Determining Recoverable Consensus Numbers

TL;DR

This paper investigates the power of deterministic, readable object types in recoverable consensus hierarchies, proving robustness and establishing bounds on their consensus and recoverable consensus numbers.

Contribution

It proves the robustness of the recoverable consensus hierarchy for deterministic, readable types and constructs types with specific consensus and recoverable consensus numbers.

Findings

Robustness of the recoverable consensus hierarchy for deterministic, readable types.

Existence of readable types with a gap of two between consensus and recoverable consensus numbers for all n ≥ 4.

Existence of non-readable types with arbitrary consensus and recoverable consensus numbers for all n > n' ≥ 1.

Abstract

Herlihy's wait-free consensus hierarchy classifies the power of object types in asynchronous shared memory systems where processes can permanently crash (i.e. stop taking steps). In this hierarchy, a type has consensus number $n$ if objects of that type can be used along with (read/write) registers to solve consensus among $n$ processes that can permanently crash, but not among $n+1$ or more processes. In systems where processes can recover after crashing, the power of an object type to solve consensus may be different. Golab's recoverable consensus hierarchy classifies the power of object types in such a system. In the recoverable consensus hierarchy, a type has recoverable consensus number $n$ if objects of that type can be used along with registers to solve consensus among $n$ processes that can recover after crashing, but not among $n+1$ or more processes. In this paper, we prove that the recoverable consensus hierarchy of deterministic, readable types is robust, i.e., if consensus can be solved among $n$ processes that can recover after crashing using a collection of objects of deterministic, readable types, then one of these types has recoverable consensus number at least $n$. This is important for comparing the relative computational power of different deterministic, readable types, because it implies that one cannot combine various objects to obtain an algorithm that is better at solving recoverable consensus than any of the individual object types. Our result can be used to show that, for all $n \geq 4$, there exists a readable type with consensus number $n$ and recoverable consensus number $n-2$. We also show that, for all $n > n' \geq 1$, there exists a non-readable type that has consensus number $n$ and recoverable consensus number $n'$.