A simple constructive computability theorem for wait-free computation

In modern shared-memory multiprocessors, processes can be halted or delayed without warning by interrupts, pre-emption, or cache misses. In such environments, it is desirable to design synchronization protocols that are wait-free: Any processes that continues to run will finish the protocol in a fixed number of steps, regardless of delays or failures by other processes. Not all synchronization problems have wait-free solutions. In this paper, we give a new, remarkably simple necessary and sufficient combinatorial condition characterizing the problems that have wait-free solutions using shared read/write memory. We associate the range of possible input and output values for any synchronization problem with a high-dimensional geometric structure called a simplicial complex. We show that a synchronization problem has a wait-free solution if and only if its input complex can be continuously "stretched and folded" to cover its output complex. The key to the new theorem is a novel "simplex agreement" protocol, allowing processes to converge asynchronously to a common simplex of a simplicial complex. The proof exploits a number of classical results from algebraic and combinatorial topology.