Understanding the Set Consensus Partial Order using the Borowsky-Gafni Simulation
We present a complete characterization of the Set Consensus Partial Order, a refinement of the Consensus Hierarchy of Herlihy. We define the (n, k) -set consensus problem as the k-set consensus problem for n processors. We then answer the question of whether an (n, k)-set consensus object (an object which solves the (n, k)-set consensus problem) can be implemented using a combination of (m, l)-set consensus objects and snapshot objects, for all possible values of n, k, m,l, creating a partial order of set consensus objects. The model we consider is the asynchronous shared memory model.
To prove our results, we use the Borowsky-Gafni Simulation technique, a powerful tool which has been used to prove several impossibility results about shared memory algorithms. Lynch and Rajsbaum gave a formal description of the basic technique, along with a proof of its correctness. We extend their results to include simulations of algorithms which access set consensus objects. Our description of the simulation, and its proof of correctness, are also in terms of I/O Automata. We need this stronger version of the simulation algorithm to obtain our results on the Set Consensus Partial Order. We state a general Simulation Theorem which specifies the properties of the simulation, and characterizes all the impossibility results that can be obtained using this technique. Our partial order result can then be derived as a special case of this theorem.
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