On the Importance of Registers for Computability

TL;DR

This paper investigates the necessity of registers in consensus hierarchies, demonstrating that many objects cannot replace registers and that initialization and object count influence consensus capabilities without registers.

Contribution

It challenges the assumption that unbounded registers are essential for consensus hierarchies and explores the impact of object types and initialization in their absence.

Findings

Queues and stacks cannot emulate registers without registers.

Initialization is crucial for synchronization power without registers.

Number of objects affects the level of solvable consensus without registers.

Abstract

All consensus hierarchies in the literature assume that we have, in addition to copies of a given object, an unbounded number of registers. But why do we really need these registers? This paper considers what would happen if one attempts to solve consensus using various objects but without any registers. We show that under a reasonable assumption, objects like queues and stacks cannot emulate the missing registers. We also show that, perhaps surprisingly, initialization, shown to have no computational consequences when registers are readily available, is crucial in determining the synchronization power of objects when no registers are allowed. Finally, we show that without registers, the number of available objects affects the level of consensus that can be solved. Our work thus raises the question of whether consensus hierarchies which assume an unbounded number of registers truly capture synchronization power, and begins a line of research aimed at better understanding the interaction between read-write memory and the powerful synchronization operations available on modern architectures.