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

Distributed Computability in Byzantine Asynchronous Systems

In this work, we extend the topology-based approach for characterizing computability in asynchronous crash-failure distributed systems to asynchronous Byzantine systems. We give the first theorem with necessary and sufficient conditions to solve arbitrary tasks in asynchronous Byzantine systems where an adversary chooses faulty processes. In our adversarial formulation, outputs of non-faulty processes are constrained in terms of inputs of non-faulty processes only. For colorless tasks, an important subclass of distributed problems, the general result reduces to an elegant model that effectively captures the relation between the number of processes, the number of failures, as well as the topological structure of the task's simplicial complexes.Comment: Will appear at the Proceedings of the 46th Annual Symposium on the Theory of Computing, STOC 201

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

On the Bit Complexity of Iterated Memory

Computability, in the presence of asynchrony and failures, is one of the central questions in distributed computing. The celebrated asynchronous computability theorem (ACT) charaterizes the computing power of the read-write shared-memory model through the geometric properties of its protocol complex: a combinatorial structure describing the states the model can reach via its finite executions. This characterization assumes that the memory is of unbounded capacity, in particular, it is able to store the exponentially growing states of the full-information protocol. In this paper, we tackle an orthogonal question: what is the minimal memory capacity that allows us to simulate a given number of rounds of the full-information protocol? In the iterated immediate snapshot model (IIS), we determine necessary and sufficient conditions on the number of bits an IIS element should be able to store so that the resulting protocol is equivalent, up to isomorphism, to the full-information protocol. Our characterization implies that $n\geq 3$ processes can simulate $r$ rounds of the full-information IIS protocol as long as the bit complexity per process is within $\Omega(r n)$ and $O(r n \log n)$. Two processes, however, can simulate any number of rounds of the full-information protocol using only $2$ bits per process, which implies, in particular, that just $2$ bits per process are sufficient to solve $\varepsilon$-agreement for arbitrarily small $\varepsilon$.Comment: 21 pages, 4 figures. To be published in 31st International Colloquium On Structural Information and Communication Complexity (SIROCCO 2024

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

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

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