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

An Introduction to the Topological Theory of Distributed Computing with Safe-consensus

AbstractThe theory of distributed computing shares a deep and fascinating connection with combinatorial and algebraic topology. One of the key ideas that facilitates the development of the topological theory of distributed computing is the use of iterated shared memory models. In such a model processes communicate through a sequence of shared objects. Processes access the sequence of objects, one-by-one, in the same order and asynchronously. Each process accesses each shared object only once. In the most basic form of an iterated model, any number of processes can crash, and the shared objects are snapshot objects. A process can write a value to such an object, and gets back a snapshot of its contents.The purpose of this paper is to give an introduction to this research area, using an iterated model based on the safe-consensus task (Afek, Gafni and Lieber, DISCʼ09). In a safe-consensus task, the validity condition of consensus is weakened as follows. If the first process to invoke an object solving a safe-consensus task returns before any other process invokes it, then the process gets back its own input; otherwise the value returned by the task can be arbitrary. As with consensus, the agreement requirement is that always the same value is returned to all processes.A safe-consensus-based iterated model is described in detail. It is explained how its runs can be described with simplicial complexes. The usefulness of the iterated memory model for the topological theory of distributed computing is exhibited by presenting some new results (with very clean and well structured proofs) about the solvability of the (n,k)-set agreement task. Throughout the paper, the main ideas are explained with figures and intuitive examples

Relating L-Resilience and Wait-Freedom via Hitting Sets

The condition of t-resilience stipulates that an n-process program is only obliged to make progress when at least n-t processes are correct. Put another way, the live sets, the collection of process sets such that progress is required if all the processes in one of these sets are correct, are all sets with at least n-t processes. We show that the ability of arbitrary collection of live sets L to solve distributed tasks is tightly related to the minimum hitting set of L, a minimum cardinality subset of processes that has a non-empty intersection with every live set. Thus, finding the computing power of L is NP-complete. For the special case of colorless tasks that allow participating processes to adopt input or output values of each other, we use a simple simulation to show that a task can be solved L-resiliently if and only if it can be solved (h-1)-resiliently, where h is the size of the minimum hitting set of L. For general tasks, we characterize L-resilient solvability of tasks with respect to a limited notion of weak solvability: in every execution where all processes in some set in L are correct, outputs must be produced for every process in some (possibly different) participating set in L. Given a task T, we construct another task T_L such that T is solvable weakly L-resiliently if and only if T_L is solvable weakly wait-free

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

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

Power and limits of distributed computing shared memory models

Due to the advent of multicore machines, shared memory distributed computing models taking into account asynchrony and process crashes are becoming more and more important. This paper visits some of the models for these systems, and analyses their properties from a computability point of view. Among them, the snapshot model and the iterated model are particularly investigated. The paper visits also several approaches that have been proposed to model crash failures. Among them, the wait-free case where any number of processes can crash is fundamental. The paper also considers models where up to t processes can crash, and where the crashes are not independent. The aim of this survey is to help the reader to better understand recent advances on what is known about the power and limits of distributed computing shared memory models and their underlying mathematics.Ce rapport est une introduction au modèles de calcul asynchrone pour les systèmes à mémoire partagée

Consensus in the Unknown-Participation Message-Adversary Model

We propose a new distributed-computing model, inspired by permissionless distributed systems such as Bitcoin and Ethereum, that allows studying permissionless consensus in a mathematically regular setting. Like in the sleepy model of Pass and Shi, we consider a synchronous, round-by-round message-passing system in which the set of online processors changes each round. Unlike the sleepy model, the set of processors may be infinite. Moreover, processors never fail; instead, an adversary can temporarily or permanently impersonate some processors. Finally, processors have access to a strong form of message-authentication that authenticates not only the sender of a message but also the round in which the message was sent. Assuming that, each round, the adversary impersonates less than 1/2 of the online processors, we present two consensus algorithms. The first ensures deterministic safety and constant latency in expectation, assuming a probabilistic leader-election oracle. The second ensures deterministic safety and deterministic liveness assuming irrevocable impersonation and eventually-stabilizing participation. The model is unrealistic in full generality. However, if we assume finitely many processes and that the set of faulty processes remains constant, the model coincides with a practically-motivated model: the static version of the sleepy model