Dynamic Asynchronous Iterations

TL;DR

This paper extends models of asynchronous iterations to dynamic systems where both the functions and participants change over time, providing new correctness conditions and formal verification in Agda.

Contribution

It introduces a formal model for dynamic asynchronous iterations, generalizing previous static models, and proves correctness conditions with a formal Agda implementation.

Findings

Proposed a new model for dynamic asynchronous iterations.

Established conditions ensuring correctness of such iterations.

Formalized the results in Agda with an accessible library.

Abstract

Many problems can be solved by iteration by multiple participants (processors, servers, routers etc.). Previous mathematical models for such asynchronous iterations assume a single function being iterated by a fixed set of participants. We will call such iterations static since the system's configuration does not change. However in several real-world examples, such as inter-domain routing, both the function being iterated and the set of participants change frequently while the system continues to function. In this paper we extend Uresin & Dubois's work on static iterations to develop a model for this class of "dynamic" or "always on" asynchronous iterations. We explore what it means for such an iteration to be implemented correctly, and then prove two different conditions on the set of iterated functions that guarantee the full asynchronous iteration satisfies this new definition of correctness. These results have been formalised in Agda and the resulting library is publicly available.