By Ivanyi A. (ed.)

Ivanyi A. (ed.) Algorithms of informatics, vol.2.. purposes (2007)(ISBN 9638759623)

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**Example text**

Fault-tolerant consensus 605 messages per processor that enters phase number i. The contribution may be smaller than 4 · 2i if a probe message gets swallowed on the way away from the processor. 9 provides an upper bound on the number of processors that enter phase number k . What is the highest phase that a processor can ever enter? The number k of processors that can be in phase i is at most n/2i−1 . So when n/2i−1 < 1, then there can be no processor that ever enters phase i. Thus no processor can enter any phase beyond phase number h = 1 + log2 n , because n < 2(h+1)−1 .

As in the reliable case, every processor takes a computation step in every round and every message sent is delivered in that round. Hence a faulty processor can behave arbitrarily and even maliciously. For example, it could send dierent messages to dierent processors. It can even appear that the faulty processors coordinate with each other. A faulty processor can also mimic the behaviour of a crashed processor by failing to send any messages from some point on. In this case, the denition of the consensus problem is the same as in the message passing model with crash failures.

Formally, ki denotes the number of instructions that processor pi has executed. Not all cuts correspond to collections of states of distributed processors that could be considered natural or consistent. For example, if a processor pi has received a message from pj and we record the state of pi in the cut by making ki appropriately large, but make kj so small that the cut contains the state of the sender before the moment when the message was sent, then we could say that such cut is not naturalthere are instructions recorded in the cut 616 13.