1. Termination Detection Algorithm for Distributed Computations Edsger W. Dijkstra W.H.J. Feijen A.J.M. van Gasteren Presented by : Charu Jain

2. Presentation and Derivation of the Algorithm System Description System Description N machines N machines Active or Passive Active or Passive Stable State Stable State

3. Initiation Machine nr.0 initiates the “probe” Machine nr.0 initiates the “probe” Ring-Minimize the signaling traffic Ring-Minimize the signaling traffic Assumption about passivity Assumption about passivity

4. Propagation Propagation of the probe Propagation of the probe System State - P System State - P Probe ends with t=0 Probe ends with t=0

5. Stipulation for concluding termination Termination Termination Invariant Invariant t=0 t=0 Info at nr.0 Info at nr.0 Invariant has to hold independent of the distribution of the activity Invariant has to hold independent of the distribution of the activity

6. Absence of Messages P0 : (A i: t<i<N : machine nr.i is passive) P0 : (A i: t<i<N : machine nr.i is passive) When active machine nr.i+1 keeps the token, when passive; it hands over the token to machine nr.i. When active machine nr.i+1 keeps the token, when passive; it hands over the token to machine nr.i. Termination Detection : Termination Detection : t=0 and additional info. available at nr.0 t=0 and additional info. available at nr.0

7. Presence of Messages P0 is falsified -> Weaker invariant P0 V P1 P0 is falsified -> Weaker invariant P0 V P1 P1: (E j: 0<=j<=t : machine nr.j is black) P1: (E j: 0<=j<=t : machine nr.j is black) A machine sending a message to a machine with a number higher than it’s own makes itself Black A machine sending a message to a machine with a number higher than it’s own makes itself Black Termination Detection Termination Detection (t=0^machine nr.0 is white) => ~P1 (t=0^machine nr.0 is white) => ~P1

8. Continued… P1 may be falsified and hence invariant P0 V P1 – Weaker invariant P0 V P1 V P2 P1 may be falsified and hence invariant P0 V P1 – Weaker invariant P0 V P1 V P2 P2 : the token is black P2 : the token is black nr. i+1 hands over a black token to nr. i if it itself is black nr. i+1 hands over a black token to nr. i if it itself is black Termination detection Termination detection Token is white => ~P2 Token is white => ~P2

9. Unsuccessful Probe A black token at nr. 0 or a black nr. 0 A black token at nr. 0 or a black nr. 0 Next probe Next probe nr. 0 makes itself white and sends a white token to nr. N-1 nr. 0 makes itself white and sends a white token to nr. N-1 Upon transmission of the token to nr. i machine nr. i+1 becomes white (P1 is not falsified) Upon transmission of the token to nr. i machine nr. i+1 becomes white (P1 is not falsified)

10. Termination A probe initiated after termination will end up with all machines white, and, hence a next probe is guaranteed to return a white token to a white machine nr.0. A probe initiated after termination will end up with all machines white, and, hence a next probe is guaranteed to return a white token to a white machine nr.0. Simplified rule – A machine sending a message makes itself black. Simplified rule – A machine sending a message makes itself black. Assumption of instantaneous messaging. Assumption of instantaneous messaging.