Parallelized sequential composition, pipelines, and hardware weak memory models

Since the introduction of the CDC 6600 in 1965 and its scoreboarding' technique processors have not (necessarily) executed instructions in program order. Programmers of high-level code may sequence independent instructions in arbitrary order, and it is a matter of significant programming abstraction and computational efficiency that the processor can be relied upon to make sensible parallelizations/reorderings of low-level instructions to take advantage of, eg., multiple ALUs. At the architectural level such reordering is typically implemented via a per-processor pipeline, into which instructions are fetched in order, but possibly committed out of order depending on local considerations, provided any reordering preserves sequential semantics from that processor's perspective. However multicore architectures, where several pipelines run in parallel, can expose these processor-level reorderings as unexpected, orweak', behaviours. Such weak behaviours are hard to reason about, and (via speculative execution) underlie at least one class of widespread security vulnerability. In this paper we introduce a novel program operator, \emph{parallelized sequential composition}, which can be instantiated with a function that controls the reordering of atomic instructions. It generalises both sequential and parallel composition, and when appropriately instantiated exhibits many of the weak behaviours of well-known hardware weak memory models. Our framework admits the application of established compositional techniques (eg. Owicki-Gries) for reasoning about weak behaviours, and is convenient for abstractly expressing properties from the literature. The semantics and theory is encoded and verified in a theorem prover, and we give an implementation of the pipeline semantics which we use to empirically show conformance against established models of ARM and RISC-V.