A Relaxation of Üresin and Dubois’ Asynchronous Fixed-Point Theory in Agda

Abstract

Üresin and Dubois' paper "Parallel Asynchronous Algorithms for Discrete Data" shows how a class of synchronous iterative algorithms may be transformed into asynchronous iterative algorithms. They then prove that the correctness of the resulting asynchronous algorithm can be guaranteed by reasoning about the synchronous algorithm alone. These results have been used to prove the correctness of various distributed algorithms, including in the fields of routing, numerical analysis and peer-to-peer protocols. In this paper we demonstrate several ways in which the assumptions that underlie this theory may be relaxed. Amongst others, we (i) expand the set of schedules for which the asynchronous iterative algorithm is known to converge and (ii) weaken the conditions that users must prove to hold to guarantee convergence. Furthermore, we demonstrate that two of the auxiliary results in the original paper are incorrect, and explicitly construct a counterexample. Finally, we also relax the alternative convergence conditions proposed by Gurney based on ultrametrics. Many of these relaxations and errors were uncovered after formalising the work in the proof assistant Agda. This paper describes the Agda code and the library that has resulted from this work. It is hoped that the library will be of use to others wishing to formally verify the correctness of asynchronous iterative algorithms.

Dynamic Asynchronous Iterations

“…However in the late 1980s Üresin & Dubois [12] came up with one of the first conditions for the convergence of discrete asynchronous iterations. Here we use the relaxed version of the conditions as proposed in [11].…”

“…A mathematical model for static asynchronous iterations was standardised by work in the 1970s and 80s [8,9,10]. The notation and terminology used here is taken from the recent paper [11] which in turn is based on that used by Üresin & Dubois [12].…”

“…Note that this definition does not forbid the data flow function β from delaying, losing, reordering or even duplicating messages (see Figure 1). Prior to recent work [11], static schedules were assumed to have two additional assumptions that guaranteed every node continued to activate indefinitely and that every pair of nodes continued to communicate indefinitely.…”

“…This essentially says that the schedule is well-behaved forever. In contrast [11] built upon the work of Üresin & Dubois and their concept of pseudocycles and relaxed this condition to only require that schedules must be well-behaved for a finite period of time. This distinction will be important in the dynamic model described later in Section 3, as a dynamic iteration will only have a finite period of time to converge before either the participants or the function being iterated changes.…”

“…As with the ACO conditions, these conditions were relaxed by [11] and the latter version is now presented.…”