The solvability of consensus in iterated models extended with safe-consensus

TL;DR

This paper investigates the solvability of consensus in shared memory models extended with safe-consensus objects, establishing impossibility results for certain models and a tight bound for others.

Contribution

It introduces three iterated models with safe-consensus and proves the conditions under which consensus can or cannot be implemented.

Findings

Consensus cannot be implemented in the first two models.

A tight bound of binom{n}{2} safe-consensus boxes is established for the third model.

Safe-consensus objects do not always enable consensus in iterated shared memory models.

Abstract

The safe-consensus task was introduced by Afek, Gafni and Lieber (DISC' 09) as a weakening of the classic consensus. When there is concurrency, the consensus output can be arbitrary, not even the input of any process. They showed that safe-consensus is equivalent to consensus, in a wait-free system. We study the solvability of consensus in three shared memory iterated models extended with the power of safe-consensus black boxes. In the first iterated model, for the $i$-th iteration, the processes write to memory, then they snapshot it and finally they invoke safe-consensus boxes. We prove that in this model, consensus cannot be implemented. In a second iterated model, processes first invoke safe-consensus, then they write to memory and finally they snapshot it. We show that this model is equivalent to the previous model and thus consensus cannot be implemented. In the last iterated model, processes write to the memory, invoke safe-consensus boxes and finally they snapshot the memory. We show that in this model, any wait-free implementation of consensus requires $\binom{n}{2}$ safe-consensus black-boxes and this bound is tight.