Decision Power of Weak Asynchronous Models of Distributed Computing.

Esparza and Reiter have recently conducted a systematic comparative study of models of distributed computing consisting of a network of identical finite-state automata that cooperate to decide if the underlying graph of the network satisfies a given property. The study classifies models according to four criteria, and shows that twenty initially possible combinations collapse into seven equivalence classes with respect to their decision power, i.e. the properties that the automata of each class can decide. However, Esparza and Reiter only show (proper) inclusions between the classes, and so do not characterise their decision power. In this paper we do so for labelling properties, i.e. properties that depend only on the labels of the nodes, but not on the structure of the graph. In particular, majority (whether more nodes carry label $a$ than $b$) is a labelling property. Our results show that only one of the seven equivalence classes identified by Esparza and Reiter can decide majority for arbitrary networks. We then study the expressive power of the classes on bounded-degree networks, and show that three classes can. In particular, we present an algorithm for majority that works for all bounded-degree networks under adversarial schedulers, i.e. even if the scheduler must only satisfy that every node makes a move infinitely often, and prove that no such algorithm can work for arbitrary networks.