Locality and Checkability in Wait-Free Computing

Abstract

This paper studies notions of locality that are inherent to the specification of a distributed task and independent of the computing environment, in a shared memory wait-free system. A locality property called projection-closed is identified, that completely characterizes tasks that are wait-free checkable. A task T = (I,O,Δ) is checkable if there exists a wait-free distributed algorithm that, given s ∈ I and t ∈ O, determines whether t ∈ Δ(s), i.e., if t is a valid output for s according to the specification of T. Moreover, determining whether a projection-closed task is wait-free solvable remains undecidable, and hence this is a rich class of tasks. A stronger notion of locality considers tasks where the outputs look identically to the inputs at every vertex (input value of a process). A task T = (I,O,Δ) is said to be locality-preserving if O is a covering complex of I. This topological property yields obstacles for wait-free solvability different in nature from the classical agreement impossibility results. On the other hand, locality-preserving tasks are projection-closed and therefore always wait-free checkable. A classification of localitypreserving tasks in term of their relative computational power is provided. A correspondence between locality-preserving tasks and subgroups of the edgepath group of an input complex shows the existence of hierarchies of locality-preserving tasks, each one containing at the top the universal task (induced by the universal covering complex), and at the bottom the trivial identity task.