Minimizing the Number of Opinions for Fault-Tolerant Distributed Decision Using Well-Quasi Orderings

Abstract

The notion of deciding a distributed language L is of growing interest in various distributed computing settings. Each process p i is given an input value x i , and the processes should collectively decide whether their set of input values x = (x i ) i is a valid state of the system w.r.t. to some specification, i.e., if x ∈ L. In non-deterministic distributed decision each process p i gets a local certificate c i in addition to its input x i . If the input x ∈ L then there exists a certificate c = (c i ) i such that the processes collectively accept x, and if x ∈ L, then for every c, the processes should collectively reject x. The collective decision is expressed by the set of opinions emitted by the processes, and one aims at minimizing the number of possible opinions emitted by each process.In this paper we study non-deterministic distributed decision in asynchronous systems where processes may crash. In this setting, it is known that the number of opinions needed to deterministically decide a language can grow with n, the number of processes in the system. We prove that every distributed language L can be non-deterministically decided using only three opinions, with certificates of size O(log α(n)) bits, where α grows at least as slowly as the inverse of the Ackerman function. The result is optimal, as we show that there are distributed languages that cannot be decided using just two opinions, even with arbitrarily large certificates.To prove our upper bound, we introduce the notion of distributed encoding of the integers, that provides an explicit construction of a long bad sequence in the well-quasi-ordering ({0, 1} * , = ) controlled by the successor function. Thus, we provide a new class of applications for wellquasi-orderings that lies outside logic and complexity theory. For the lower bound we use combinatorial topology techniques.

Redundancy in distributed proofs

“…Finally, verification mechanisms a la proof-labeling schemes were used in other contexts, including the congested clique [34], wait-free computing [26], failure detectors [27], anonymous networks [22], and mobile computing [9,25]. For more references to work related to distributed verification, or distributed decision in general, see the survey [18].…”

Perfect failure detection with very few bits

“…For the upper bound, i.e., for the design of the micro-perfect failure detector µP , we use a combination of failure detector techniques, with the notion of distributed encoding of the integers recently introduced in [6]. A distributed encoding of the integers is a distributed structure that encodes each positive integer n by a word w…”

Perfect Failure Detection with Very Few Bits

“…We show how to use Higman's lemma to prove that no failure detector outputting a constant number of bits at each process can achieve perfect failure detection. For the upper bound, i.e., for the design of the micro-perfect failure detector µP , we use a combination of failure detector techniques, with the notion of distributed encoding of the integers recently introduced in [6]. A distributed encoding of the integers is a distributed structure that encodes each positive integer n by a word w (n)…”

“…For more applications and related work on wqos, including rewriting systems, tree embeddings, lossy channel systems, and graph minors, see recent work [10,19]. The notion of distributed encoding of the integers was proposed in [6] to show that every one-shot system specification can be wait-free runtime monitored non-deterministically using only three opinions.…”

“…[6]). There exists a distributed encoding of the integers (Σ, f ) such that, for every integer n ≥ 1, |Σ n | ≤ α(n) where α : N → N grows as least as slowly as the inverse-Ackermann function.…”