312 research outputs found
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 Relative Power of Composite Loop Agreement Tasks
Loop agreement is a family of wait-free tasks that includes set agreement and
simplex agreement, and was used to prove the undecidability of wait-free
solvability of distributed tasks by read/write memory. Herlihy and Rajsbaum
defined the algebraic signature of a loop agreement task, which consists of a
group and a distinguished element. They used the algebraic signature to
characterize the relative power of loop agreement tasks. In particular, they
showed that one task implements another exactly when there is a homomorphism
between their respective signatures sending one distinguished element to the
other. In this paper, we extend the previous result by defining the composition
of multiple loop agreement tasks to create a new one with the same combined
power. We generalize the original algebraic characterization of relative power
to compositions of tasks. In this way, we can think of loop agreement tasks in
terms of their basic building blocks. We also investigate a category-theoretic
perspective of loop agreement by defining a category of loops, showing that the
algebraic signature is a functor, and proving that our definition of task
composition is the "correct" one, in a categorical sense.Comment: 18 page
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, -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
Tight Bounds for Connectivity and Set Agreement in Byzantine Synchronous Systems
In this paper, we show that the protocol complex of a Byzantine synchronous
system can remain -connected for up to rounds,
where is the maximum number of Byzantine processes, and .
This topological property implies that rounds are
necessary to solve -set agreement in Byzantine synchronous systems, compared
to rounds in synchronous crash-failure systems. We
also show that our connectivity bound is tight as we indicate solutions to
Byzantine -set agreement in exactly synchronous
rounds, at least when is suitably large compared to . In conclusion, we
see how Byzantine failures can potentially require one extra round to solve
-set agreement, and, for suitably large compared to , at most that
Topological Characterization of Consensus Solvability in Directed Dynamic Networks
Consensus is one of the most fundamental problems in distributed computing.
This paper studies the consensus problem in a synchronous dynamic directed
network, in which communication is controlled by an oblivious message
adversary. The question when consensus is possible in this model has already
been studied thoroughly in the literature from a combinatorial perspective, and
is known to be challenging. This paper presents a topological perspective on
consensus solvability under oblivious message adversaries, which provides
interesting new insights. Our main contribution is a topological
characterization of consensus solvability, which also leads to explicit
decision procedures. Our approach is based on the novel notion of a
communication pseudosphere, which can be seen as the message-passing analog of
the well-known standard chromatic subdivision for wait-free shared memory
systems. We further push the elegance and expressiveness of the "geometric"
reasoning enabled by the topological approach by dealing with uninterpreted
complexes, which considerably reduce the size of the protocol complex, and by
labeling facets with information flow arrows, which give an intuitive meaning
to the implicit epistemic status of the faces in a protocol complex
A generalized asynchronous computability theorem
We consider the models of distributed computation defined as subsets of the
runs of the iterated immediate snapshot model. Given a task and a model
, we provide topological conditions for to be solvable in . When
applied to the wait-free model, our conditions result in the celebrated
Asynchronous Computability Theorem (ACT) of Herlihy and Shavit. To demonstrate
the utility of our characterization, we consider a task that has been shown
earlier to admit only a very complex -resilient solution. In contrast, our
generalized computability theorem confirms its -resilient solvability in a
straightforward manner.Comment: 16 pages, 5 figure
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