Synchronization power depends on the register size

Though it is common practice to treat synchronization primitives for multiprocessors as abstract data types, they are in reality machine instructions on registers. A crucial theoretical question with practical implications is the relationship between the size of the register and its computational power. We wish to study this question and choose as a first target the popular compare-&-swap operation (which is the basis for many modern multiprocessor architectures). Our main results are: 1. We show that multi-valued consensus among n processes can be solved using a compare&swap register that can hold ≈log n/log log n values. That is, n = (k-1)! where k is the number of values in the register, so the register has only O(log log n) bits. 2. We prove that there is a dependency between register size and processes' ability to solve multi-valued consensus. The key to the proof is a novel method of reducing a multi-valued decision task with limited size compare-&-swap registers to the set-consensus problem with read/write registers, allowing us to build on the recent powerful impossibility results of [2, 9, 18]. 3. We further use the reduction method to prove a tight tradeoff between the space and time necessary to solve multi-valued consensus with a compare-&-swap register. Specifically, we show that any algorithm for multi-valued consensus among n processes with a k value compare-&-swap register, where k≥log n/log log n, must have a run that accesses the register Θ (log_k n) times. The results of this paper suggest that a complexity hierarchy for multiprocessor synchronization operations should be based on the space complexity of synchronization registers and not on the number of so called `synchronization objects.'