By Ivanyi A. (ed.)

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Additional info for Algorithms of informatics, vol. 2

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It follows that processor pi never receives a message from pk with timestamp greater than t − 1, which by updating rules in lines 24-26 means that processor pk never receives a message < m, t > from j, which contradicts the liveness property of ssf broadcast service. To prove Total Order property it is sufficient to prove that for every processor pi and messages m, m sent by processors pk , pl with timestamps t, t respectively, each of the triples (m, t, k), (m , t , l) are accepted according to the lexicographic order of (t, k), (t , l).

The value of logical time assigned to instruction x is defined as the pair (counter, id), where counter is the value of the variable counter right after the instruction has been executed, and id is the identifier of the processor. The values of logical time form a total order, where pairs are compared lexicographically. This logical time is also called Lamport time. We define tx to be a quotient counter + 1/(id + 1), which is an equivalent way to represent the pair. 21 For any execution, logical time satisfies three conditions: (i) if an instruction x is performed by a processor before an instruction y is performed by the same processor, then the logical time of x is strictly smaller than that of y, (ii) any two distinct instructions of any two processors get assigned different logical times, 614 13.

Then all nonfaulty processors finish phase g with the same preference. Proof Suppose all nonfaulty processors use the majority value received from the king for their preference. Since the king is nonfaulty, it sends the same message and hence all the nonfaulty preferences are the same. Suppose a nonfaulty processor pi uses its own majority value v for its preference. Thus pi receives more than n/2 + f messages for v in the first round of phase g. Hence every processor, including pg receives more than n/2 messages for v in the first round of phase g and sets its majority value to v.

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