Reductions and Extension-Based Proofs

ABSTRACTIn the theory of distributed computing, the notion of a reduction is a common tool for proving impossibility results. If task T reduces to task S, and T is impossible to solve, then so is S. Extension-based proofs demonstrate the impossibility of solving a task in a wait-free manner by constructing an infinite execution. It is known that extension-based proofs are limited in power: there is no extension-based proof of the impossibility of a wait-free protocol in the NIS model for k-set agreement among n > k ≥ 2 processes. We introduce augmented extension-based proofs, which generalize extension-based proofs by providing a new type of query. For a general class of reductions, we prove that, if T reduces to S, and T has an augmented extension-based proof that it is impossible to solve in the NIS model, then so does S. Since there is also no augmented extension-based proof of the impossibility of a wait-free protocol in the NIS model for k-set agreement among n > k ≥ 2 processes, it follows that there are no (augmented) extension-based proofs of the impossibility of wait-free protocols in the NIS model for a number of other distributed computing problems.