Sergio Rajsbaum

A theory of clock synchronization (extended abstract)

EXTENDED ABSTRACT Boaz Patt-Shamir* Sergio Raj sbaumt Laboratory for Computer Science Massachusetts Institute of Technology

Set consensus using arbitrary objects (preliminary version)

In the (N, k)-consensus task, each process in a group starts with a private input value, communicates with the others by apply ingoperations to shared objects} and then halts after choosing a private output value. Each process is required to choose some processs 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 asyn-chronous 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 solv-ability 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 A4 < N. Our proofs exploit a number of techniques from algebraic topology.

Colorless Wait-Free Computation

We outline the basic connection between distributed computing and combinatorial topology in terms of two formal models: a conventional operational model , in which systems consist of communicating state machines whose behaviors unfold over time, and the combinatorial model , in which all possible behaviors are captured statically using topological notions. We start with one particular system model (shared memory) and focus on a restricted (but important) class of problems (so-called “colorless” tasks).

Byzantine-Resilient Colorless Computation

We consider the Byzantine failure model, in which a faulty process can display arbitrary, even malicious, behavior. A faulty process may fall silent; it may also lie about its input, or it may lie about the information it has received from other processes.

Immediate Snapshot Subdivisions

Throughout this book, we have relied on the fact that Ch Δ n , the standard chromatic subdivision of the n n -simplex Δ n Δ n defined in Chapter 3 , is indeed a subdivision of Δ n Δ n . In this chapter, we give a rigorous proof of this claim.

Two-Process Systems

This chapter is an introduction to how techniques and models from combinatorial topology can be applied to distributed computing by focusing exclusively on two-process systems. It explores several distributed computing models, still somewhat informally, to illustrate the main ideas.

Solvability of Colorless Tasks in Different Models

This chapter explores the circumstances under which colorless tasks can be solved in different communication models, satisfying different fault-tolerance requirements. We consider both shared memory and message-passing models, wait-free and t t -resilient protocols, and protocols that work against adversaries.

Elements of Combinatorial Topology

This chapter defines the basic notions of topology needed to formulate the language we use to describe distributed computation.

Chapter 9 – Manifold Protocols

Theoretical distributed computing is primarily concerned with classifying tasks according to their difficulty. Which tasks can be solved in a given distributed computing model? We consider here two important tasks: set agreement and weak symmetry breaking. It turns out that the immediate snapshot protocols of Chapter 8 cannot solve these tasks. Moreover, we will identify a broader class of protocols called manifold protocols xa0, that cannot solve kk-set agreement. (The impossibility proof for weak symmetry breaking is more complicated and is deferred to Chapter 12.)

Chapter 13 – Task Solvability in Different Models

The principal technical idea introduced in this chapter is a proof that if the single-layer complex is shellable , a combinatorial property defined later, multilayer compositions preserve connectivity under certain easily checkable conditions. These are theorems of combinatorial topology, independent of any model of computation.