Reductions in Distributed Computing Part I: Consensus and Atomic   Commitment Tasks
  We introduce several notions of reduction in distributed computing, and investigate reduction properties of two fundamental agreement tasks, namely Consensus and Atomic Commitment.   We first propose the notion of reduction "a la Karp'', an analog for distributed computing of the classical Karp reduction. We then define a weaker reduction which is the analog of Cook reduction. These two reductions are called K-reduction and C-reduction, respectively.   We also introduce the notion of C*-reduction which has no counterpart in classical (namely, non distributed) systems, and which naturally arises when dealing with symmetric tasks.   We establish various reducibility and irreducibility theorems with respect to these three reductions. Our main result is an incomparability statement for Consensus and Atomic Commitment tasks: we show that they are incomparable with respect to the C-reduction, except when the resiliency degree is 1, in which case Atomic Commitment is strictly harder than Consensus. A side consequence of these results is that our notion of C-reduction is strictly weaker than the one of K-reduction, even for unsolvable tasks. 
Bernadette Charron-Bost