Strongly Terminating Early-Stopping k-Set Agreement in Synchronous Systems with General Omission Failures

The k-set agreement problem is a generalization of the consensus problem: considering a system made up of n processes where each process proposes a value, each non-faulty process has to decide a value such that a decided value is a proposed value, and no more than k different values are decided. It has recently be shown that, in the crash failure model, $\min(\lfloor \frac{f}{k}\rfloor+2,\lfloor \frac{t}{k}\rfloor +1)The k-set agreement problem is a generalization of the consensus problem: considering a system made up of n processes where each process proposes a value, each non-faulty process has to decide a value such that a decided value is a proposed value, and no more than k different values are decided. It has recently be shown that, in the crash failure model, min(?\fracfk?+2,?\fractk?+1)\min(\lfloor \frac{f}{k}\rfloor+2,\lfloor \frac{t}{k}\rfloor +1) is a lower bound on the number of rounds for the non-faulty processes to decide (where t is an upper bound on the number of process crashes, and f, 0≤f≤t, the actual number of crashes).     

On the computability power and the robustness of set agreement-oriented failure detector classes

Solving agreement problems deterministically, such as consensus and k-set agreement, in asynchronous distributed systems prone to an unbounded number of process failures has been shown to be impossible. To circumvent this impossibility, unreliable failure detectors for the crash failure model have been widely studied. These are oracles that provide information on failures. The exact nature of such information is defined by a set of abstract properties that a particular class of failure detectors satisfy. The weakest class of such failure detectors that allow to solve consensus is Ω. This paper considers failure detector classes from the literature that solve k-set agreement in the crash failure model, and studies their relative power. It shows that the family of failure detector classes (1 ≤ x ≤ n), and (0 ≤ y ≤ n), can be “added” to provide a failure detector of the class Ω ^z (1 ≤ z ≤ n, a generalization of Ω). It also characterizes the power of such an “addition”, namely, , can construct Ω ^z iff y + z > t, and can construct Ω ^z iff x + z > t + 1, where t is the maximum number of processes that can crash in a run. As an example, the paper shows that, while allows solving 2-set agreement (but not consensus) and allows solving t-set agreement (but not (t − 1)-set agreement), a system with failure detectors of both classes can solve consensus for any value of t. More generally, the paper studies the failure detector classes , and Ω ^z , and shows which reductions among these classes are possible and which are not. The paper also presents a message-passing Ω ^k -based k-set agreement protocol and shows that Ω ^k is not enough to solve (k − 1)-set agreement. In that sense, it can be seen as a step toward the characterization of the weakest failure detector class that allows solving the k-set agreement problem. An extended abstract of this paper has appeared in the proceedings of PODC 2006 [20]. This work has been supported partially by a grant from LAFMI (Franco-Mexican Lab in Computer Science), the European Network of Excellence ReSIST and PAPIIT-UNAM.     

Narrowing power vs efficiency in synchronous set agreement: Relationship, algorithms and lower bound

The k-set agreement problem is a generalization of the uniform consensus problem: each process proposes a value, and each non-faulty process has to decide a value such that a decided value is a proposed value, and at most k different values are decided. It has been shown that any algorithm that solves the k-set agreement problem in synchronous systems that can suffer up to t crash failures requires rounds in the worst case. It has also been shown that it is possible to design early deciding algorithms where no process decides and halts after rounds, where f is the number of actual crashes in a run (0≤f≤t).This paper explores a new direction to solve the k-set agreement problem in a synchronous system. It considers that the system is enriched with base objects (denoted has [m,?]_SA objects) that allow solving the ?-set agreement problem in a set of m processes (m<n). The paper makes several contributions. It first proposes a synchronous k-set agreement algorithm that benefits from such underlying base objects. This algorithm requires rounds, more precisely, rounds, where . The paper then shows that this bound, that involves all the parameters that characterize both the problem (k) and its environment (t, m and ?), is a lower bound. The proof of this lower bound sheds additional light on the deep connection between synchronous efficiency and asynchronous computability. Finally, the paper extends its investigation to the early deciding case. It presents a k-set agreement algorithm that directs the processes to decide and stop by round . These bounds generalize the bounds previously established for solving the k-set agreement problem in pure synchronous systems.     

Synchronous condition-based consensus

The condition-based approach studies restrictions on the inputs to a distributed problem, called conditions, that facilitate its solution. Previous work considered mostly the asynchronous model of computation. This paper studies conditions for consensus in a synchronous system where processes can fail by crashing. It describes a full classification of conditions for consensus, establishing a continuum between the asynchronous and synchronous models, with the following hierarchy where includes all conditions (and in particular the trivial one made up of all possible input vectors). For a condition , we have:

Interface synthesis and protocol conversion

Given deterministic interfaces P and Q, we investigate the problem of synthesising an interface R such that P composed with R refines Q. We show that a solution exists iff P and are compatible, and the most general solution is given by , where is the interface P with inputs and outputs interchanged. Remarkably, the result holds both for asynchronous and synchronous interfaces. We model interfaces using the interface automata formalism of de Alfaro and Henzinger. For the synchronous case, we give a new definition of synchronous interface automata based on Mealy machines and show that the result holds for a weak form of nondeterminism, called observable nondeterminism. We also characterise solutions to the synthesis problem in terms of winning input strategies in the automaton , and the most general solution in terms of the most permissive winning strategy. We apply the solution to the synthesis of converters for mismatched protocols in both the asynchronous and synchronous domains. For the asynchronous case, this leads to automatic synthesis of converters for incompatible network protocols. In the synchronous case, we obtain automatic converters for mismatched intellectual property blocks in system-on-chip designs. The work reported here is based on earlier work on interface synthesis in Bhaduri (Third international symposium on automated technology for verification and analysis, ATVA 2005, pp 338–353, 2005) for the asynchronous case, and Bhaduri and Ramesh (Sixth international conference on application of concurrency to system design, ACSD 2006, pp 208–216) for the synchronous one.     

Of Choices, Failures and Asynchrony: The Many Faces of Set Agreement

Set agreement is a fundamental problem in distributed computing in which processes collectively choose a small subset of values from a larger set of proposals. The impossibility of fault-tolerant set agreement in asynchronous networks is one of the seminal results in distributed computing. In synchronous networks, too, the complexity of set agreement has been a significant research challenge that has now been resolved. Real systems, however, are neither purely synchronous nor purely asynchronous. Rather, they tend to alternate between periods of synchrony and periods of asynchrony. Nothing specific is known about the complexity of set agreement in such a “partially synchronous” setting. In this paper, we address this challenge, presenting the first (asymptotically) tight bound on the complexity of set agreement in such systems. We introduce a novel technique for simulating, in a fault-prone asynchronous shared memory, executions of an asynchronous and failure-prone message-passing system in which some fragments appear synchronous to some processes. We use this simulation technique to derive a lower bound on the round complexity of set agreement in a partially synchronous system by a reduction from asynchronous wait-free set agreement. Specifically, we show that every set agreement protocol requires at least $\lfloor\frac{t}{k}\rfloor + 2$ synchronous rounds to decide. We present an (asymptotically) matching algorithm that relies on a distributed asynchrony detection mechanism to decide as soon as possible during periods of synchrony. From these two results, we derive the size of the minimal window of synchrony needed to solve set agreement. By relating synchronous, asynchronous and partially synchronous environments, our simulation technique is of independent interest. In particular, it allows us to obtain a new lower bound on the complexity of early deciding k-set agreement complementary to that of Gafni et al. (in SIAM J. Comput. 40(1):63–78, 2011), and to re-derive the combinatorial topology lower bound of Guerraoui et al. (in Theor. Comput. Sci. 410(6–7):570–580, 2009) in an algorithmic way.     

Renaming in synchronous message passing systems with Byzantine failures

We study the renaming problem in a fully connected synchronous network with Byzantine failures. We show that when the original namespace of the processors is unbounded, this problem cannot be solved in an a priori bounded number of rounds for , where n is the size of the network and t is the number of failures. On the other hand, for n > 3t, we present a Byzantine renaming algorithm that runs in O(lg n) rounds. In addition, we present a fast, efficient strong renaming algorithm for n > t, which runs in rounds, where N ₀ is the value of the highest identifier among all the correct processors.     

Tight bounds for k-set agreement with limited-scope failure detectors

In a system with limited-scope failure detectors, there are q disjoint clusters of processes such that some correct process in each cluster is never suspected by any process in that cluster. The failure detector class S_x,q satisfies this property all the time, while ⋄S_x,q satisfies it eventually. This paper gives the first tight bounds for the k-set agreement task in asynchronous message-passing models augmented with failure detectors from either the S_x,q or ⋄S_x,q classes. For S_x,q, we show that any k-set agreement protocol that tolerates f failures must satisfy f < k + x − q if q < k and f < x otherwise, where x is the combined size of the q disjoint clusters if q < k (or the k largest, otherwise). This result establishes for the first time that the protocol of Mostèfaoui and Raynal for the S_x = S_x,1 failure detector is optimal. For ⋄S_x,q, we show that any k-set agreement protocol that tolerates f failures must satisfy if q < k and otherwise, where n + 1 is the total number of processes. We give a novel protocol that matches our lower bound, disproving a conjecture of Mostèfaoui and Raynal for the ⋄S_x = ⋄S_x,1 failure detector. Our lower bounds exploit techniques borrowed from Combinatorial Topology, demonstrating for the first time that this approach is applicable to models that encompass failure detectors.     

On the complexity of approximating k-set packing

Given a k-uniform hypergraph, the Maximum k -Set Packing problem is to find the maximum disjoint set of edges. We prove that this problem cannot be efficiently approximated to within a factor of unless P = NP. This improves the previous hardness of approximation factor of by Trevisan. This result extends to the problem of k-Dimensional-Matching.     

Uniform stabilization for linear systems with persistency of excitation: the neutrally stable and the double integrator cases

Consider the controlled system dx/dt = Ax + α(t)Bu where the pair (A, B) is stabilizable and α(t) takes values in [0, 1] and is persistently exciting, i.e., there exist two positive constants μ, T such that, for every t ≥ 0, ${\int_t^{t+T}\alpha(s){\rm d}s \geq \mu}Consider the controlled system dx/dt = Ax + α(t)Bu where the pair (A, B) is stabilizable and α(t) takes values in [0, 1] and is persistently exciting, i.e., there exist two positive constants μ, T such that, for every t ≥ 0, . In particular, when α(t) becomes zero the system dynamics switches to an uncontrollable system. In this paper, we address the following question: is it possible to find a linear time-invariant state-feedback u = Kx, with K only depending on (A, B) and possibly on μ, T, which globally asymptotically stabilizes the system? We give a positive answer to this question for two cases: when A is neutrally stable and when the system is the double integrator. Notation  A continuous function is of class , if it is strictly increasing and is of class if it is continuous, non-increasing and tends to zero as its argument tends to infinity. A function is said to be a class -function if, for any t ≥ 0, and for any s ≥ 0. We use |·| for the Euclidean norm of vectors and the induced L ₂-norm of matrices.     

Efficient Solution of A, x ( k )?= b ( k ) Using A ?1

In this work, we consider the problem of solving , , where b ^(k+1) = f(x ^(k)). We show that when A is a full matrix and , where depends on the specific software and hardware setup, it is faster to solve for by explicitly evaluating the inverse matrix A ⁻¹ rather than through the LU decomposition of A. We also show that the forward error is comparable in both methods, regardless of the condition number of A.     

A new bound for the D0L sequence equivalence problem

The D0L sequence equivalence problem consists of deciding, given two morphisms , and a word , whether or not g ⁱ (w) = h ⁱ (w) for all i ≥ 0. We show that in case of smooth and loop-free morphisms, to decide the D0L sequence equivalence problem, it suffices to consider the terms of the sequences with , where n is the cardinality of X.     

Regular Random k-SAT: Properties of Balanced Formulas

We consider a model for generating random k-SAT formulas, in which each literal occurs approximately the same number of times in the formula clauses (regular random and k-SAT). Our experimental results show that such regular random k-SAT instances are much harder than the usual uniform random k-SAT problems. This is in agreement with other results that show that more balanced instances of random combinatorial problems are often much more difficult to solve than uniformly random instances, even at phase transition boundaries. There are almost no formal results known for such problem distributions. The balancing constraints add a dependency between variables that complicates a standard analysis. Regular random 3-SAT exhibits a phase transition as a function of the ratio α of clauses to variables. The transition takes place at approximately α = 3.5. We show that for α > 3.78 with high probability (w.h.p.) random regular 3-SAT formulas are unsatisfiable. Specifically, the events hold with high probability if Pr when n → ∞. We also show that the analysis of a greedy algorithm proposed by Kaporis et al. for the uniform 3-SAT model can be adapted for regular random 3-SAT. In particular, we show that for formulas with ratio α < 2.46, a greedy algorithm finds a satisfying assignment with positive probability.     

Partial synchrony based on set timeliness

We introduce a new model of partial synchrony for read-write shared memory systems. This model is based on the simple notion of set timeliness—a natural generalization of the seminal concept of timeliness in the partially synchrony model of Dwork et?al. (J. ACM 35(2):288–323, 1988). Despite its simplicity, the concept of set timeliness is powerful enough to define a family of partially synchronous systems that closely match individual instances of the t-resilient k-set agreement problem among n processes, henceforth denoted (t, k, n)-agreement. In particular, we use it to give a partially synchronous system that is synchronous enough for solving (t, k, n)-agreement, but not enough for solving two incrementally stronger problems, namely, (t + 1, k, n)-agreement, which has a slightly stronger resiliency requirement, and (t, k ? 1, n)-agreement, which has a slightly stronger agreement requirement. This is the first partially synchronous system that separates these sub-consensus problems. The above results show that set timeliness can be used to study and compare the partial synchrony requirements of problems that are strictly weaker than consensus.     

Lower bound for scalable Byzantine Agreement

We consider the problem of computing Byzantine Agreement in a synchronous network with n processors, each with a private random string, where each pair of processors is connected by a private communication line. The adversary is malicious and non-adaptive, i.e., it must choose the processors to corrupt at the start of the algorithm. Byzantine Agreement is known to be computable in this model in an expected constant number of rounds. We consider a scalable model where in each round each uncorrupt processor can send to any set of log n other processors and listen to any set of log n processors. We define the loss of an execution to be the number of uncorrupt processors whose output does not agree with the output of the majority of uncorrupt processors. We show that if there are t corrupt processors, then any randomised protocol which has probability at least 1/2 + 1/ logn of loss less than requires at least f rounds. This also shows that lossless protocols require both rounds, and for at least one uncorrupt processor to send messages during the protocol.     

On set consensus numbers

It is conjectured that the only way a failure detector (FD) can help solving n-process tasks is by providing k-set consensus for some ${k\in\{1,\ldots,n\}}$ among all the processes. It was recently shown by Zieli??ski that any FD that allows for solving a given n-process task that is unsolvable read-write wait-free, also solves (n ? 1)-set consensus. In this paper, we provide a generalization of Zieli??ski??s result. We show that any FD that solves a colorless task that cannot be solved read-write k-resiliently, also solves k-set consensus. More generally, we show that every colorless task ${\mathcal{T}}$ can be characterized by its set consensus number: the largest ${k\in\{1,\ldots,n\}}$ such that ${\mathcal{T}}$ is solvable (k ? 1)-resiliently. A task ${\mathcal{T}}$ with set consensus number k is, in the failure detector sense, equivalent to k-set consensus, i.e., a FD solves ${\mathcal{T}}$ if and only if it solves k-set consensus. As a corollary, we determine the weakest FD for solving k-set consensus in every environment, i.e., for all assumptions on when and where failures might occur.