The Synchronization Power of Atomic Bitwise Operations

In a distributed system, processes must reach a certain level of synchronization to solve a common problem. The strongest form of synchronization can be reached through consensus: all the processes must agree on a common value that has been proposed by one of them. Consensus is universal in shared memory systems: any type of shared object can be implemented using it. Unfortunately, consensus is impossible to solve using only shared registers when processes can crash. To circumvent this impossibility, one can use stronger objects, for example Test&Set or Compare&Swap. The synchronization power of these objects can be measured using the concept of Consensus Number: the maximum number of processes for which they can solve consensus in a crash-prone system. Bitwise AND, OR and XOR operations are very widely used, but have received little attention in the distributed setting. Because bitwise operations are available in most modern processors, they can constitute a valuable tool for synchronization in distributed systems. It is then natural to consider the level of synchronization that these operations can achieve. This paper introduces shared AND/OR and AND/OR/XOR registers. A shared AND/OR register consists of an array of x bits and offers three atomic operations: AND and OR operations, which take an array of x bits as parameter and change the state of the register by applying the corresponding bitwise operation, and a read operation which returns the content of the array. A shared AND/OR/XOR register additionally offers a XOR operation. We show that shared AND/OR registers of x bits have consensus number lfloor (x+1)/2 rfloor, by presenting an algorithm that solves consensus using these registers, and by proving that consensus cannot be solved for n processes using AND/OR registers that have strictly less than 2n-1 bits. We then show that shared AND/OR/XOR registers of x bits have consensus number x using a similar technique