Closed schedulers: Constructions and applications to consensus protocols

  • Ronit Lubitch
  • Shlomo Moran
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 647)


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.


Initial Configuration Atomic Step Impossibility Result Consensus Protocol Infinite Path 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    H. Attiya, A. Bar-Noy, D. Dolev, D. Peleg, and R. Reischuk. Renaming in an asynchronous environment. Journal of the ACM, 37(3):524–548, 1990.Google Scholar
  2. 2.
    O. Biran, S. Moran, and S. Zaks. A combinatorial characterization of the distributed 1-solvable tasks. Journal of Algorithm, (11):420–440, 1990.Google Scholar
  3. 3.
    S. Chaudhuri. Agreement is harder than consensus: Set consensus problems in totally asynchronous systems. In Proceedings of 9-th PODG Conference, pages 311–324, 1990.Google Scholar
  4. 4.
    König. D. Theorie der endlichen und unendlichen graphen. Liepzig 1936. reprinted by Chelsea, 1950.Google Scholar
  5. 5.
    D. Dolev, C. Dwork, and L. Stockmeyer. On the minimal synchronism needed for distributed consensus. Journal of the ACM, 34(1):77–97, January 1987.Google Scholar
  6. 6.
    M. J. Fischer, N. A. Lynch, and M. S. Paterson. Impossibility of distributed consensus with one faulty process. Journal of the ACM, 32(2):374–382, April 1985.Google Scholar
  7. 7.
    M.C Loui and H.H Abu-Amara. Memory requirements for agreement among unreliable asynchronous processes. Advances in Computing Research, 4:163–183, 1987.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Ronit Lubitch
    • 1
  • Shlomo Moran
    • 1
  1. 1.Dept. of Computer ScienceTechnionHaifaIsrael

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