A natural way to measure the power of a distributed-computing model is to characterize the set of tasks that can be solved in it. In general, however, the question of whether a given task can be solved in a given model is undecidable, even if we only consider the wait-free shared-memory model. In this paper, we address this question for restricted classes of models and tasks. We show that the question of whether a collection C of (l, j)-set consensus objects, for various l (the number of processes that can invoke the object) and j (the number of distinct outputs the object returns), can be used by n processes to solve wait-free k-set consensus is decidable. Moreover, we provide a simple O(n^2) decision algorithm, based on a dynamic programming solution to the Knapsack optimization problem. We then present an adaptive wait-free set-consensus algorithm that, for each set of participating processes, achieves the best level of agreement that is possible to achieve using C. Overall, this gives us a complete characterization of a read-write model defined by a collection of set-consensus objects through its set-consensus power.
@InProceedings{delportegallet_et_al:LIPIcs.OPODIS.2016.7, author = {Delporte-Gallet, Carole and Fauconnier, Hugues and Gafni, Eli and Kuznetsov, Petr}, title = {{Set-Consensus Collections are Decidable}}, booktitle = {20th International Conference on Principles of Distributed Systems (OPODIS 2016)}, pages = {7:1--7:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-031-6}, ISSN = {1868-8969}, year = {2017}, volume = {70}, editor = {Fatourou, Panagiota and Jim\'{e}nez, Ernesto and Pedone, Fernando}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2016.7}, URN = {urn:nbn:de:0030-drops-70761}, doi = {10.4230/LIPIcs.OPODIS.2016.7}, annote = {Keywords: Decidability, distributed tasks, set consensus, Knapsack problem} }
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