Weak System Models for Fault-Tolerant Distributed Agreement Problems

This thesis1 investigates various aspects of weak system models for agreement problems in fault-tolerant distributed computing. In Part I we provide an introduction to the context of this work, discuss related literature and describe the basic system assumptions. In Part II of this thesis, we introduce the Asynchronous Bounded-Cycle (ABC) model, which is entirely time-free. In contrast to existing system models, the ABC model does not require explicit time-based synchrony bounds, but rather stipulates a graph-theoretic synchrony condition on the relative lengths of certain causal chains of messages in the space-time graph of a run. We compare the ABC model to other models in literature, in particular to the classic models by Dwork, Lynch, and Stockmeyer. Despite Byzantine failures, we show how to simulate lock-step rounds, and therefore make consensus solvable, and prove the correctness of a clock synchronization algorithm in the ABC model. We then present the technically most involved result of this thesis: We prove that any algorithm working correctly in the partially synchronous Θ-Model by Le Lann and Schmid, also works correctly in the time-free ABC model. In the proof, we use a variant of Farkas’ Theorem of Linear Inequalities and develop a non-standard cycle space on directed graphs in order to guarantee the existence of a certain message delay transformation for finite prefixes of runs. This shows that any time-free safety property satisfied by an algorithm in the Θ-Model also holds in the ABC model. By employing methods from point-set topology, we can extend this result to liveness properties. In Part III, we shift our attention to the borderland between models where consensus is solvable and the purely asynchronous model. To this end, we look at the k-set agreement problem where processes need to decide on at most k distinct decision values. We introduce two very weak system models Manti and Msink and prove that consensus is impossible in these models. Nevertheless, we show that (n−1)-set agreement is solvable in Manti and Msink, by providing algorithms that implement the weakest failure detector L. We also discuss how models Manti and Msink relate to the f -source models by Aguilera et al. for solving consensus. In the subsequent chapter, we present a novel failure detector L(k) that generalizes L, and analyze an algorithm for solving k-set agreement with L(k), which works even in systems without unique process identifiers. Moreover, We explore the relationship between L(k) and existing failure detectors for k-set agreement. Some aspects of L(k) relating to anonymous systems are also discussed. This research has been supported by the Austrian Science Foundation (FWF) projects P17757 and P20529.