Strong Equivalence Relations for Iterated Models

The Iterated Immediate Snapshot model (IIS), due to its elegant geometrical representation, has become standard for applying topological reasoning to distributed computing. Its modular structure makes it easier to analyze than the more realistic (non-iterated) read-write Atomic-Snapshot memory model (AS). It is known that AS and IIS are equivalent with respect to \emph{wait-free task} computability: a distributed task is solvable in AS if and only if it solvable in IIS. We observe, however, that this equivalence is not sufficient in order to explore solvability of tasks in \emph{sub-models} of AS (i.e. proper subsets of its runs) or computability of \emph{long-lived} objects, and a stronger equivalence relation is needed. In this paper, we consider \emph{adversarial} sub-models of AS and IIS specified by the sets of processes that can be \emph{correct} in a model run. We show that AS and IIS are equivalent in a strong way: a (possibly long-lived) object is implementable in AS under a given adversary if and only if it is implementable in IIS under the same adversary. %This holds whether the object is one-shot or long-lived. Therefore, the computability of any object in shared memory under an adversarial AS scheduler can be equivalently investigated in IIS

The solvability of consensus in iterated models extended with safe-consensus

The safe-consensus task was introduced by Afek, Gafni and Lieber (DISC'09) as a weakening of the classic consensus. When there is concurrency, the consensus output can be arbitrary, not even the input of any process. They showed that safe-consensus is equivalent to consensus, in a wait-free system. We study the solvability of consensus in three shared memory iterated models extended with the power of safe-consensus black boxes. In the first model, for the $i$-th iteration, processes write to the memory, invoke safe-consensus boxes and finally they snapshot the memory. We show that in this model, any wait-free implementation of consensus requires $\binom{n}{2}$ safe-consensus black-boxes and this bound is tight. In a second iterated model, the processes write to memory, then they snapshot it and finally they invoke safe-consensus boxes. We prove that in this model, consensus cannot be implemented. In the last iterated model, processes first invoke safe-consensus, then they write to memory and finally they snapshot it. We show that this model is equivalent to the previous model and thus consensus cannot be implemented.Comment: 49 pages, A preliminar version of the main results appeared in the SIROCCO 2014 proceeding

Read-Write Memory and k-Set Consensus as an Affine Task

The wait-free read-write memory model has been characterized as an iterated \emph{Immediate Snapshot} (IS) task. The IS task is \emph{affine}---it can be defined as a (sub)set of simplices of the standard chromatic subdivision. It is known that the task of \emph{Weak Symmetry Breaking} (WSB) cannot be represented as an affine task. In this paper, we highlight the phenomenon of a "natural" model that can be captured by an iterated affine task and, thus, by a subset of runs of the iterated immediate snapshot model. We show that the read-write memory model in which, additionally, $k$-set-consensus objects can be used is, unlike WSB, "natural" by presenting the corresponding simple affine task captured by a subset of $2$-round IS runs. Our results imply the first combinatorial characterization of models equipped with abstractions other than read-write memory that applies to generic tasks