# Closed schedulers: Constructions and applications to consensus protocols

## Abstract

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.

## Keywords

Initial Configuration Atomic Step Impossibility Result Consensus Protocol Infinite Path## Preview

Unable to display preview. Download preview PDF.

## References

- 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.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.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.König. D. Theorie der endlichen und unendlichen graphen. Liepzig 1936. reprinted by Chelsea, 1950.Google Scholar
- 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.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.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