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

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 deter-ministically 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 well-quasi-orderings that lies outside logic and complexity theory. For the lower bound we use combinatorial topology techniques