Near-optimal self-stabilising counting and firing squads

Abstract

Consider a fully-connected synchronous distributed system consisting of n nodes, where up to f nodes may be faulty and every node starts in an arbitrary initial state. In the synchronous C-counting problem, all nodes need to eventually agree on a counter that is increased by one modulo C in each round for given C > 1. In the self-stabilising firing squad problem, the task is to eventually guarantee that all non-faulty nodes have simultaneous responses to external inputs: if a subset of the correct nodes receive an external "go" signal as input, then all correct nodes should agree on a round (in the not-too-distant future) in which to jointly output a "fire" signal. Moreover, no node should generate a "fire" signal without some correct node having previously received a "go" signal as input.We present a framework reducing both tasks to binary consensus at very small cost. For example, we obtain a deterministic algorithm for self-stabilising Byzantine firing squads with optimal resilience f < n/3, asymptotically optimal stabilisation and response time O(f ), and message size O(log f ). As our framework does not restrict the type of consensus routines used, we also obtain efficient randomised solutions, and it is straightforward to adapt our framework for other types of permanent faults.1 Current affiliation of JR.

Topology-Specific Synthesis of Self-Stabilizing Parameterized Systems with Constant-Space Processes

“…Most existing approaches [16,8,23,11,31,5] for the synthesis of Parameterized Systems (PSs) synthesize from temporal logic specifications and/or make assumptions about synchrony, fairness and complete knowledge of the network for each process. Moreover, most existing methods focus on synthesis for either safety properties or local liveness properties (e.g., progress of a thread); they do not address self-stabilization under asynchronous semantics with no fairness where convergence (i.e., recovery from any state) should be achieved through the collaboration of all processes.…”

“…Bloem et al [5] use bounded synthesis and parameterized synthesis to extend Dolev et al's approach for other problems. Lenzen and Rybicki [31] provide an SS and Byzantine-tolerant solution for the counting problem with near-optimal stabilization time and message sizes. What the aforementioned methods have in common is that they are based on bounded/parameterized synthesis from temporal logic specifications (using SMT solvers), and they make assumptions about synchrony, fairness and complete knowledge of the network for each process.…”

Efficient Counting with Optimal Resilience

“…Are there algorithms that satisfy (1)-(3), but need to store and communicate substantially fewer than log 2 f bits? This question has been partially answered in follow-up work [25], showing that O(log f ) bits suffice. However, no non-trivial lower bound is known, so it remains open whether o(log f ) bits suffice.…”

Synthesizing Parameterized Self-stabilizing Rings with Constant-Space Processes

“…Existing synthesis methods can be classified into problem-specific and general approaches. The problem-specific methods focus on generating a parameterized solution for a specific problem (e.g., counting [11,25], consensus [4], sorting [8], etc.). General methods [20,16] for the synthesis of parameterized systems are mainly specification-based in that they provide a decision procedure for extracting the skeleton of symmetric processes from their temporal logic specifications.…”

“…Dolev et al [11] present a verification-based method to generate synchronous and constant-space counting algorithms that are self-stabilizing under Byzantine faults. Lenzen and Rybicki [25] provide an SS and Byzantine-tolerant solution for the counting problem with near-optimal stabilization time and message sizes. What the aforementioned methods have in common is that they synthesize from temporal logic specifications and/or make assumptions about synchrony, fairness and complete knowledge of the network for each process.…”