Deterministic computations whose history is independent of the order of asynchronous updating

TL;DR

This paper investigates the conditions under which the history of state updates in asynchronous networks remains consistent regardless of update order, introducing the concept of commutativity as a sufficient condition for invariant histories.

Contribution

It establishes that invariant histories are generally undecidable but provides a simple, practical sufficient condition called commutativity for ensuring order-independent outcomes.

Findings

Invariant histories are typically undecidable.

Commutativity guarantees order-independent update results.

Provides a practical criterion for designing asynchronous systems.

Abstract

Consider a network of processors (sites) in which each site x has a finite set N(x) of neighbors. There is a transition function f that for each site x computes the next state \xi(x) from the states in N(x). But these transitions (updates) are applied in arbitrary order, one or many at a time. If the state of site x at time t is \eta(x,t) then let us define the sequence \zeta(x,0), \zeta(x,1), ... by taking the sequence \eta(x,0), \eta(x,1), ..., and deleting repetitions. The function f is said to have invariant histories if the sequence \zeta(x,i), (while it lasts, in case it is finite) depends only on the initial configuration, not on the order of updates. This paper shows that though the invariant history property is typically undecidable, there is a useful simple sufficient condition, called commutativity: For any configuration, for any pair x,y of neighbors, if the updating would change both \xi(x) and \xi(y) then the result of updating first x and then y is the same as the result of doing this in the reverse order.