Geometric and combinatorial views on asynchronous computability

Abstract

We show that the protocol complex formalization of fault-tolerant protocols can be directly derived from a suitable semantics of the underlying synchronization and communication primitives, based on a geometrization of the state space. By constructing a one-to-one relationship between simplices of the protocol complex and (di)homotopy classes of (di)paths in the latter semantics, we describe a connection between these two geometric approaches to distributed computing: protocol complexes and directed algebraic topology. This is exemplified on atomic snapshot, iterated snapshot and layered immediate snapshot protocols, where a well-known combinatorial structure, interval orders, plays a key role. We believe that this correspondence between models will extend to proving impossibility results for much more intricate fault-tolerant distributed architectures.

On Extending Brandt's Speedup Theorem from LOCAL to Round-Based Full-Information Models

“…It has also been extended to mobile computing [1], to dynamic environments [23], and to Byzantine failures [34]. Moreover, a topological description of concurrent programming has been developed [15,25]. It is however only recently that distributed network computing has been approached through the lens of combinatorial topology [9,20], specifically applied to local computing.…”

Time-reversal homotopical properties of concurrent systems

“…They give semantics, for the left one X, to program U.S || U.S, and for the one on the right Y , to program U.S || S.U , where S stands for a scan operation and U stands for an update operation in shared memory, see e.g. [21]. As topological spaces, X and Y are homeomorphic spaces, homotopic to a wedge of two circles.…”