Termination Detection

1. Termination Detection Presented by: Yonni Edelist

2. The model • A finite graph of machines (nodes) and communication channels (edges), in which a distributed computation is taking place • Machines can be either ‘active’ or ‘passive’ • Only active machines may send messages • Passive machine may become active only when it receives a message • Active machine may become passive at any time

3. The model- example 0 2 3 1 6 5 4

4. The problem • If, in the computation, a finite number of messages are sent, the graph will reach a stable state in which no more messages are sent • One machine, say machine no. 0, wants to detect that the graph has reached that state • We will build an algorithm which enables machine 0 to detect termination

5. example • We denote messages sent by the termination detection algorithm ‘signals’ 0 2 3 1 6 5 4

6. Termination detection in a ring

7. The model • N machines, 0..N-1 • Machine 0 initiates the detection algorithm • Does so by sending a signal (a token) to machine 1 • Machine i can propagate the token to machine i+1 • We denote the machine currently holding the token ‘t’

8. System Invariant • We describe an invariant, which will be true for every configuration of the system (for every value of t) • Machine 0 will know that all other machines are passive based on: • t=0 • The invariant • (possibly) further information available at machine 0

9. First Invariant- P0 • First step- we assume that there are no messages • Meaning: an active node may become passive, but not vice-versa • P0: (∀ i: 0≤i<t machine i is passive)

10. P0- Example 0 3 1 2 Red outline represents an active machine

11. Rule 0 • In order to preserve the invariant, we must follow the rule: • Rule 0: Machine i transmits the token only if it’s passive • (remember that a passive machine can’t become active again)

12. Rule 0- Example 0 3 1 2

13. But what if a message is sent? 0 3 1 2

14. Invariant P1 • If machine i<t receives a message, it becomes active, thus falsifying P0 • Message must have been sent by machine j≥t • P1 is established whenever P0 is falsified in the above manner P0: (∀ i: 0≤i<t machine i is passive)

15. Invariant P1 (cont.) • A machine can be either black or white • P1: (∃ j: t ≤ j ≤ N: machine j is black) • Rule 1: When machine j sends a message to machine i<j, it becomes black

16. Rule 1- Example 0 3 1 2

17. Invariant P1 (cont.) • Now P= P0 ∨ P1 is not falsified when a message is sent to i < j. • Machine 0 knows: • (t=0 ∧ machine 0 is white)  ¬P1 • P0: (∀ i: 0≤i<t machine i is passive) • Therefore- machine 0 can still detect termination

18. But… 0 3 1 2

19. Invariant P2 • P1 may be falsified when a black machine hands the token over • We want to ensure that every token propagation that falsifies P1 establishes P2 • We define that the token may be either black or white

20. Invariant P2 (cont) • P2: the token is black • And the accompanying rule: • Rule 2: A black machine transmits a black token. A white machine doesn’t change the colour of the token

21. Invariant P2 (cont) • Machine 0 knows: • (t=0 ∧ machine 0 is white)  ¬P1 - Token is white  ¬P2 • P0: (∀ i: 0≤i<t machine i is passive) • Therefore- machine 0 can still detect termination

22. Invariant P2- Example 0 3 1 2

23. Rule 3 • The algorithm may finish unsuccessfully • Rule 3: after an unsuccessful round, machine 0 initiates another round

24. But… 0 3 1 2

25. Rule 4 • Since a round that begun with a black token is doomed, we adopt: • Rule 4: at the beginning of a new round, machine 0 makes itself white and transmits a white token

26. Still not good enough 0 3 1 2

27. Rule 5 • Whitening a black machine i may falsify only P1, and only in the case i > t • Rule 5: when transmitting the token to machine i+1, machine i makes itself white P1: (∃ j: t ≤ j ≤ N: machine j is black)

28. Final example 0 3 1 2