Causality

1. Causality

2. The “happens before” relation happens before (causes)

3. A message is sent from to

4. transitivity and

6. “happens before” is a partial order Parallel events:

7. “happens before” is a partial order Parallel events:

8. We can move parallel events The happens before relation doesn’t change

9. We want to find a mechanism that captures the “happens before” relationship so that we use causality in various computation problems

10. Logical Clocks 0 1 2 3 4 5 0 2 3 4 1 In each process, the logical clock increases by 1 with each local event

11. Logical clocks are piggy-packed on messages 0 1 2 5 4 0 1 2 3 4

12. 1 2 3 7 8 2 4 5 6 1 2 3 4 5 6 7 Logical clocks, seem that they capture the happens before relation

13. 1 2 3 7 8 2 4 5 6 1 2 3 4 5 6 7 Example:

14. 1 2 3 7 8 2 4 5 6 1 2 3 4 5 6 7 However, logical clocks cannot capture parallelism Parallel events

15. We need another mechanism that can capture the parallelism of events

16. Vector Clocks Process entry Process entry

17. Each process increases its entry at each event increment

18. Each process increases its entry at each event increment

19. vector clocks are piggy-packed on messages The maximum of each entry

20. vector clocks are piggy-packed on messages

21. vector clocks are piggy-packed on messages

23. Comparison of vector clocks We write if for all

24. Examples:

25. Incomparable vector clocks We write If neither nor

26. Examples:

27. Vector clocks capture causality If then If then

28. By examining the vector clocks we can determine the order of events If then If then

33. Cuts

34. 2 1 3 4 5 1 2 3 4 1 2 3 4 5 6 7

35. 2 1 3 4 5 1 2 3 4 1 2 3 4 5 6 7 Cut: consists from an event from each process

36. 2 1 3 4 5 1 2 3 4 1 2 3 4 5 6 7 no messages cross the cut Consistent cut:

37. 2 1 3 4 5 1 2 3 4 1 2 3 4 5 6 7 messages can cross from left to right of the cut Consistent cut:

38. 2 1 3 4 5 1 2 3 4 1 2 3 4 5 6 7 messages cross from right to left of the cut Inconsistent cut:

39. 2 1 3 4 5 1 2 3 4 1 2 3 4 5 6 7 messages cross from right to left of the cut Inconsistent cut:

40. Maximal Consistent Cut Consider some (inconsistent) cut Maximal Consistent Cut of : A consistent cut such that and contains most recent events

41. 2 1 3 4 5 1 2 3 4 1 2 3 4 5 6 7 (inconsistent cut)

42. 2 1 3 4 5 1 2 3 4 1 2 3 4 5 6 7 maximal consistent cut (inconsistent cut)

43. Theorem: For every cut , there is a unique maximal consistent cut Proof: Proof by contradiction

44. Assume for contradiction there are two (or more) maximal cuts of maximal consistent cuts

45. It cannot be that and don’t cross not maximal!

46. and cross

47. and cross new cut

48. and cross impossible since is consistent

49. and cross impossible since is consistent

50. and cross Contradiction! not maximal! not maximal! is consistent End of proof