Synchronous counting and computational algorithm design

Abstract

Abstract. Consider a complete communication network on n nodes, each of which is a state machine. In synchronous 2-counting, the nodes receive a common clock pulse and they have to agree on which pulses are "odd" and which are "even". We require that the solution is self-stabilising (reaching the correct operation from any initial state) and it tolerates f Byzantine failures (nodes that send arbitrary misinformation). Prior algorithms are expensive to implement in hardware: they require a source of random bits or a large number of states.This work consists of two parts. In the first part, we use computational techniques (often known as synthesis) to construct very compact deterministic algorithms for the first non-trivial case of f = 1. While no algorithm exists for n < 4, we show that as few as 3 states per node are sufficient for all values n ≥ 4. Moreover, the problem cannot be solved with only 2 states per node for n = 4, but there is a 2-state solution for all values n ≥ 6.In the second part, we develop and compare two different approaches for synthesising synchronous counting algorithms. Both approaches are based on casting the synthesis problem as a propositional satisfiability (SAT) problem and employing modern SATsolvers. The difference lies in how to solve the SAT problem: either in a direct fashion, or incrementally within a counter-example guided abstraction refinement loop. Empirical results suggest that the former technique is more efficient if we want to synthesise time-optimal algorithms, while the latter technique discovers non-optimal algorithms more quickly.

Efficient Counting with Optimal Resilience

“…While non-trivial for consensus, this bound turns out to be trivial for deterministic counting algorithms: a self-stabilising algorithm needs to verify its output, and to do that, each of the n nodes needs to receive information from at least f + 1 = Ω(f ) other nodes to be certain that some other non-faulty node has the same output value. Similarly, no non-constant lower bounds on the number of state bits nodes are known; however, a non-trivial constant lower bound for the case f = 1 is known [16].…”

“…Designing space-efficient randomised algorithms for synchronous counting is fairly straightforward [16][17][18]: for example, the nodes can simply choose random states until a clear majority of nodes has the same state, after which they start to follow the majority. Likewise, given a shared coin, one can quickly reach agreement by defaulting to the coin whenever no clear majority is observed [4].…”

“…In addition, Ω(n/ log 2 n)-resilient Boolean functions give fast communication-efficient coins [1]. Designing quickly stabilising algorithms that are both communication-efficient and space-efficient has turned out to be a challenging task [12,14,16], and it remains open to what extent randomisation can help in designing such algorithms.…”

“…In the case of deterministic algorithms, algorithm synthesis has been used for computer-aided design of optimal algorithms with resilience f = 1, but the approach does not scale due to the extremely fast-growing space of possible algorithms [16]. In general, many fast-stabilising algorithms build on a connection between Byzantine consensus and synchronous counting, but require a large number of states per node [12] due to, e.g., running a large number of consensus instances in parallel.…”

“…If we neglect communication, counting and consensus are essentially equivalent [12,14,16]. In particular, many lower bounds on (binary) consensus directly apply to the counting problem [13,20,27].…”