The consensus number of a shift register equals its width

2025-05-03

AI Summary

This paper investigates the computational power of shift registers in distributed consensus, focusing on how their consensus number depends on bit-width and operation type. Using wait-free computability analysis, linear register modeling, and a classification of register types, we rigorously establish that: (i) the consensus number of a w-bit logical left/right shift register is exactly w—constituting the first demonstration of a general-purpose hardware primitive that fully spans all levels of the consensus hierarchy; (ii) arithmetic right-shift registers with width ≥2 have infinite consensus number, breaking the conventional paradigm of finite consensus numbers; and (iii) these results generalize to w-dimensional shift registers over multi-symbol alphabets. Our work systematically characterizes the fundamental impact of shift operations on synchronization primitive capabilities, providing a theoretical foundation for hardware-assisted concurrent algorithm design.

Technology Category

• Machine Learning: Hardware-aware ML
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Multiagent Systems: Agreement, Argumentation & Negotiation

Application Category

• Security and Privacy: Large-scale security measurements
• Responsible Web: Consent frameworks and practices on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Determines consensus number of w-bit shift registers
• Generalizes result to w-wide shift registers
• Compares logical and arithmetic shift operations

Innovation

• w-bit shift register consensus equals width
• generalizes to larger alphabet shift registers
• arithmetic right shift solves multi-process consensus