Wait-Free Solvability of Equality Negation Tasks

We introduce a family of tasks for n processes, as a generalization of the two process equality negation task of Lo and Hadzilacos (SICOMP 2000). Each process starts the computation with a private input value taken from a finite set of possible inputs. After communicating with the other processes using immediate snapshots, the process must decide on a binary output value, 0 or 1. The specification of the task is the following: in an execution, if the set of input values is large enough, the processes should agree on the same output; if the set of inputs is small enough, the processes should disagree; and in-between these two cases, any output is allowed. Formally, this specification depends on two threshold parameters k and l, with k<l, indicating when the cardinality of the set of inputs becomes "small" or "large", respectively. We study the solvability of this task depending on those two parameters. First, we show that the task is solvable whenever k+2 <= l. For the remaining cases (l = k+1), we use various combinatorial topology techniques to obtain two impossibility results: the task is unsolvable if either k <= n/2 or n-k is odd. The remaining cases are still open

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

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

Locally Solvable Tasks and the Limitations of Valency Arguments

An elegant strategy for proving impossibility results in distributed computing was introduced in the celebrated FLP consensus impossibility proof. This strategy is local in nature as at each stage, one configuration of a hypothetical protocol for consensus is considered, together with future valencies of possible extensions. This proof strategy has been used in numerous situations related to consensus, leading one to wonder why it has not been used in impossibility results of two other well-known tasks: set agreement and renaming. This paper provides an explanation of why impossibility proofs of these tasks have been of a global nature. It shows that a protocol can always solve such tasks locally, in the following sense. Given a configuration and all its future valencies, if a single successor configuration is selected, then the protocol can reveal all decisions in this branch of executions, satisfying the task specification. This result is shown for both set agreement and renaming, implying that there are no local impossibility proofs for these tasks