Asynchronous Network Computing with Process Failures By Sindhu Karthikeyan.

1. Chapter 21 Asynchronous Network Computing with Process Failures By Sindhu Karthikeyan.

2. 21.1 The Network Model Theorem 21.1 If A is asynchronous broadcast system with a reliable broadcast channel, then there is an asynchronous send/receive system B with reliable FIFO send/receive channels that has the same user interface as A and that �simulates� A, as follows. For every Execution a of B, there is an execution a� of A such that the following conditions hold: 1. a and a� are indistinguishable to U (the composition of users Ui ). 2. For each i, a stopi occurs in a exactly if it does in a� . Moreover, if a is fair, then a� is also fair.

3. Proof Sketch. . System B has one processor Qi for each processor Pi of A. . Each Qi is responsible for simulating Pi, and participating in the simulation of the broadcast channel. . Qi simulates a broadcast msg bcast(m)i output of Pi by performing send(m,t)i,j outputs for all j ? i, where t = local-integer valued tag (which starts from 1 and increments with each successive broadcast), and also performing an internal step simulating receive(m)i,i. . If Qi receives a message (m,t) sent by Qj, it helps in simulation of Pj�s broadcast by relaying the message- it sends (m,t,j) to all processors other than i and j. . Qi collects tagged messages which was broadcasted by Pj, j ? i, which are received directly from Qj or by relays. . Qi is also allowed to perform an internal step simulating a receive(m)j,i, Qi can do this only when Qi has a message (m,t) originally broadcast by Pj, Qi has already relayed (m,t,j) to all processors other than I and j, and Qi has already simulated receivej,i events for message from Pj with all tag values less than t.

4. Some key facts for the proof are as follows: No Qi simulates a receive(m)j,i unless it has succeeded in sending the corresponding (m,t) to all the other processors, and thus it has guaranteed that all processors will receive (m,t) from j. Although Qi can receive messages broadcasted by Pj out of order, the tags allow Qi to sort these messages into proper order. 3. If a message with tag t is sent by any processor Qi then it must be that message originating at Pi with all the other messages having smaller tag values have been previously sent to all processors.

5. Impossibility of Agreement in the presence of Faults stopi init(v) i decide(v) i