Identifiers in Registers

Abstract

We propose a formal model of distributed computing based on register automata that captures a broad class of synchronous network algorithms. The local memory of each process is represented by a finitestate controller and a fixed number of registers, each of which can store the unique identifier of some process in the network. To underline the naturalness of our model, we show that it has the same expressive power as a certain extension of first-order logic on graphs whose nodes are equipped with a total order. Said extension lets us define new functions on the set of nodes by means of a so-called partial fixpoint operator. In spirit, our result bears close resemblance to a classical theorem of descriptive complexity theory that characterizes the complexity class pspace in terms of partial fixpoint logic (a proper superclass of the logic we consider here).

Descriptive complexity for distributed computing with circuits

“…This is because there is no guarantee on what the set of IDs in the current graph (or distributed system) is, and in a directed graph, we cannot even scan through the graph to find out. On the other hand, the logic in [3] can express, e.g., that the current node has the largest ID, which we cannot do. Of course, with a non-uniform formula allocation function, the circuit-style paradigm can even specify non-computable properties.…”

“…Distributed computing with identifiers has been studied from the point of view of logic earlier in [3]. The paper [3] approaches identifiers via a uniform logical characterization of a certain class of algorithms using IDs, while our work is based on the circuit-style paradigm with formulas and circuits being given based on model size. Thus, the two approaches are not comparable in any uniquely obvious way.…”

“…Thus, the two approaches are not comparable in any uniquely obvious way. Nevertheless, one simple difference between our work and [3] is that we treat IDs bit by bit as concrete bit strings. Thus, we can express, e.g., that the current ID has a bit 1 without implying that the current node cannot have the smallest ID in the system.…”