The consensus number of a shift register equals its width

TL;DR

This paper determines the exact consensus number of w-bit shift registers supporting logical shifts, showing they match their width, and contrasts this with arithmetic shift registers that can solve consensus universally for certain widths.

Contribution

It precisely characterizes the consensus number of shift registers with logical and arithmetic shifts, expanding understanding of their computational power in distributed systems.

Findings

Logical shift registers have a consensus number equal to their width.

Arithmetic shift registers can solve consensus for any fixed number of processes if width ≥ 2.

Results generalize to larger alphabets.

Abstract

The consensus number of a w-bit register supporting logical left shift and right shift operations is exactly w, giving an example of a class of types, widely implemented in practice, that populates all levels of the consensus hierarchy. This result generalizes to w-wide shift registers over larger alphabets. In contrast, a register providing arithmetic right shift, which replicates the most significant bit instead of replacing it with zero, is shown to solve consensus for any fixed number of processes as long as its width is at least two.