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 pi is given an input value xi, and the processes should collectively decide whether their set of input values x = (xi)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 pi gets a local certificate ci in addition to its input xi. If the input x ∈ L then there exists a certificate c = (ci)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. In this paper we study non-deterministic distributed decision in systems where asynchronous processes may crash. 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 ┌log α(n) ┐+1 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.