We give necessary and sufficient combinatorial conditions characterizing the tasks that can be solved by asynchronous processes, of which all but one can fail, that communicate by reading and writing a shared memory. We introduce a new formalism for tasks, based on notions from classical algebraic and combinatorial topology, in which a task's possible input and output values are each associated with high-dimensional geometric structures called simplicial complexes. We characterize computability in terms of the topological properties of these complexes. This characterization has a surprising geometric interpretation: a task is solvable if and only if the complex representing the task's allowable inputs can be mapped to the complex representing the task's allowable outputs by a function satisfying certain simple regularity properties. Our formalism thus replaces the "operational" notion of a wait-free decision task, expressed in terms of interleaved computations unfolding ...

and insights. 1 Introduction For many years, researchers studying synchronous message-passing systems have considered algorithms composed of rounds of computation. In each round, a process sends a message to the others and then waits to receive messages from the other processes. The synchronous nature of the system ensures that, by the end of the round, each process receives all messages sent to it in that round by correct processes. In the parlance of Elrad and Frances [1] then, each round of a synchronous system is a communication-closed-layer.

In the (N; k)-consensus task, each process in a group starts with a private input value, communicates with the others by applying operations to shared objects, and then halts after choosing a private output value. Each process is required to choose some process's input value, and the set of values chosen should have size at most k. This problem, first proposed by Chaudhuri in 1990, has been extensively studied using asynchronous read/write memory. In this paper, we investigate this problem in a more powerful asynchronous model in which processes may communicate through objects other than read/write memory, such as test&set variables. We prove two general theorems about the solvability of set consensus using objects other than read/write registers. The first theorem addresses the question of what kinds of shared objects are needed to solve (N; k)-consensus, and the second addresses the question of what kinds of tasks can be solved by N processes using (M; j)-consensus objects, for M N...

Topological methods have yielded a variety of lower bounds and impossibility results for distributed computing. In this paper, we introduce a new tool for proving impossibility results, which is based on a core theorem of algebraic topology, the acyclic carrier theorem, and unifies, generalizes and extends earlier results.

We take a significant step toward unifying the synchronous, semisynchronous, and asynchronous message-passing models of distributed computation. The key idea is the concept of a pseudosphere, a new combinatorial structure in which each process from a set of processes is independently assigned a value from a set of values. Pseudospheres have a number of nice combinatorial properties, but their principal interest lies in the observation that the behavior of protocols in the three models can be characterized as simple unions of pseudospheres, where the exact structure of these unions is determined by the timing properties of the model. We use this pseudosphere construction to derive new and remarkably succinct proofs of bounds on consensus and k-set agreement in the asynchronous and synchronous models, as well as the first lower bound on wait-free k-set agreement in the semi-synchronous model. To appear in the 16th Annual ACM Symposium on Principles of Distributed Computing (PODC98), Puer...

This paper presents a self-contained study of wait-free solvable tasks. A new necessary and sufficient condition for wait-free solvability is proved, providing a characterization of the wait-free solvable tasks. The necessary condition is used to prove tight bounds on renaming and k-set consensus. The framework is based on topology, but uses only elementary combinatorics, and does not rely on algebraic or geometric arguments.

This document is a set of course notes from an informal tutorial to be presented at UCLA in August 1994. These notes are intended to be informative, even provocative, but are not intended to be balanced, comprehensive, or authoritative. All reasonable suggestions for improvement will be enthusiastically acknowledged in future revisions. 1 Introduction

. Models and techniques borrowed from classical algebraic topology have recently yielded a variety of new lower bounds and impossibility results for distributed and concurrent computation. This paper explains the basic concepts underlying this approach, and shows how they apply to a simple distributed problem. 1 Introduction The problem of coordinating concurrent processes remains one of the central problems of distributed computing. Coordination problems arise at all scales in distributed and concurrent systems, ranging from synchronizing data access in tightly-coupled multiprocessors, to allocating data paths in networks. Coordination is difficult because modern multiprocessor systems are inherently asynchronous: processes may be delayed without warning for a variety of reasons, including interrupts, pre-emption, cache misses, communication delays, or failures. These delays can vary enormously in scale: a cache miss might delay a process for fewer than ten instructions, a page fault...

This paper discusses results that say what cannot be computed in certain environments or when insucient resources are available. A comprehensive survey would require an entire book. As in Nancy Lynch's excellent 1989 paper, \A Hundred Impossibility Proofs for Distributed Computing" [86], we shall restrict ourselves to some of the results we like best or think are most important. Our aim is to give you the avour of the results and some of the techniques that have been used. We shall also mention some interesting open problems and provide an extensive list of references. The focus will be on results from the past decade.

This paper studies the time complexity of solving decision problems in