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

Üresin & 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 counter-example. 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.