A new formalism is given for read-modify-write (RMW) synchronization operations. This formalism is used to extend the memory reference combining mechanism introduced in the NYU Ultracomputer, to arbitrary RMW operations. A formal correctness proof of this combining mechanism is given. General requirements for the practicality of combining are discussed. Combining is shown to be practical for many useful memory access operations. This includes memory updates of the form mem_val := mem_val op val, where op need not be associative, and a variety of synchronization primitives. The computation involved is shown to be closely related to parallel prefix evaluation. © 1988, ACM. All rights reserved.