Synchronous condition-based consensus

Abstract

The condition-based approach studies restrictions on the inputs to a distributed problem, called conditions, that facilitate its solution. Previous work considered mostly the asynchronous model of computation. This paper studies conditions for consensus in a synchronous system where processes can fail by crashing. It describes a full classification of conditions for consensus, establishing a continuum between the asynchronous and synchronous models, with the following hierarchy ${\cal S}_t^{[-t]}\subset\cdots\subset {\cal S}_t^{[0]}\subset\cdots\subset {\cal S}_t^{[t]}$ where ${\cal S}_t^{[t]}$ includes all conditions (and in particular the trivial one made up of all possible input vectors). For a condition $C \in{\cal S}_t^{[d]}$, $-t \leq d \leq t$, we have:

• For values of $d \leq 0$ consensus is solvable in an asynchronous system with t failures, and we obtain the known hierarchy of conditions that allows solving asynchronous consensus with more and more efficient protocols as we go from d = 0 to d = −t.

• For values of $d \leq 0$ consensus is solvable in an asynchronous system with t failures, and we obtain the known hierarchy of conditions that allows solving asynchronous consensus with more and more efficient protocols as we go from d = 0 to d = −t.

• For values of d<0 consensus is known not solvable in an asynchronous system with t failures, but we obtain a hierarchy of conditions that allows solving synchronous consensus with protocols that can take more and more rounds, as we go from d = 0 to d = t.

• d = 0 is the borderline case where consensus can be solved in an asynchronous system with t failures, and can be solved optimally in a synchronous system.

After having established the complete hierarchy, the paper concentrates on the two last items: $0\leq d\leq t$. The main result is that the necessary and sufficient number of rounds needed to solve uniform consensus for a condition $C \in {\cal S}_t^{[d]}$ (such that $C\notin {\cal S}_t^{[d-1]}$) is d +1.

In more detail, the paper presents a generic synchronous early-deciding uniform consensus protocol that enjoys the following properties. Let f be the number of actual crashes, I the input vector and $C \in {\cal S}_t^{[d]}$ the condition the protocol is instantiated with. The protocol terminates in two rounds when $I\in C$ and $f\leq t-d$, and in at most d +1 rounds when $I\in C$ and $f>t-d$. (It also terminates in one round when $I\in C$ and $d=f=0$.) Moreover, whether I belongs or not to C, no process requires more than min$(t+1,f+2)$ rounds to decide. The paper then proves a corresponding lower bound stating that at least d +1 rounds are necessary to get a decision in the worst case when $I\in C$ (for $C \in {\cal S}_t^{[d]}$ and $C \notin {\cal S}_t^{[d-1]}$).