1. © 2007 P. Kouznetsov On the Weakest Failure Detector Ever Petr Kouznetsov (Max Planck Institute for SWS) Joint work with: Rachid Guerraoui (EPFL) Maurice Herlihy (Brown) Nancy Lynch (MIT) Calvin Newport (MIT)

2. 2 © 2007 P. Kouznetsov Big picture Choosing a model:  Optimistic model: the system is very efficient but likely to fail  Conservative model: the system is very robust but inefficient (or impossible to implement) What is the right model?

3. 3 © 2007 P. Kouznetsov Synchrony assumptions  Asynchronous read-write shared memory model: no bounds on relative processing speed  Very appealing in practice!  Too conservative: most problems are not solvable [ FLP85, LA87; HS,SZ,BG93]; (solvable in synchronous systems though)

4. 4 © 2007 P. Kouznetsov So what do we need exactly? What is the minimal amount of synchrony that circumvents some asynchronous impossibility?  “minimal amount of synchrony”? - The weakest failure detector

5. 5 © 2007 P. Kouznetsov Model p FD qr Asynchronous read-write shared-memory system with failure detectors

6. 6 © 2007 P. Kouznetsov Comparing failure detectors Failure detector D is weaker than failure detector D ’ if there exists an algorithm that emulates D using D ’ p D’D qr DD

7. 7 © 2007 P. Kouznetsov The weakest non-trivial failure detector A failure detectors X that is: non-trivial: circumvents some asynchronous impossibility weaker than any non-trivial failure detector The “easiest” non-trivial problem?

8. 8 © 2007 P. Kouznetsov A Very Weak Failure Detector Y outputs a non-empty set of process ids Eventually, the same set U is output at every correct process: U is not the current set of correct processes Example:  Π={p,q,r}, C={p,q}  Y outputs {p},{q},{p,r},{q,r},{p,q,r}

9. 9 © 2007 P. Kouznetsov Y is non-trivial Theorem 1 Y solves (N-1)-set agreement Every process in P 1,…,P N proposes a value and must decide on some proposed value so that:  At most N-1 distinct values are decided (!) not solvable in asynchronous systems [HS93,BG93,SZ93]

10. 10 © 2007 P. Kouznetsov Set agreement is almost solvable  If N-1 or less distinct values are proposed, e.g., if N-1 or less processes participate k-convergence [YNG98]  Y should handle the case when N values are around

11. 11 © 2007 P. Kouznetsov Citizens and gladiators  Split the system into Gladiators (the stable output of Y) and Citizens (all the rest)  Gladiators eliminate at least one value using (G-1)-convergence or adopt a value from Citizens Y Π-YΠ-Y

12. 12 © 2007 P. Kouznetsov Correctness Eventually, Gladiators are not the set of correct processes ⇨ At least one gladiator is faulty, or at least one Citizen is correct ⇨ Gladiators commit on G-1 values or adopt a value from a citizen ⇨ At least one process gives up its value ⇨ at most N-1 values survive!

13. 13 © 2007 P. Kouznetsov Y is minimal Theorem 2 Y is weaker than any stable non- trivial failure detector D D is stable if, eventually, the same value is permanently output at every correct process (e.g., P, ⃟P, Ω, Ω k )

14. 14 © 2007 P. Kouznetsov Minimality proof: toy example Consider a “faithful” failure detector D that solves a wait-free impossible problem P: in every execution E, D outputs the same value v that depends only on correct(E) Claim 1 For all v, there is a non-empty set of processes C such that v cannot be output by D when C is the set of correct processes Suppose not: v is valid for any C => D can be replaced with a “dummy” that always outputs v --- a contradiction!

15. 15 © 2007 P. Kouznetsov Minimality proof: general case Consider any non-trivial stable D Claim 2 For all v, there exists an infinite execution E in which v cannot be the only value output by D Reduction:  As long as D is stable on v: use E(v) to extract Y

16. 16 © 2007 P. Kouznetsov Conclusions  Y is the weakest non-trivial stable failure detector (can be generalized to the f-resilient case – Y f )  (N-1)-set agreement is the easiest non-trivial problem?

17. 17 © 2007 P. Kouznetsov Future  Establishing the “weakest ever” result in the most general class of failure detectors (not Y!) Y is not the weakest: an unstable “composition” of Ω n and Y is even weaker! [Chen et al., Zielinski, …]

18. 18 © 2007 P. Kouznetsov Thank you!

19. 19 © 2007 P. Kouznetsov k-convergence [YNG98] Processes propose values and commit on or adopt one of the proposed values:  If a process commits, then at most k values are committed or adopted  If k or less values proposed, every process commits (!) wait-free solvable for any k (!!) (N-1)-convergence almost solves (N-1)-agreement! But termination is an issue in case all N values are around – that’s where Y is of use!

20. 20 © 2007 P. Kouznetsov Minimality proof: general case Consider any non-trivial stable D Claim 2 For all v, there exists an infinite execution E in which v cannot be the only value output by D Reduction:  As long as D is stable: Locate a faulty process in a finite prefix of E (including all steps of faulty(E) ) Or, output correct(E)  Y is extracted!

21. 21 © 2007 P. Kouznetsov Generalization to f-resilience f-resilient impossible problems: can be solved when less or f fail but cannot when f fail  Y f output a set of size ≥N-f  Eventually, the same set U is permanently output at every correct process Y f is the weakest stable failure detector to circumvent an f-resilient impossibility

22. 22 © 2007 P. Kouznetsov Big picture Addressing the WFD question contributes to:  Understanding complexity and computability bounds of distributed abstractions  Establishing a clean classification of problems in distributed computing “WFD ever” corresponds to the easiest non-trivial problem in distributed computing