1,365 research outputs found
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
Continuous Tasks and the Asynchronous Computability Theorem
The celebrated 1999 Asynchronous Computability Theorem (ACT) of Herlihy and Shavit characterized distributed tasks that are wait-free solvable and uncovered deep connections with combinatorial topology. We provide an alternative characterization of those tasks by means of the novel concept of continuous tasks, which have an input/output specification that is a continuous function between the geometric realizations of the input and output complex: We state and prove a precise characterization theorem (CACT) for wait-free solvable tasks in terms of continuous tasks. Its proof utilizes a novel chromatic version of a foundational result in algebraic topology, the simplicial approximation theorem, which is also proved in this paper. Apart from the alternative proof of the ACT implied by our CACT, we also demonstrate that continuous tasks have an expressive power that goes beyond classic task specifications, and hence open up a promising venue for future research: For the well-known approximate agreement task, we show that one can easily encode the desired proportion of the occurrence of specific outputs, namely, exact agreement, in the continuous task specification
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 -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 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, -set-consensus objects can
be used is, unlike WSB, "natural" by presenting the corresponding simple affine
task captured by a subset of -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
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
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