Closed schedulers: Constructions and applications to consensus protocols
Analyzing distributed protocols in various models often involves a careful analysis of the set of admissible runs, for which the protocols should behave correctly. In particular, the admissible runs assumed by a t-resilient protocol are runs which are fair for all but at most t processors. In this paper we define closed sets of runs, and suggest a technique to prove impossibility results for t-resilient protocols, by restricting the corresponding sets of admissible runs to smaller sets, which are closed, as follows:
For each protocol PR and for each initial configuration c, the set of admissible runs of PR which start from c defines a tree in a natural way: the root of the tree is the empty run, and each vertex in it denotes a finite prefix of an admissible run; a vertex u in the tree has a son v iff v is also a prefix of an admissible run, which extends u by one atomic step.
The tree of admissible runs described above may contain infinite paths which are not admissible runs. A set of admissible runs is closed if for every possible initial configuration c, each path in the tree of admissible runs starting from c is also an admissible run. Closed sets of runs have the simple combinatorial structure of the set of paths of an infinite tree, which makes them easier to analyze.
We introduce a unified method for constructing closed sets of admissible runs by using a model-independent construction of closed schedulers. We use this construction to provide unified proofs of impossibility results in various models of asynchronous computations. One of our results generalizes a known impossibility result in a non-trivial way.
KeywordsInitial Configuration Atomic Step Impossibility Result Consensus Protocol Infinite Path
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