In the renaming task n + 1 processes start with unique input names taken from a large space and must choose unique output names taken from a smaller name space, 0, 1, . . . , K . To rule out trivial solutions, a protocol must be anonymous: the value chosen by a process can depend on its input name and on the execution, but not on the specific process id. Attiya et al. showed in 1990 that renaming has a wait-free solution when K ≥ 2n. Several proofs of a lower bound stating that no such protocol exists when K < 2n have been published. We presented in the ACM PODC 2008 conference the following two results. First, we presented the first completely combinatorial lower bound proof stating that no such a protocol exists when K < 2n. This bound holds for infinitely many values of n. Second, for the other values of n, we proved that the lower bound for K < 2n is incorrect, exhibiting a wait-free renaming protocol for K = 2n − 1. More precisely, we presented a theorem stating that there exists a wait-free renaming protocol for K < 2n if and only if the set of integers { n+1 i+1 |0 ≤ i ≤ n−1 2 } are relatively prime. This paper is the first part of the full version of the results presented in the ACM PODC 2008 conference. It includes only the lower bound. Namely, we show here that no protocol for renaming exists when K < 2n, if n is such that { n+1 i+1 |0 ≤ i ≤ n−1 2 } are not relatively prime. We prove This is the first part of the full version of the work presented in the ACM PODC 2008 [9]. Partly supported by PAPIIT-DGAPA project IN116808 and Macroproyecto para las Tecnologías de la Información, UNAM.A. Castañeda (B) · S. Rajsbaum this result using the known equivalence of K -renaming for K = 2n − 1 and the weak symmetry breaking task. In this task processes have no input values and the output values are 0 or 1, and it is required that in every execution in which all processes participate, at least one process decides 0 and at least one process decides 1. The full version of the upper bound appears in a companion paper [10].

In the renaming task n+1 processes start with unique input names from a large space and must choose unique output names taken from a smaller name space, namely 0, 1, . . . , K.To rule out trivial solutions, a protocol must be anonymous: the value chosen by a process can depend on its input name and on the execution, but not on the specific process id.Attiya et al. showed in 1990 that renaming has a waitfree solution when K ≥ 2n. Several proofs of a lower bound stating that no such protocol exists when K < 2n have been published. In this paper we prove that, for certain values of n, this lower bound is incorrect, exhibiting a wait-free renaming protocol for K = 2n − 1. For the other values of n, we present the first completely combinatorial lower bound proof stating that no such protocol exists when K < 2n.More precisely, our main theorem states that there exists a wait-free renaming protocol for K < 2n if and only if the set of integers { n+1 i+1 ¡ : 0 ≤ i ≤ n−1 2 } are relatively prime. Thus, such protocol exists for six processes, and not for less. The proof of the theorem uses combinatorial topology techniques, both for the lower bound and to derive the renaming protocol.

Exploring the power of shared memory communication objects and models, and the limits of distributed computability are among the most exciting research areas of distributed computing. In that spirit, this paper focuses on a problem that has received considerable interest since its introduction in 1987, namely the renaming problem. It was the rst non-trivial problem known to be solvable in an asynchronous distributed system despite process failures. Many algorithms for renaming and variants of renaming have been proposed, and sophisticated lower bounds have been proved, that have been a source of new ideas of general interest to distributed computing. It has consequently acquired a paradigm status in distributed fault-tolerant computing. In the renaming problem, processes start with unique initial names taken from a large name space and decide new names such that no two processes decide the same new name and the new names are from a name space as small as possible. This paper presents an introduction to the renaming problem in shared memory systems, for non-expert readers. It describes both algorithms and lower bounds. Also, it discusses strong connections relating renaming and other important distributed problems such as set agreement and symmetry breaking.

Tasks and objects are two predominant ways of specifying distributed problems where processes should compute outputs based on their inputs. Roughly speaking, a task specifies, for each set of processes and each possible assignment of input values, their valid outputs. In contrast, an object is defined by a sequential specification. Also, an object can be invoked multiple times by each process, while a task is a one-shot problem. Each one requires its own implementation notion, stating when an execution satisfies the specification. For objects, linearizability is commonly used, while tasks implementation notions are less explored. The article introduces the notion of interval-sequential object, and the corresponding implementation notion of interval-linearizability , to encompass many problems that have no sequential specification as objects. It is shown that interval-sequential specifications are local , namely, one can consider interval-linearizable object implementations in isolation and compose them for free, without sacrificing interval-linearizability of the whole system. The article also introduces the notion of refined tasks and its corresponding satisfiability notion. In contrast to a task, a refined task can be invoked multiple times by each process. Also, objects that cannot be defined using tasks can be defined using refined tasks. In fact, a main result of the article is that interval-sequential objects and refined tasks have the same expressive power and both are complete in the sense that they are able to specify any prefix-closed set of well-formed executions. Interval-linearizability and refined tasks go beyond unifying objects and tasks; they shed new light on both of them. On the one hand, interval-linearizability brings to task the following benefits: an explicit operational semantics, a more precise implementation notion, a notion of state, and a locality property. On the other hand, refined tasks open new possibilities of applying topological techniques to objects.

The unbeatability of a consensus protocol, introduced by Halpern, Moses and Waarts in [14], is a stronger notion of optimality than the accepted notion of early stopping protocols. Using a novel knowledge-based analysis, this paper derives the first practical unbeatable consensus protocols in the literature, for the standard synchronous message-passing model with crash failures. These protocols strictly dominate the best known protocols for uniform and for nonuniform consensus, in some case beating them by a large margin. The analysis provides a new understanding of the logical structure of consensus, and of the distinction between uniform and nonuniform consensus. Finally, the first (early stopping and) unbeatable protocol that treats decision values "fairly" is presented. All of these protocols have very concise descriptions, and are shown to be efficiently implementable. 4. Early stopping protocols for consensus are traditionally one-sided, preferring to decide on 0 (or on 1) if possible. deciding on a predetermined value (say, 0) if possible, we present an An unbeatable (and early stopping) majority consensus protocol Opt Maj is presented, that prefers the majority value.5. We identify the notion of a hidden path as being crucial to decision in the consensus task. If a process identifies that no hidden path exists, then it can decide. In the fastest early-stopping protocols, a process decides after the first round in which it does not detect a new failure. By deciding based on the nonexistence of a hidden path, our unbeatable protocols can stop up to t − 3 rounds faster than the best early stopping protocols in the literature.We now sketch the intuition behind, our unbeatable consensus protocols.In the standard version of consensus, every process i starts with an initial value v i ∈ {0, 1}, and the following properties must hold in every run r:(Nonuniform) Consensus:Decision: Every correct process must decide on some value, Validity: If all initial values are v then the correct processes decide v, and Agreement: All correct processes decide on the same value.The connection between knowledge and distributed computing was proposed in [13] and has been used in the analysis of a variety of problems, including consensus (see [9] for more details and references). In this paper, we employ simpler techniques to perform a more direct knowledgebased analysis. Our approach is based on a simple principle recently formulated by Moses in [17], called the knowledge of preconditions principle (KoP), which captures an essential connection between knowledge and action in distributed and multi-agent systems. Roughly speaking, the KoP principle says that if C is a necessary condition for an action α to be performed by process i, then K i (C) -i knowing C -is a necessary condition for i performing α. E.g., it is not enough for a client to have positive credit in order to receive cash from an ATM; the ATM must know that the client has positive credit.Problem specifications typically state or imply a variety of necessary conditions...

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