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 (, j)-set consensus objects, for various (the number of processes

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... 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. 1 Introduction A plethora of models of computation were proposed for distributed environments. The models vary in timing assumptions they make, types of failures they assume, and communication primitives they employ. It is hard to say a priori whether one model provides more power to the programmer than the other. A natural way to measure this power is to characterize the set of distributed tasks that can be solved in a model. In general, however, the question of whether a given task can be solved in the popular wait-free read-write model, i.e., tolerating asynchrony and failures of arbitrary subsets of processes, is undecidable [13]. Of course, in models in which processes can additionally access arbitrary objects, the question is not decidable either. However, many natural models have been shown to be characterized by their power to solve set consensus [10].