Erez Druk Seminar in Distributed Algorithms, 10/6/13

1. Linearizable Implementations Do Not Suffice For Randomized Distributed ComputationWojciechGolab, Lisa Higham, Philipp Woelfel ErezDruk Seminar in Distributed Algorithms, 10/6/13

2. Overview • Motivation • Preliminaries • Main Theorem • Good to Know • Conclusion • Questions

3. Motivation • Linearizability • Simplicity • Correctness • Intuitive • Local & Composable • Useful

4. Motivation • Randomization • Simplicity • Achieves the impossible • Natural • Cool

5. The Big Question • Can we combine the two? NO!

6. An Example • is a register initialized to • An adversary tries to minimize the value that reads

7. An Example • When is atomic • The expected value of ’s is at least

8. An Example • When is implemented via atomic bits [Vid88] 1 1 0 0 0 0

9. An Example • When is implemented via atomic bits [Vid88] 0 1 1 1 1 1 0 0 0 0 0 0/1 Start reading Finish reading • The expected value of ’s is

10. Another Example • have access to a shared snapshot object • An adversary tries to minimize the some of values that observes

11. Another Example • Atmoic • The expected value of ’s is

12. Another Example • Linearizableimplementation [AADGMS93] • Wait for two identical collects, or • Burrow from a process which updated twice • Scan • Write the whole snapshot

13. Another Example • Linearizable implementation [AADGMS93] • The expected value of ’s is

14. Some definitions • A coin flip vector is an infinite binary vector • An adversary is a function from past coin flips to the set of processes • An history is a sequence of steps of processes • An history is sequential if it is sequential • An history is linearizable if there is a sequential history that agrees with it • denotes the prefix-closure of a set of histories • is the history formed by running algorithm with adversary and coin flips vector

15. Some definitions • and are equivalent if for any there exists a sequential history that is a linearization of both and • For convenience we denote

16. Goal • Fix the following theorem • Let be an algorithm that uses atomic objects and be obtained from by replacing some objects with linearizable implementations. Then for any there exists such that and are equivalent

17. Strong Linearizability • A set of histories is strongly linearizable if there exists a function mapping of histories in to sequential histories such that • is a linearization of • is prefix preserving • A shared object is strongly linearizable if its set of histories is strongly linearizable • From here on is obtained from by replacing some object with their strongly linearizable implementation

18. Main Theorem • For every adversary there exists an adversary such that and are equivalent.

19. For every adversary there exists an adversary such that and are equivalent. Main Theorem- Proof • Need to find such that and are simultaneously linearizable • is already sequential • How can we guarantee that is a linearization of ? (hint: use the assumptions!) • Let • Need such that • Define to be the schedule of

20. For every adversary there exists an adversary such that and are equivalent. Main Theorem- Proof : :

21. For every adversary there exists an adversary such that and are equivalent. Main Theorem- Proof • Done? • What is ? • Locality – A set of strongly linearizable objects is strongly linearizable

22. Locality Lemma • Highly Technical • Induction on • are the histories of length at most • is good for

23. Locality Lemma - from With : : With : Define

24. For every adversary there exists an adversary such that and are equivalent. Main Theorem- Proof • Done? • is not allowed to cheat

25. For every adversary there exists an adversary such that and are equivalent. Main Theorem- Proof : : : :

26. For every adversary there exists an adversary such that and are equivalent. Main Theorem- Proof • Fix • Move coin flips to the earliest point possible : : : :

27. For every adversary there exists an adversary such that and are equivalent. Main Theorem- Proof • is good if is used • Done? Yes!

28. Strong Linearizability is Necessary • If and are equivalent then is strongly linearizable

29. Necessity • If and are equivalent then is strongly linearizable • Should linearize • and have the same linearization • is already linearized • Define

30. Necessity • If and are equivalent then is strongly linearizable • Done? • must map all : : : :

31. Necessity • If and are equivalent then is strongly linearizable • Done? • depends on the choice of • Lemma: No it doesn’t • The theorem follows

32. Good to Know • Composability • Weak Adversaries • Feasibility • Proof techniques

33. Conclusion • Linearizability & Randomization • Examples • Strong Linearizability • Sufficient • Necessary • Local & Composable • Feasible

34. (Easy) Questions?

35. Thanks For Listening!