses D to maintain variable outputp for every process p
every history TDD’ of is a history of D’
if algorithm A requires D’, but only D is available, A can use TDD’
if exists TDD’ – D provides at least as much info as D’
D’ is weaker than D
D’ is reducible to D
D  D’
reducibility relation is transitive
if D>D’ and D’>D then D  D’: D and D’ are equivalent
reducibility and equivalence applies to classes of detectors as well

7. Relation between Weak and Strong Completeness
4/28/2019
Relation between Weak and Strong Completeness
observe that strongly complete detectors trivially emulate weak, thus P  Q, S  W, P  Q, S  W
however, weakly complete detectors can also emulate strong ones
in the algorithm TDD’ each process broadcasts the list of suspects
TDD’
transforms weak into strong completeness
preserves perpetual (weak and strong) accuracy
preserves eventual (weak and strong) accuracy
Thus,
P  Q, S  W, P  Q, S  W
need to consider only strongly complete detectors
need to implement only weekly complete deterctors

8. Consensus with Failure Detectors
4/28/2019
Consensus with Failure Detectors
primitives at each process
propose(v) propose a value v for consensus
decide(v) decide on a consensus value v
properties
termination – each correct process eventually decides on a value
uniform integrity – each process decides at most once
agreement – no two correct processes decide differently
uniform agreement – no two (correct or faulty) processes decide differently
uniform validity – if a process decides on v, then some process proposed v

9. Solving Consensus Using S
4/28/2019
Solving Consensus Using S
tolerates up to n-1 crashes, satisfies uniform agreement
three phases
first – n-1 rounds of disseminating each process’ value
second – processes agreeing on the vector of values
correctness proof
c – correct process that is never suspected
Theorem: the algorithm solves consensus using S

10. Solving Consen-sus using S
4/28/2019
Solving Consen-sus using S
assumptions
majority of processes are correct
each process knows the id of coordinator at round r
Theorem: the algorithm solves consensus using S

11. Corollaries and Other Results
4/28/2019
Corollaries and Other Results
from detector classes equivalence
consensus is solvable with W with up to n-1 crashes
consensus is solvable using W with less than n/2 crashes
other results
consensus is not solvable even with P with if maximum number of crashes is at least n/2 crashes
W is the weakest failure detector to solve consensus with less than n/2 crashes
that is for any detector D, there exists TDW